This thread is up for change, and addition. If anybody feels strong enough to give a solid contribution his or her comments are most welcomed. However, please include the following: Title of the topic, Formulas, Descriptions to explain the formulas, a sample question, and links if possible. Please avoid repeating a discussion that is already delved into, unless you wish to complement it in which case do mention it.
_________________________
Physics I
TOPIC 1: LINEAR MOTION
Definition:
LINEAR MOTION (Technical Definition): Any motion in a one dimensional frame that is traceable through a straight line. A linear motion is one in which the destination and the point of origin can be aligned through a straight line.
LINEAR MOTION (Common Definition): Any motion that is not curved or twisted. In a sense, any motion that is similar to a bullet's motion.
____________________________
Concepts:
Linear motion is divided into two groups:
1) motion with constant acceleration: any motion whose acceleration is constant and the force causing it is constant.
2) motion with variable acceleration: any motion whose acceleration is not constant and it is due to the fact that the force causing it is not constant.
In physics books, and for college students primarily the linear motion means a motion with constant acceleration. Motion with variable acceleration is often not studied. It however does not mean it does not exist.
Before one can delve into studying this area of Mechanics one has to distinguish between a series of vocabulary:
Distance vs. Displacement: Distance is not the same as displacement. Displacement is a vector quantity while distance is a scalar quantity. Distance is dependent upon the path we take while displacement is independent of the path we take and is about the relative position of an object after its travel. The picture below explains it:
In the picture above, If one moves from A to B to C and then back to A, his displacement is d = 0 meters while the distance he has gone is AB + BC + CA
Speed vs. Velocity: The difference between speed and velocity is the same as the difference between distance and displacement. Velocity is displacement divided by time and speed is distance divided by time. Again speed is a magnitude and not a vector while the velocity is a vector. For instance when two cars simultaneously move from point B to A in the picture above both can move at the same speed but their displacements would have opposite signs since one is move in a direction opposite to the other.
Acceleration vs. Deceleration: Deceleration is a wrong use of the word acceleration. Acceleration is the change of velocity over time. It can either be negative (meaning the velocity is decreasing) or positive (meaning the velocity is increasing) or zero (meaning the velocity is constant). Deceleration is a special case of acceleration in which a < 0.
Motion vs. frames: In Mechanics we rarely deal with frames of reference. Ideally we assume the universe is a fixed point compared to which we measure the speed of other objects. In reality however, as Dr. Einstein made clear, there are frames of reference. Two cars moving at the same speed would seem to be stationary to each other but they might be moving at a speed compared to a stationary observer.
____________________________
Formulas:
Legend: X is displacement, a is acceleration, V is velocity, g is gravitational acceleration, t is time.
Non-Differential:
(1) V = Δ X / t
(2) a = Δ V / t (this implies that if an object moves at a constant speed its acceleration is zero)
(3) X = V1t + 1/2a(t^2)
(4) V2^2 = V1^2 + 2 a X
In free fall replace a with g;
Differential format of the same equation:
V = dX/dt
a = dV/dt
X = ∫ V dt
V = ∫ a dt
X = V1t + 1/2a(t^2)
V2^2 = V1^2 + 2 a X
In free fall replace a with g;
_____________________________
How to Use the Formulas:
The formulas above are useless if you do not know when or where to use them. Usually in problems involving the linear motion are not that obvious at first. It takes practice, diligence, and an desire for challenges in life.
Lets evaluate the formulas: Formula (1) and (2) are simply not that useful. They are more the mathematical depiction of the definition of velocity and acceleration.
(3) and (4) however are the real deal. Do take note that (3) and (4) can give you X, V1 [initial velocity] , t , V2 [secondary velocity] if all the other variables are ready. So although the formal format is made to look like they are aimed to measure X and V2 only they in fact can be used to find any one variable as long as the rest are known.
____________________________
Problem Solving Steps:
Read the Question
Ask yourself what variable are given to you among X, t, V1, V2, and a.
Write the variables in the SI (System Internationale) Units. Do not under any condition leave them in British or American units unless otherwise specified.
Write down the formula (3) and (4)
See which variables are missing. If you have three variables you are most likely asked for the forth; look to see which equation (3) or (4) can accommodate that need.
Calculate the result, maintaining the significant figure agreement.
BOX your answer and make sure the units are present.
If time allows go back and check your answers.
Try to make sense of the answer: If a car moves at a speed of 30 m/s and you accelerate at a constant rate of 2 m/s/s, and you find that the secondary velocity of your car is 5000 m/s then you would know that logically something is terrible awry.
____________________________
Recommended English Texts with special attention to linear motion:
US:
College Physics , Sixth Ed. , Serway/Faughn. [Thomsons publication]
EU:
Taschenbuch der Physik, Fachbuchverlag Leipzig im Carl Hanser Verlag
______________________________
Online simulator for linear motion:
**Note: Great tools for comparison of your answers with actual results using a simulator.
http://jersey.uoregon.edu/vlab/
______________________________
Sample Problem:
A car is moving at 50 m/s in a highway, when suddenly a dear jumps in front of the car. The driver suddenly pushes the breaks and comes to stop in 2 seconds before hitting the dear. The dear escapes. What was the acceleration of the car before the driver hit the break? what was the acceleration during the breaking time? What was the distance the car moved after breaking?
Solution:
First read the question
Visualize it
Write down all the variables you have elicited from the question:
V1 = 50 m/s , t = 2 seconds, a = ? , X = ? , V2 = 0 (since the car came to a stop)
Write down (3) and (4)
you have three variables {V1,t,V2} so look to see which equation out of (3) and (4) can give you the missing variable
Equations (4) and (3) both wont help us since both have one missing variable so we need to either find a or t first
since (3) and (4) don't help us look at (1) and (2); we can use (2) to find "a" so we have:
a = Δ V / t = V2 - V1 / t = 0 - 50 / 2 = - 25 m/s^2
So now we have a and we can use the (4) to find X, so we have:
V2^2 = V1^2 + 2 a X or (V2^2 - V1^2)/ 2 a = X
X = ( 0 - 625) / 2 * (-25) = 25 meters
The question also asks for the acceleration before breaking and after the car has come to stop; since before the breaking the car is moving at a constant velocity, the acceleration is zero and since the car comes to a complete stop and has no motion it also has a zero acceleration after coming to stop.
____________________________________
[END OF TOPIC 1]
*******************************************************************
___________
________________
____________________
*******************************************************************
TOPIC 2: PROJECTILE MOTION
___________________________
Definition:
Technical-kind of definition: A projectile motion is any motion in a two dimensional gravitational field with constant velocity in one direction and acceleration in the other perpendicular direction
Normal definition: Any kind of motion that has a curve like nature to it.
_______________________________
Formulas:
If a projectile is initially launched at a speed of V at an angle of a, from the level ground then it will follow a curvature that is symmetrical in nature. Symmetry is very important because there are usually two kinds of projectiles:
Asymmetrical Projectile: For instance a plane throws a package down, or a person jumps from a high cliff into the sea, or water pours down from an elevation. This kind of projectile is Asymmetrical because it is 1/4 of a circle in its nature. The below diagram shows an ideal (meaning the only forces are the one creating the initial velocity and the force of the gravity) asymmetrical and symmetrical projectile motion:
the formulas are almost the same for the both kind but usually the normal physics textbooks fail to explain the subtle differences or perhaps they do so on intention. In any case, the formulas are three major kinds:
(1) Vx = X / t [constant velocity in the x-direction]
(2) Y = Vy1t + 1/2 g t^2 [if the ball goes up g is negative and if down g is positive]
(2') Vyt^2 = vy1^2 + 2gY
(3) Vyt = Vy1 - gt , where t is time, Vy1 is the vertical component of the initial velocity and g is the acceleration
(4) Notes: In symmetrical projectile the highest point is the very same point at which the momentary vertical velocity of the object is zero meaning Vyt = 0 ; Using (3) this can give us a clue as to the time it takes for an object to reach its highest point in the projectile. 0 = Vy1 - gt , SO t = Vy1/g but in SYMMETRICAL projectile the projectile looks very symmetrical so the time it takes for the object to reach the top of the curve is equal to the time it takes that same object to reach the ground therefore the overall time from the moment the projectile is fired to the very moment it hits the ground can be expressed as 2t and that means that the time it takes for a projectile to hit its target is 2t and since t =Vy1/g we have
(5) T (total)= 2t = 2Vy1/g
(6) You are never given the horizontal or the vertical component of the velocity but you are always given the initial velocity of projectile and the angle of projection so u simply derive the components from the Pythagorean methods or Sin , Cos laws:
(6) Vy1 = Sin a * V [notice that I labeled it Vy1 because Vyt is constantly changing, and its value at any time t is given by (3)]
(7) Vx = Cos a * V [notice I didn't label the Vx so it is constant throughout the whole projectile]
(8) It is not recommended but I usually think that it is a better idea to mix the formulas (1) and (2) and (6) and (7). The result is derivation of the following formula You don't have to commit this to memory but it is a challenge for you to try and see how it was derived in the first place:
(9) Y = tan(a)*X + 1/2*g*X^2/ V * Cos (a) , where Y is the height a symmetrical projectile can achieve if it is shot with an initial velocity of V and an angle of projection of (a) and X is the distance from the point of firing to the landing place or in other words the horizontal distance.
______________________________________
Problem solving steps involving projectile
Read the question
Visualize the question (I mean practically imagine yourself as the projectile)
Record the known variables (including x, t, Vy1, Vx1 , V initial, angle a, Y, and
wind resistance or other non-common variables)
Convert all the variables to SI *(System Internationale) Units (most ppl miss this)
Write down formulas (1),(2), (6), and (7).
First , just calculate Vy1 and Vx from (6), and (7).
Second, see which of the formulas (either (1), or (2) ) is solvable.
Third, solve either (1) or (2) and find a variable that can help solve the other [for instance if you can solve (1) first and you find per se t, then use this value of t and solve for (2)
Forth, do your calculations METICULOUSLY and accurately double checking
Do not lose sight of time; a good problem solver should get answers for questions in 4 minutes or less. (In my class I put a restriction of 1.5 minutes per each question only)
Write the answer and BOX THEM.
Make sure the Significant figures agree, and that the answer has UNITS.
Reason to see whether the answer makes sense (for instance if you drop a disk at 10 m/s if you get that it hits the ground at 4000 m/s then something went terribly , terribly wrong so you can go back and find the fault and correct it if time allows).
______________________________________________________________
Recommended English Texts with detailed attention to projectile:
US:
College Physics , Sixth Ed. , Serway/Faughn. [Thomsons publication]
EU:
Taschenbuch der Physik, Fachbuchverlag Leipzig im Carl Hanser Verlag
_______________________________________________________________
Great online Simulator for Projectile:
http://galileo.phys.virginia.edu/classes/1.../jarapplet.html
Online Games where you can apply Projectile and have fun with it:
www.jippii.co.uk , Go to Blue Game House, and Challenge your opponents to Modern Game [Cannon]
________________________________________________________________
Ok, now lets finish this by solving your sample:
QUOTE
A projectile is launched at 65 m/s at an angle of 40*. Find:
a) the final velocity (as two components and as a magnitude and direction)
B) the total time of flight
c) the maximum height
d) the horizontal range
a) the final velocity (as two components and as a magnitude and direction)
B) the total time of flight
c) the maximum height
d) the horizontal range
Lets apply the methods of steps I mentioned above:
Read the question (Check)
Visualize (Check)
Record variables in SI units: (Check)
V = 65 m/s , a = 40 degrees , Y= ? , X =? , T (total) =?
Use formula (6) and (7) and find Vy1 and Vx:
Vy1 = V * Sin a = 65 * Sin 40 = 42 m/s
Vx = V * Cos a = 65 * Cos 40 = 50 m/s
Use formulas (1) and (2) and see which one you can solve for first (Check)
Vx = X / T [cant solve]
Y = Vy1T - 1/2gT^2 [cant solve]
so we are stuck here but here is the trick, as u remember as the top of the projectile the vertical velocity is zero so Vyt = 0:
now we use (3) to find t : Vyt = Vy1 - gt , So 0 =42 - 9.80* t So t = 4.3 seconds
Since this is the time it takes the projectile to reach the top of its path the actual total time is T (total) = 2t = 8.6 seconds
Now that we have T, and Vx we can use (1) to find X
X = Vx * T = 8.6 * 50 = 430 m
now we also can use (2') to find Y so we have:
Vyt = 0 and Vy1 = 42 m/s
so
Vyt^2 = Vy1^2 - 2gY , So 0^2 = 42^2 - 2 *9.8*Y SO Y= 88.2 m
OK, now we want to find the final velocity and to do that we need Vx which we already have and Vyt at the very moment the projectile hits the ground:
since the total time of the projectile was calculated to be t = 8.6 seconds, and since the time from when the projectile is on the top of the curve to when it hits the ground is half this amount of 4.3 seconds so we have
g = V 2 - v 1 / t so V2 = gt + v1 = 4.3*9.8 + 0 = 42 m/s
Vx = 50 m/s = constant all the time in the flight
so V final =radical{ Vx^2 + Vyt^2 }= radical (50^2 + 42^2 ) = 65 m/s
Why did I use V final =radical{ Vx^2 + Vyt^2 }? well, because the Horizontal and the vertical components are always perpendicular and the outcome is a vector sum of the two horizontal and the vertical vector and since it has the shape of a right triangle I used the Pythagorean theorem to solve it.
Also do notice that the final speed of this symmetrical projectile is the same as its initial speed. It is because this is a conserved system meaning that the only forces in work are the force of the gravity and no friction, or air resistance.
OK , so lets put discreet answers to each of your questions:
a)V final = 65 m/s
B)T (total) = 2t = 8.6 seconds
c)Y= 88.2 m
d) X= 430 m
___________________________________
[END OF TOPIC 2]
*****************************************************************
_____________
_________________
_____________________
*****************************************************************
TOPIC 3: NEWTONIAN PHYSICS
Definition:
Technical Definition: Any kind of assessment that involves force vectors and free body diagrams.
Normal Definitions:All those questions that deal with force, acceleration and Newton's laws.
_______________________________
Newton's Laws:
1) All entities (objects, systems, frames, particles, etc.) in motion tend to stay moving, unless acted upon otherwise. (This also means that all objects moving at zero speed or not moving would also stay that way if not acted upon by any external force).
2) If a constant or net force is applied on an object of mass "m", the object would gain an acceleration of quantity "a" but only when the following equation is present:
F(net) = m a *Note: do consider the fact that this is net force and not a single force; Also net force is relative to a direction for instance we can have Fx, Fy, Fz or etc.
3) For every action there is an equal but opposite reaction. In other words if you hit your head at the wall, the wall will hit you with the same strength. In more technical terms, the force applied to an atomic surface, would be retaliated with a force of equal magnitude in the opposite direction back at the inflicter. (Kind of like an eye for an eye).
4) Newton's law of Universal Gravitation: All objects of mass attract each other. This attraction results in a force that is directly proportional to the size of the participating masses, and inversely proportional to the square of the distance between then. In lamen terms, the bigger and the close the more the attraction the smaller and the farther the less. This gives rise to the following formula:
F = G m1 * m2 / d^2
5) On the surface of the earth the formula above is approximated to W ~ mg, and in physics book it is often accepted to be W = mg since the gravitational acceleration due to earth is not much affected by individual masses in short distances. This formula as you know is the weight of an object with mass of "m."
___________________________________
Formulas:
(1) ∑ F = ma or a = ∑ F / m
(2) F = G m1 * m2 / d^ 2 where d is the distance between he centre of gravity of two masses with a distance of d meters between them.
(3) W = mg , or weight equals mass times the gravitational acceleration.
This formula is derived from the Universal gravitational formula. In a sense W is the force of earth with the mass M(earth) on a person with the mass m, so we can re-write:
W = mg = G m * M(earth) / d ^ 2 , where d is the distance of a person from the centre of the earth and G is a constant of the value G = 6.67 * 10^(-11) N. m^2 / kg^2. This would implicitly give rise to our next equation:
(4) g = G M(earth) / d ^ 2 ; Your physics teachers usually consider this to be a constant at the sea level on earth and approximate it to be g = 9.80 m/s/s .
(5) all connected systems experience the same acceleration.
(6) an Ideal pulley has no mass, and does not alter the tension of the cables around it.
(7) an Ideal cable is not elastic and would experience a constant tension of T.
(8) Any object that moves at a constant velocity (even if it means the object is stationary or has a zero velocity) experiences an acceleration equal to zero. According to (1) the ∑ F = m a = m * 0 = 0 . In short, objects that are stable and do not move or move at a constant speed have a net force of Zero or ∑F = 0.
(9)In the real world, no two surfaces are ideally flat and so any motion between to objects in contact, would experience a resistance termed friction. Friction depends upon the normal on a surface and a constant that is unique to each material known as the coefficient of friction. In other words:
F ≈ μ * n where n is the normal on the surface and "μ" is a special constant that changes depending on the surfaces involved.
(10) Friction is of two kinds: 1. Static Friction: Frictional force on an object that is not in motion , and 2.Kinetic Friction: Frictional force on an object that is in motion.
(11) Kinetic friction is a constant value depending on normal and the coefficient of Kinetic friction or μk. Static friction on the other hand is not a constant value and can be variable up until a maximum. In other words:
F (static) ≤ μs . n ; where n is the normal and μs is the coefficient of static friction.
____________________________________________
Problem Solving steps involving a Newtonian Problem:
Read the question
Draw a FREE BODY DIAGRAM showing all the forces
If the system is moving then the net force is not zero and if the system is stable simply assume a = 0 and F (net) = 0.
Observe whether the friction exists or if you deal with an ideal system with nonconservative forces put friction equal to zero or avoid it altogether.
Use (1) and (3), keeping (8). (9).and (10) in mind.
Find the variables that you are asked for
BOX your answer
Check your answer for validity
If time allows, come back for a re-validation
_____________________________________________
Recommended Texts with detailed attention to Newtonian Physics:
US:
Newtonian Physics; Online Ed. Various Authors.
<http://www.lightandmatter.com/area1book1.html>
EU:
RF Linear Accelerators; von Thomas Wangler
<http://www.amazon.de/exec/obidos/ASIN/0471168149/ref=si_1_6/028-1487989-4297342>
______________________________________________
Sample Problem:
In the following figure, the masses m1 and m2 are connected together via an ideal cable, and the pulley is ideal. Find the acceleration of the system and the tension. The surface is ideal and frictionless.
Read the Question
Draw the Body Diagram to something Like I did above
seperate the system into mass m1 and mass m2 system.
Ask yourself is this system moving or is it stable? Obviously since the question is asking for acceleration you must expect it to be accelerating and so net force is not zero and the system has an acceleration.
use equation (1) to write the expression for mass 1:
the forces acting on mass m1 are the weight of the object which is mg downward, the normal out of the desk upward. These two forces repel each other and cancel each other out. We can simply ignore them as I did in the figure since there is no friction but I recommend you draw them. The only remaining force is the tension in the cable T, but this tension is not level and has an angle of 30 degrees. This means that two vector components work on the mass:
T sin 30 and T cos 30 , out of which the only one causing any kind of motion is T cos 30 or the Tension component parallel to the surface and so the only force acting on the system that causes motion is T cos 30. Now using (1) we write the equation (1) for this mass:
∑ F = ma so T cos30 = m1 * a (1')
we leave this for now since we can not solve this for T or a since we have two variables.
the second mass also only has T or the cable tension plus its weight. Since I know that box is moving down I know the weight is more than the tension so the net force is:
∑ F = ma or m2g - T = m2 a (2')
now I have two equations and two variables and I can simply solve for either one:
T = m1 * a / cos30 using this we put in (2') and have:
m2g - (m1a/cos30) = m2a or a = ( m2 g )/(m2 - m1/cos30)
this follows that T = m2a - m2g
____________________________________________
********************************************************************
[END OF THIS TOPIC]
_________
______________
___________________
********************************************************************
TOPIC 4: MECHANICAL ENERGY
Definitions:
Conservative Forces: Any force whose magnitude is independent of the route it takes and depends on relative positioning to a constant frame. In other words force of gravity, kinetic energy, and inter-atomic attractive forces, Van Der Waals attraction forces, or any force in general whose magnitude has nothing to do with the path it takes. For instance a person standing 2 meters up the stairs has an equal potential energy as a person standing 2 meters on top of a hill. Path is not of value but the absolute position.
Non-Conservative Forces: Specifically used to refer to friction on surfaces, friction in air, or other not very understood forces (such as membrane attractions and etc.). These forces are path independent and not state dependent. Unlike potential energy it does matter how much you drag a bag on the floor. If you drag it from A and B the amount of hit produced would differ depending on the path you take. Look at the following example:
As you can see the longer path produces more friction induced heat than the shorter path:
Potential Energy: One of the Conservative energies that is defined based on a point of reference (usually the ground) and is defined as a tendency to do work.
Kinetic Energy: Energy of the moving object. The energy associated with a moving object. It is directly proportional to the mass and the square of the velocity of an object.
Work: Work is defined as the movement of a force. Work and its concept is physics is not the same as our concept of work. We consider carrying a bucket of water working but physics says no you did no work! In fact work in physics is not often the same as energy. A man who is holding a weight without moving it does no work although he expends considerable chemical energy and tolerates tension in his muscle and tissue, while his blood circulates and his heat is dissipated to the surrounding. Work is often the same as force times displacement, however the angle of the force vs. displacement is very important.
____________________________
Formulas:
(1)W = Fd Cos a where a is the angel between the applied force F and the displacement of d.
(2)PE = mgh where a mass of "m" is h meters away from a point of reference (for instance the floor) and is experiencing a gravitational constant of g. Since we already proved previously that g = G M(earth) / d^2 , we can say that potential energy for an object that is very far from the surface of the earth is:
(3) U= PE = G m * M(earth) / d
(4)KE = 1/2 m v^2
When the Mechanical energy is conserved, we can write:
(5)PEi + KEi = PEf + KEf where i stands for initial and f for final
When the Mechanical energy is not conserved which means there is friction involved or any other nonconservative forces we have:
(6)Wnc = (Mechanical energy final) - (Mechanical energy initial) = (KEf + PEf) - (KEi + PEi) ; where Wnc is the work done by non-conservative forces namely friction
The above formula is very useful in Mechanics since almost all questions could be somewhat solved by assumption of conservation of mechanical energy through the non-conservative forces.
(7)Power is defined as work or energy expended over time. so P = W / t or P = E /t
(8) since P = W /t = F . d / t = F * (d/t) = F v ; where v is the velocity by which a force is moved.
_______________________________________________
Problem Solving Steps:
Read the question
Record the variables such as Force, times, distance, velocity, power, PE, KE, Frictional force, constant of friction, and acceleration in SI units.
Write down (6) if there is friction present, or (5) if there is no friction to solve for questions that ask for the final kinetic energy or potential having given you the initial values of the two and one final value. write down (1) for simple work problems, and make use of the rest upon demand.
calculate the missing variables
Write yours answers with the proper SI units and BOX them.
If possible reason your way through your answers to see whether they make sense or not.
If time allows come back and re-check.
______________________________________________
Online Simulations:
http://jersey.uoregon.edu/vlab/Work/
_______________________________________________
Sample Problem:
A coin is dropped from a height "h" from the top of a building. What is its velocity when it hits the ground assuming the air friction is minimal?
Well, PEi = mgh and KEi = 0
PEf = 0 and KEf = 1/2 mv^2
so (5) PEi + KEi = PEf + KEf
so 1/2 m v^2 = mgh
so v = radical ( 2gh)
___________________________________________________
[End of this Topic]
*******************************************************************
___________
_______________
__________________
*******************************************************************
THE SCIENCE BEHIND ROCKET PROPULSION:
(A tribute to Dr. Wherner Van Braun, and Dr. Goddard for their contributions to NASA and Space Age)
People often confuse the movement of rockets with movements of entities such as cars and trains. The truth is the movement of a rocket is nothing like a car or a train. A car or a train pushes against a track or the surface and it is the force of friction that allows a car to move. In other words without static friction a cars wheels simply wont be able to turn or without static friction the wheels or a train can not push against a track and just like a man on an ice sheet they would move but stay in one place simply because friction is just too little.
For rockets however the movement is not because of friction but due to CONSERVATION OF MOMENTUM between the rocket itself and the ejected fuel!
It is simple lets assume a hypothetical rocket. Suppose that at some time "t" the momentum of the rocket and its fuel is (M + Δm) v , where M is the mass of the rocket body minus its fuel and Δm is the mass of the fuel that wants to eject.
Obviously before any fuel is ejected the momentum of the rocket is
(M + Δm) v
Now, assume that during a short time interval of "Δt" , the rocket ejects fuel of mass Δm and the rocket's speed therefore increases to v + Δv. If the fuel is ejected with a speed of v(e) (for v(fuel) ) relative to the rocket, the speed of the fuel relative to a stationary frame of reference is v - v(e). Thus, if we equate the total initial momentum of the system with the total final momentum (known as conservation of momentum), we have:
Initial momentum of the rocket and fuel = momentum of rocket + momentum of fuel
OR
(M + Δm) v = M (v + Δv) + Δm (v - v(e))
Simplifying the above expression gives us:
M Δv = Δm v(e)
Furthermore the increase Δm in the exhaust mass corresponds to an equal decrease in the rocket mass, so that:
Δm = - ΔM
This results together with the method of calculus (I am not going into that here) can be used to obtain the following important equation:
Δv = v(e) ln (Mi / Mf)
Where, Mi = initial mass of the rocket plus the fuel
Mf = final mass of rocket plus its remaining fuel (just mass of target is all of the fuel is expelled)
Δv = increase in speed;
This is the basic expression for rocket propulsion. it tells us that the increase in speed is proportional to to the exhaust speed v(e), and to the natural log of Mi/Mf. Because the maximum ratio of Mi/Mf for a single-stage rocket is about 10:1(NO rocket can have a higher ratio, not yet!), the increase in speed can reach v(e) ln 10 = 2.3 v(e) or about twice the exhaust speed!!
Therefore the speed of any space ship is no more than twice the speed by which it pumps its gas and ejects it! Therefore for a rocket to move the exhaust speed should be relatively high. Currently typical exhaust speeds are several Km/s only! (Hey maybe scientists like you can change that ;) ).
Another concept in rockets is THRUST, that I am sure you heard a lot about but this is what physics mean when it talks about THRUST. It is defined as the force exerted on the rocket by ejected exhaust gas. We can obtain an expression for the instantaneous thrust from the equation Δm = - ΔM .
Instantaneous Thrust =F(Thrust) = M*a = | v(e) ΔM / ΔT |
Here obviously we see that the thrust increases as the exhaust speed increases and as the rate of change of mass ΔM /Δt (the burn rate) increase.
__________________________________
__________________________________
TOPIC 4: FLUIDS, SOLIDS, GASSES
Definitions:
Fluid: Any substance that has a fixed volume but has no shape and conforms to the shape of its container.
Solids: Any substance that has a defined mass and defined volume. Solids are of two kinds: 1)crystalline structure solids that follow a specific molecular formation and have a cubic structure, and 2)amorphous solids that have no particular intermolecular bonding and therefore have an erratic structure such as glass or frozen magma.
Gases: Gases have an infinite volume fitting the volume of the container and they poses no volume by their own. An IDEAL gas has no volume (hence no Van Der Waals forces) and it is often found in conditions of high temperature, low pressure and no intermolecular forces. A non-ideal gas however has considerable mass of its own, has intermolecular attractive Van der Waals or dipole attraction and is often found in normal conditions. Any non-Ideal gas can be made ideal at very high temperature and low pressures.
Plasma: At very high pressure, and high temperature the boundary between liquid and gas breaks apart and we are left with a substance that is like both yet neither one. Core of the sun, where there is abundant pressure and heat has its mass in plasma. Plasma can be achieved for any substance once the pressure and temperature go beyond the critical point.
Pressure: it is defined as force over area.
Elastic Modulus: It is a concept arising from the atomic nature of all substance. Due to the fact that no one entity can have all its atoms packed together as to allow no space in between, atomic structures are subject to packing or unpacking resulting in compression or stretching. In general Elastic Modulus is defined as a ratio of applied stress over a specific resultant strain. There are three kinds of elastic modulus: 1)YOUNG'S MODULUS, 2)SHEAR MODULUS, 3)BULK MODULUS.
_____________________________________
Formulas:
Young's Modulus: Y = P / (ΔL /L) , where the pressure applied is P and the change in length caused by this pressure is ΔL and L is the original length.
Shear's Modulus: S = P / (x / h) where x is the horizontal distance the force moves the object and h is the height of the object.
Bulk's Modulus: B = ΔP / (ΔV/V) where Δ V is the change of the volume.
Density = Mass / Volume
Pressure: P = F / A
Pressure in Liquids (A special case of pressure):
Imagine a container that is cylindrical in shape filled with a liquid of density "ρ", and a Volume of "V" and it is located on the surface of earth where the gravitational pull is g = 9.80 m/s/s and the container is filled "h" meters up with the liquid. The following is derivation of the liquid pressure:
[The figure]
P = F / A = mg / A = ρVg / A
and since V = A * h in any cylinder we have:
P = ρVg / A = ρ*A*h*g / A = ρhg
So the pressure in a liquid depends only on its height on the container and the gravitational pull and the density of the liquid and is independent of the shape of the container.
A1v1 = A2v2 This it the flow rate equation or in other words conservation of flow rate for an ideal liquid (fluid). This means that if a fluid is moving at a velocity of v1 in a pipe of the cross-sectional area of A1, it would move at a new speed of v2, should the tube change its cross-sectional area to A2.
DANIEL Bernoulli's Equation:
1/2 ρv^2 + ρgh + P = constant also called conservation of mechanical energy in ideal liquids
*Note: in any system you have to take points that are at the same level before applying this equation.
Buoyant Force:
Buoyant force is not an independent force but is derived from difference in pressure from the bottom of an object to its top.
Lets say an object of density "ρ'" is located "d" meters down the surface of a liquid of density "ρ" relative to its upper surface so the upper surface experience a force of water pressure equal to
P1= ρ*g*d
Also assuming that the object itself is "h" meters in heights then the lower part is h +d meters down the surface and experiences a Pressure of
P2 = ρ*g* (d+h)
obviously P2 > P1
so ΔP = P2 - P1 = ρ*g* (d + h - d) = ρ*g*h
and then,
ΔP = ΔF / A = ρ*g*h
so we have
ΔF = ρ*g*h*A = ρ*g*V ; here V is the volume of the object that is submerged in the liquid, g is the gravity constant, and ρ is the density of the liquid.
Also since we have:
ΔF = ρ*g*V and ρ' = m / V so V = m / ρ' we have,
ΔF = ΔF = ρ*g*Vρ*g* (m / ρ') = (ρ * ρ')(mg) = W(object)* (ρ * ρ')
so the buyant force is also equal to the weight of the object times the multiple of the densities of object and the liquid medium. Buoyant force is also equal to the displaced weight of the liquid since again:
ΔF = ρ*g*V = (ρ * V) g = m(liquid) * g = W (liquid) (This derivative is also referred to as the Archimedes Principal)
Pascal's Principal:
F1/A1 =F2/A2 ; this applies in a closed system of connected tubes.
Terminal velocity: [Ultra-Centrifugation]
If an object falls through a viscous medium (such as air) as is the case for a person he or she will eventually attain a terminal velocity:
As you can see the person is pushed down by a force of gravity W = mg
He also experiences a Buoyant force of B = ρ*g*V where V is his volume and ρ is the density of the air.
As he falls he picks up speed and as he goes faster the resistive R air friction increases. Air friction is R = kv where k is a constant unique to an object's shape, and size and v is the velocity of the object.
So since at the beginning his speed is not fast enough he would accelerate since the resistive force is not large enough to compensate for his weight:
R + B < mg there fore a ≠ 0
but as he picks up speed, the R increases until:
R + B = mg ; at that time he is moving at a constant speed before a = 0
so we have:
kv + ρ*g*V = mg
and
v = g ( m - ρV) / k
if the person himself also has a constant density of "ρ'" then we have:
v = g (ρ'V - ρV) / k = gV/k (ρ' -ρ)
This is the magnitude of the terminal velocity for an object of density ρ' in a medium of density ρ and a resistive constant of k.
Surface Tension: S = F / L ; where L is the length over which the force F acts.
Reynold's Number: RN = ρvd / η ; where v is the velocity, and η is the viscosity of the liquid, and d is the radius of the tube through which this fluid flows and ρ is the density of the liquid.
Ideal Gas Law:
PV = nRT ; where P is pressure in atm, V is volume in Litter, n is the number of moles, R is equal to .08021, T is temperature in Kelvin.
Boyle's Law: PV = constant
Charles' Law: V/T = constant
Kinetic Energy of a gas:
KE = 3/2*Kb*T ; where T is the temperature in Kelvin, Kb is Ludwig Boltzmann's constant [kb = 1.38E-23 J/K]
Average speed (rms) of a gas:
u(rms) = √ 3RT / MM ; where MM is the molecular mass in g/mol, T is the temperature in Kelvin, and R is the gas constant equal to 8.31 J/mol * K.
Internal Energy of the monoatomic gas:
U = 3/2 nRT
_____________________________________________
Problem Solving Steps:
This is a very diverse category of discussion. Usually the most important formulas are the Buoyancy force, and Pascal's equation. I recommend reading the question clearly, identifying whether the question is about a liquid, gas or solid and whether about pressure, motion, or liquid motion. From there on, any one of the formulas above is applicable. Of course as usual, read the question carefully, and try to re-evaluate at the end.
_____________________________________________
Sample Problem: [REAL LIFE APPLICATION]
How to identify a pure gold item: Lets say you buy a gold crown of mass "m" that you assume is pure gold but you want to test it. Here is how it is done using buoyancy:
First weigh the item in air (if you could in vacuum to avoid buoyancy of air)
you find that the object weighs W = mg
You then immerse the object in water holding it with a thread connected to a balance.
Obviously since the crown is not moving we have the net force equal to zero so we have:
B + T = W
ρ*g*V + T = mg or T = mg - ρ*g*V
also we know that the buoyant force on the crown is equal to the volume of the water displaced by the crown we have:
B = ρ*g*V(crown) so we can have then:
T = mg - ρ*g*V(crown) and since density of crown is ρ' = m / V
we have:
T = mg - ρ*g* (m/ρ ')
re-writing the equation above gives us:
(Wρ) / (W - T) = ρ'
so if you measure the weight or "W" of the crown in air and then find its weight when completely submerged in water to be T then you can find its density from the formula above. Now if:
If your salesman cheated on you and added some impurity your estimated ρ' would be off from the actual value for gold and if it is close or the same you got a sweet deal.
__________________________________
[End of This Topic]
**********************************************************
_________________
___________
______
*********************************************************
TOPIC 5: ELECTRICITY (DC)
Definitions:
Current: Flow of charged particles due to a potential difference from one point to the next. Charge always moves from a place of high potential to a point of law potential. Current is conventionally taken to be the movement of the positive charge but it is in fact the electrons that move in the metallic Crystal lattice from a place of high negative potential to a place of low negative potential. Currents conventionally move from the positive end of the battery to the negative end (long length in drawing).
Charge: Anything that has an excess or lack of electron.
Circuit: A connection that allows movement of electrons in a predetermined fashion from a source of high potential to a source of low potential.
Resistance: The impediment of current movement or the impediment of amount of charge allowed to pass through any conducting wire.
Conservation of Charge: Charge lost from a source is gained by another charge but the net amount of charge in universe is always conserved. This was discovered by Benjamin Franklin along with his countless other contribution to electricity principals.
Conduction: Charging a neutral object by bringing it into contact with an object of negative or positive charge and therefore inducing the flow of electrons from the neutral object to the positive object or to induce the flow of electrons from the charged negative object to our neutral object. Conduction demands 1)direct contact, and 2) conductivity in both charged and non-charged object. Conduction does not occur in insulators or even semi-conductors under normal conditions.
Induction: To create a net charge in a neutral object without ANY contact between the two objects. The only condition is that the object being charged has to be a conductor. The effector (object causing the induction, could or could not be a conductor). Look at the picture below for samples of Induction. Induction works due to the principle of attraction of opposite charges.
Polarization: This is Induction done in a non-conductor. The result is that charges or ions oriented to have their positive end close to the external negative charge and vice versa.
Capacitors: Two parallel plates separate by a distance among them often filled with a non conducting material or air or vacuum. Capacitors develop Voltage gradually and therefore impede current in a circuit.
Resistors: Component of Circuit that impede the current.
______________________________
Formulas:
Sir Charles Coloumbs Law in vacuum:
F = K Q1 * Q2 / d^2 , where Q1 and Q2 are respectively the charges involved and d is the distance between them and K is a constant equal to 9 * 10^9 N.m^2 /C^2
Note: Like charges repel each other and opposites attract.
Sir Charles Coloumbs Law in a space filled with matter:
F = K Q1 * Q2 / º d^2
where º is the permittivity of the matter.
Electric Field Strength:
E = F / Q where F is the electric force or force created by the electric field and E is the magnitude of the electric field. Q is the charge over which the electric field effects the force F.
Electric field strength when the source is a single point charge:
E = F / Q = K Q Q(source) / d^2 / Q = K Q(source) / d^2 where d is the distance in meters away from the charge source or Q source.
Electric field strength for a constant electric field (like a plate charge):
E = F / Q = V/d where V is the voltage across the plates seperate by a distance of d.
Electric Field lines: Hypothetical spatial lines representing the vector presence of the electric field. Field lines are drawn as if the electric field is acting on a positive charge on space. They also radiate outward from any positive charge particle and inward toward any negative charge.
Equipotential lines: Lines in space where the potential difference is zero. No charge would move from one equipotential line to the next or vice versa.
Electric Flux This is very fundamental concept in electronics and is defined as the number of electric field lines that penetrate any surface A at any time in space. Electric flux is defined by the following formula /sign:
Ô = E . A where E is the electric field strength and A is the area over which the field acts.
Electric Flux for a tilted area:
Ô = E A Cos a where a is the angle between the normal on the page and the electric field lines
or
Ô = E A Sin b where b is the angle between the page and a page that is perpendicular to the electric field.
Johann Carl Friedrich Gauss's Law:
This law applies to any point charge constricted into a close spacial membrane called the Gaussian surface. Take a point charge constricted in a Gaussian membrane formed in shape of a sphere:
**Picture property of http://atom.physics.calpoly.edu. All other images are produced by Ulrich Drude and you can manipulate them or use them in commercial or non commercial settings.
As you can in the picture above, the electric field lines penetrate our hypothetical Gaussian surface that is a sphere. The blue sphere has a radius of "r", and the charge has a charge of Q. OK lets star writing some formulas:
E = K Q / r^2 (the expression of electric field on the surface)
Ô = E A = K Q A / r^2
and
A = 4ër^2 the area of a sphere.
so we have now Ô = E A = 4ëK Q
we also know that permittivity of the free space º = 1 / 4ëK so we have:
Ô = 4ëKQ = Q / º , where º = 9 * 10^(-12) C^2/Nm^2
With the help of integration Gauss's law can be extended to include any point charge in any surface whether symmetrical or not. Ideally however symmetrical surfaces answer better than random ones.
Potential Difference and Potential energy for constant Electric field:
V = E d , and PE = Q V , so
PE = Q E d where Q is the charge E is the electric field to which it is expose, and d is the distance between the plates that created the electric charge.
Potential of a point charge:
V = k Q(source) /r
Potential energy of a point charge:
PE = Q V = k Q(source) Q / r
Capacitance:
C = Q / V where Q is the charge on EACH PLATE, and V is the voltage created by the voltage. C has units of C / V or Faraday (F).
C = א º A / d where º is the permittivity of free space 9 * 10^(-12)
, and where א is the dielectric constant which is unique to each substance.
The above formula can be proven:
V = E d and C = Q / V so we have:
C = Q / E .d = Q/E * 1/d and we also have Ô = E A = Q / º, So we have:
C = E A º / e * 1/d = A º /d
Stored energy in a Capacitor:
W = 1/2 C V^2
Resistance:
R = ρ L / A where L is the length of the resister through which the current moves, A is the cross-sectional area, and ρ is a constant unique to each conductor called the resistivity factor.
__________________
()_________________() <-- A
<----------L-------------->
Gorge Simon Ohm's Law
V = RI
Circuit Laws:
Series: 1) Resistance : Add them up ; 2)Capacitor : 1/C = 1/C1 + 1/C2+ ...
Parallel: 1)Resistance : 1/R = 1/R1 + 1/R2 +... ; 2)Capacitor: add them all up.
Paul Drude's Drift Speed Derivation:
I = n*v*Q*A ,
where A is the area of the wire, n is the number of mobile charge carriers per unit of space volume, and v is the drift velocity or average velocity of the charge carrier.
Temperature Variation of Resistance:
R = R1 (1+ a (T2 - T1) ) where a is a constant and 2 refers to the secondary state of the system while 1 refers to the primary state of the system.
Superconductivity:
Every conductor ideally has certain internal Resistance in it which impedes the movement of electrons. Dutch were the first people to notice Superconductivity in Mercury at low temperature. Thanks to work of J. George Benorz of Germany, and K. Alex Muller of IBM Zurich Lab, temperature for superconductivity has been raised to an almost applicable level.
1)But what is Superconductivity?
It is when a metal or an alloy at a certain critical temperature loses all internal resistivity and allows a current to move about unimpeded.
2)How is it explained?
BCS theory explains superconductivity in the following way:
(Excerpt from http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bcs.html)
Two electrons approach each other in the normal temperature. They collapse and leave off resulting in a random collision. We decrease the temperature until we reach the critical temperate. At this point two electrons come into contact. The electron is a fermion so it has a state of -,+1/2 Quantum spin number. The resultant attraction between opposite spins brings the electrons together and produces a union that is a Boson!
This Boson is called a "cooper pair" and it has an integer spin quantum number and can condensate into a very small space. The Bosonic effect decrease the internal energy and this decreases the resistance resulting in unimpeded flow.
QUOTE (->
| QUOTE |
| A projectile is launched at 65 m/s at an angle of 40*. Find: a) the final velocity (as two components and as a magnitude and direction) B) the total time of flight c) the maximum height d) the horizontal range |
Lets apply the methods of steps I mentioned above:
Read the question (Check)
Visualize (Check)
Record variables in SI units: (Check)
V = 65 m/s , a = 40 degrees , Y= ? , X =? , T (total) =?
Use formula (6) and (7) and find Vy1 and Vx:
Vy1 = V * Sin a = 65 * Sin 40 = 42 m/s
Vx = V * Cos a = 65 * Cos 40 = 50 m/s
Use formulas (1) and (2) and see which one you can solve for first (Check)
Vx = X / T [cant solve]
Y = Vy1T - 1/2gT^2 [cant solve]
so we are stuck here but here is the trick, as u remember as the top of the projectile the vertical velocity is zero so Vyt = 0:
now we use (3) to find t : Vyt = Vy1 - gt , So 0 =42 - 9.80* t So t = 4.3 seconds
Since this is the time it takes the projectile to reach the top of its path the actual total time is T (total) = 2t = 8.6 seconds
Now that we have T, and Vx we can use (1) to find X
X = Vx * T = 8.6 * 50 = 430 m
now we also can use (2') to find Y so we have:
Vyt = 0 and Vy1 = 42 m/s
so
Vyt^2 = Vy1^2 - 2gY , So 0^2 = 42^2 - 2 *9.8*Y SO Y= 88.2 m
OK, now we want to find the final velocity and to do that we need Vx which we already have and Vyt at the very moment the projectile hits the ground:
since the total time of the projectile was calculated to be t = 8.6 seconds, and since the time from when the projectile is on the top of the curve to when it hits the ground is half this amount of 4.3 seconds so we have
g = V 2 - v 1 / t so V2 = gt + v1 = 4.3*9.8 + 0 = 42 m/s
Vx = 50 m/s = constant all the time in the flight
so V final =radical{ Vx^2 + Vyt^2 }= radical (50^2 + 42^2 ) = 65 m/s
Why did I use V final =radical{ Vx^2 + Vyt^2 }? well, because the Horizontal and the vertical components are always perpendicular and the outcome is a vector sum of the two horizontal and the vertical vector and since it has the shape of a right triangle I used the Pythagorean theorem to solve it.
Also do notice that the final speed of this symmetrical projectile is the same as its initial speed. It is because this is a conserved system meaning that the only forces in work are the force of the gravity and no friction, or air resistance.
OK , so lets put discreet answers to each of your questions:
a)V final = 65 m/s
B)T (total) = 2t = 8.6 seconds
c)Y= 88.2 m
d) X= 430 m
___________________________________
[END OF TOPIC 2]
*****************************************************************
_____________
_________________
_____________________
*****************************************************************
TOPIC 3: NEWTONIAN PHYSICS
Definition:
Technical Definition: Any kind of assessment that involves force vectors and free body diagrams.
Normal Definitions:All those questions that deal with force, acceleration and Newton's laws.
_______________________________
Newton's Laws:
1) All entities (objects, systems, frames, particles, etc.) in motion tend to stay moving, unless acted upon otherwise. (This also means that all objects moving at zero speed or not moving would also stay that way if not acted upon by any external force).
2) If a constant or net force is applied on an object of mass "m", the object would gain an acceleration of quantity "a" but only when the following equation is present:
F(net) = m a *Note: do consider the fact that this is net force and not a single force; Also net force is relative to a direction for instance we can have Fx, Fy, Fz or etc.
3) For every action there is an equal but opposite reaction. In other words if you hit your head at the wall, the wall will hit you with the same strength. In more technical terms, the force applied to an atomic surface, would be retaliated with a force of equal magnitude in the opposite direction back at the inflicter. (Kind of like an eye for an eye).
4) Newton's law of Universal Gravitation: All objects of mass attract each other. This attraction results in a force that is directly proportional to the size of the participating masses, and inversely proportional to the square of the distance between then. In lamen terms, the bigger and the close the more the attraction the smaller and the farther the less. This gives rise to the following formula:
F = G m1 * m2 / d^2
5) On the surface of the earth the formula above is approximated to W ~ mg, and in physics book it is often accepted to be W = mg since the gravitational acceleration due to earth is not much affected by individual masses in short distances. This formula as you know is the weight of an object with mass of "m."
___________________________________
Formulas:
(1) ∑ F = ma or a = ∑ F / m
(2) F = G m1 * m2 / d^ 2 where d is the distance between he centre of gravity of two masses with a distance of d meters between them.
(3) W = mg , or weight equals mass times the gravitational acceleration.
This formula is derived from the Universal gravitational formula. In a sense W is the force of earth with the mass M(earth) on a person with the mass m, so we can re-write:
W = mg = G m * M(earth) / d ^ 2 , where d is the distance of a person from the centre of the earth and G is a constant of the value G = 6.67 * 10^(-11) N. m^2 / kg^2. This would implicitly give rise to our next equation:
(4) g = G M(earth) / d ^ 2 ; Your physics teachers usually consider this to be a constant at the sea level on earth and approximate it to be g = 9.80 m/s/s .
(5) all connected systems experience the same acceleration.
(6) an Ideal pulley has no mass, and does not alter the tension of the cables around it.
(7) an Ideal cable is not elastic and would experience a constant tension of T.
(8) Any object that moves at a constant velocity (even if it means the object is stationary or has a zero velocity) experiences an acceleration equal to zero. According to (1) the ∑ F = m a = m * 0 = 0 . In short, objects that are stable and do not move or move at a constant speed have a net force of Zero or ∑F = 0.
(9)In the real world, no two surfaces are ideally flat and so any motion between to objects in contact, would experience a resistance termed friction. Friction depends upon the normal on a surface and a constant that is unique to each material known as the coefficient of friction. In other words:
F ≈ μ * n where n is the normal on the surface and "μ" is a special constant that changes depending on the surfaces involved.
(10) Friction is of two kinds: 1. Static Friction: Frictional force on an object that is not in motion , and 2.Kinetic Friction: Frictional force on an object that is in motion.
(11) Kinetic friction is a constant value depending on normal and the coefficient of Kinetic friction or μk. Static friction on the other hand is not a constant value and can be variable up until a maximum. In other words:
F (static) ≤ μs . n ; where n is the normal and μs is the coefficient of static friction.
____________________________________________
Problem Solving steps involving a Newtonian Problem:
Read the question
Draw a FREE BODY DIAGRAM showing all the forces
If the system is moving then the net force is not zero and if the system is stable simply assume a = 0 and F (net) = 0.
Observe whether the friction exists or if you deal with an ideal system with nonconservative forces put friction equal to zero or avoid it altogether.
Use (1) and (3), keeping (8). (9).and (10) in mind.
Find the variables that you are asked for
BOX your answer
Check your answer for validity
If time allows, come back for a re-validation
_____________________________________________
Recommended Texts with detailed attention to Newtonian Physics:
US:
Newtonian Physics; Online Ed. Various Authors.
<http://www.lightandmatter.com/area1book1.html>
EU:
RF Linear Accelerators; von Thomas Wangler
<http://www.amazon.de/exec/obidos/ASIN/0471168149/ref=si_1_6/028-1487989-4297342>
______________________________________________
Sample Problem:
In the following figure, the masses m1 and m2 are connected together via an ideal cable, and the pulley is ideal. Find the acceleration of the system and the tension. The surface is ideal and frictionless.
Read the Question
Draw the Body Diagram to something Like I did above
seperate the system into mass m1 and mass m2 system.
Ask yourself is this system moving or is it stable? Obviously since the question is asking for acceleration you must expect it to be accelerating and so net force is not zero and the system has an acceleration.
use equation (1) to write the expression for mass 1:
the forces acting on mass m1 are the weight of the object which is mg downward, the normal out of the desk upward. These two forces repel each other and cancel each other out. We can simply ignore them as I did in the figure since there is no friction but I recommend you draw them. The only remaining force is the tension in the cable T, but this tension is not level and has an angle of 30 degrees. This means that two vector components work on the mass:
T sin 30 and T cos 30 , out of which the only one causing any kind of motion is T cos 30 or the Tension component parallel to the surface and so the only force acting on the system that causes motion is T cos 30. Now using (1) we write the equation (1) for this mass:
∑ F = ma so T cos30 = m1 * a (1')
we leave this for now since we can not solve this for T or a since we have two variables.
the second mass also only has T or the cable tension plus its weight. Since I know that box is moving down I know the weight is more than the tension so the net force is:
∑ F = ma or m2g - T = m2 a (2')
now I have two equations and two variables and I can simply solve for either one:
T = m1 * a / cos30 using this we put in (2') and have:
m2g - (m1a/cos30) = m2a or a = ( m2 g )/(m2 - m1/cos30)
this follows that T = m2a - m2g
____________________________________________
********************************************************************
[END OF THIS TOPIC]
_________
______________
___________________
********************************************************************
TOPIC 4: MECHANICAL ENERGY
Definitions:
Conservative Forces: Any force whose magnitude is independent of the route it takes and depends on relative positioning to a constant frame. In other words force of gravity, kinetic energy, and inter-atomic attractive forces, Van Der Waals attraction forces, or any force in general whose magnitude has nothing to do with the path it takes. For instance a person standing 2 meters up the stairs has an equal potential energy as a person standing 2 meters on top of a hill. Path is not of value but the absolute position.
Non-Conservative Forces: Specifically used to refer to friction on surfaces, friction in air, or other not very understood forces (such as membrane attractions and etc.). These forces are path independent and not state dependent. Unlike potential energy it does matter how much you drag a bag on the floor. If you drag it from A and B the amount of hit produced would differ depending on the path you take. Look at the following example:
As you can see the longer path produces more friction induced heat than the shorter path:
Potential Energy: One of the Conservative energies that is defined based on a point of reference (usually the ground) and is defined as a tendency to do work.
Kinetic Energy: Energy of the moving object. The energy associated with a moving object. It is directly proportional to the mass and the square of the velocity of an object.
Work: Work is defined as the movement of a force. Work and its concept is physics is not the same as our concept of work. We consider carrying a bucket of water working but physics says no you did no work! In fact work in physics is not often the same as energy. A man who is holding a weight without moving it does no work although he expends considerable chemical energy and tolerates tension in his muscle and tissue, while his blood circulates and his heat is dissipated to the surrounding. Work is often the same as force times displacement, however the angle of the force vs. displacement is very important.
____________________________
Formulas:
(1)W = Fd Cos a where a is the angel between the applied force F and the displacement of d.
(2)PE = mgh where a mass of "m" is h meters away from a point of reference (for instance the floor) and is experiencing a gravitational constant of g. Since we already proved previously that g = G M(earth) / d^2 , we can say that potential energy for an object that is very far from the surface of the earth is:
(3) U= PE = G m * M(earth) / d
(4)KE = 1/2 m v^2
When the Mechanical energy is conserved, we can write:
(5)PEi + KEi = PEf + KEf where i stands for initial and f for final
When the Mechanical energy is not conserved which means there is friction involved or any other nonconservative forces we have:
(6)Wnc = (Mechanical energy final) - (Mechanical energy initial) = (KEf + PEf) - (KEi + PEi) ; where Wnc is the work done by non-conservative forces namely friction
The above formula is very useful in Mechanics since almost all questions could be somewhat solved by assumption of conservation of mechanical energy through the non-conservative forces.
(7)Power is defined as work or energy expended over time. so P = W / t or P = E /t
(8) since P = W /t = F . d / t = F * (d/t) = F v ; where v is the velocity by which a force is moved.
_______________________________________________
Problem Solving Steps:
Read the question
Record the variables such as Force, times, distance, velocity, power, PE, KE, Frictional force, constant of friction, and acceleration in SI units.
Write down (6) if there is friction present, or (5) if there is no friction to solve for questions that ask for the final kinetic energy or potential having given you the initial values of the two and one final value. write down (1) for simple work problems, and make use of the rest upon demand.
calculate the missing variables
Write yours answers with the proper SI units and BOX them.
If possible reason your way through your answers to see whether they make sense or not.
If time allows come back and re-check.
______________________________________________
Online Simulations:
http://jersey.uoregon.edu/vlab/Work/
_______________________________________________
Sample Problem:
A coin is dropped from a height "h" from the top of a building. What is its velocity when it hits the ground assuming the air friction is minimal?
Well, PEi = mgh and KEi = 0
PEf = 0 and KEf = 1/2 mv^2
so (5) PEi + KEi = PEf + KEf
so 1/2 m v^2 = mgh
so v = radical ( 2gh)
___________________________________________________
[End of this Topic]
*******************************************************************
___________
_______________
__________________
*******************************************************************
THE SCIENCE BEHIND ROCKET PROPULSION:
(A tribute to Dr. Wherner Van Braun, and Dr. Goddard for their contributions to NASA and Space Age)
People often confuse the movement of rockets with movements of entities such as cars and trains. The truth is the movement of a rocket is nothing like a car or a train. A car or a train pushes against a track or the surface and it is the force of friction that allows a car to move. In other words without static friction a cars wheels simply wont be able to turn or without static friction the wheels or a train can not push against a track and just like a man on an ice sheet they would move but stay in one place simply because friction is just too little.
For rockets however the movement is not because of friction but due to CONSERVATION OF MOMENTUM between the rocket itself and the ejected fuel!
It is simple lets assume a hypothetical rocket. Suppose that at some time "t" the momentum of the rocket and its fuel is (M + Δm) v , where M is the mass of the rocket body minus its fuel and Δm is the mass of the fuel that wants to eject.
Obviously before any fuel is ejected the momentum of the rocket is
(M + Δm) v
Now, assume that during a short time interval of "Δt" , the rocket ejects fuel of mass Δm and the rocket's speed therefore increases to v + Δv. If the fuel is ejected with a speed of v(e) (for v(fuel) ) relative to the rocket, the speed of the fuel relative to a stationary frame of reference is v - v(e). Thus, if we equate the total initial momentum of the system with the total final momentum (known as conservation of momentum), we have:
Initial momentum of the rocket and fuel = momentum of rocket + momentum of fuel
OR
(M + Δm) v = M (v + Δv) + Δm (v - v(e))
Simplifying the above expression gives us:
M Δv = Δm v(e)
Furthermore the increase Δm in the exhaust mass corresponds to an equal decrease in the rocket mass, so that:
Δm = - ΔM
This results together with the method of calculus (I am not going into that here) can be used to obtain the following important equation:
Δv = v(e) ln (Mi / Mf)
Where, Mi = initial mass of the rocket plus the fuel
Mf = final mass of rocket plus its remaining fuel (just mass of target is all of the fuel is expelled)
Δv = increase in speed;
This is the basic expression for rocket propulsion. it tells us that the increase in speed is proportional to to the exhaust speed v(e), and to the natural log of Mi/Mf. Because the maximum ratio of Mi/Mf for a single-stage rocket is about 10:1(NO rocket can have a higher ratio, not yet!), the increase in speed can reach v(e) ln 10 = 2.3 v(e) or about twice the exhaust speed!!
Therefore the speed of any space ship is no more than twice the speed by which it pumps its gas and ejects it! Therefore for a rocket to move the exhaust speed should be relatively high. Currently typical exhaust speeds are several Km/s only! (Hey maybe scientists like you can change that ;) ).
Another concept in rockets is THRUST, that I am sure you heard a lot about but this is what physics mean when it talks about THRUST. It is defined as the force exerted on the rocket by ejected exhaust gas. We can obtain an expression for the instantaneous thrust from the equation Δm = - ΔM .
Instantaneous Thrust =F(Thrust) = M*a = | v(e) ΔM / ΔT |
Here obviously we see that the thrust increases as the exhaust speed increases and as the rate of change of mass ΔM /Δt (the burn rate) increase.
__________________________________
__________________________________
TOPIC 4: FLUIDS, SOLIDS, GASSES
Definitions:
Fluid: Any substance that has a fixed volume but has no shape and conforms to the shape of its container.
Solids: Any substance that has a defined mass and defined volume. Solids are of two kinds: 1)crystalline structure solids that follow a specific molecular formation and have a cubic structure, and 2)amorphous solids that have no particular intermolecular bonding and therefore have an erratic structure such as glass or frozen magma.
Gases: Gases have an infinite volume fitting the volume of the container and they poses no volume by their own. An IDEAL gas has no volume (hence no Van Der Waals forces) and it is often found in conditions of high temperature, low pressure and no intermolecular forces. A non-ideal gas however has considerable mass of its own, has intermolecular attractive Van der Waals or dipole attraction and is often found in normal conditions. Any non-Ideal gas can be made ideal at very high temperature and low pressures.
Plasma: At very high pressure, and high temperature the boundary between liquid and gas breaks apart and we are left with a substance that is like both yet neither one. Core of the sun, where there is abundant pressure and heat has its mass in plasma. Plasma can be achieved for any substance once the pressure and temperature go beyond the critical point.
Pressure: it is defined as force over area.
Elastic Modulus: It is a concept arising from the atomic nature of all substance. Due to the fact that no one entity can have all its atoms packed together as to allow no space in between, atomic structures are subject to packing or unpacking resulting in compression or stretching. In general Elastic Modulus is defined as a ratio of applied stress over a specific resultant strain. There are three kinds of elastic modulus: 1)YOUNG'S MODULUS, 2)SHEAR MODULUS, 3)BULK MODULUS.
_____________________________________
Formulas:
Young's Modulus: Y = P / (ΔL /L) , where the pressure applied is P and the change in length caused by this pressure is ΔL and L is the original length.
Shear's Modulus: S = P / (x / h) where x is the horizontal distance the force moves the object and h is the height of the object.
Bulk's Modulus: B = ΔP / (ΔV/V) where Δ V is the change of the volume.
Density = Mass / Volume
Pressure: P = F / A
Pressure in Liquids (A special case of pressure):
Imagine a container that is cylindrical in shape filled with a liquid of density "ρ", and a Volume of "V" and it is located on the surface of earth where the gravitational pull is g = 9.80 m/s/s and the container is filled "h" meters up with the liquid. The following is derivation of the liquid pressure:
[The figure]
P = F / A = mg / A = ρVg / A
and since V = A * h in any cylinder we have:
P = ρVg / A = ρ*A*h*g / A = ρhg
So the pressure in a liquid depends only on its height on the container and the gravitational pull and the density of the liquid and is independent of the shape of the container.
A1v1 = A2v2 This it the flow rate equation or in other words conservation of flow rate for an ideal liquid (fluid). This means that if a fluid is moving at a velocity of v1 in a pipe of the cross-sectional area of A1, it would move at a new speed of v2, should the tube change its cross-sectional area to A2.
DANIEL Bernoulli's Equation:
1/2 ρv^2 + ρgh + P = constant also called conservation of mechanical energy in ideal liquids
*Note: in any system you have to take points that are at the same level before applying this equation.
Buoyant Force:
Buoyant force is not an independent force but is derived from difference in pressure from the bottom of an object to its top.
Lets say an object of density "ρ'" is located "d" meters down the surface of a liquid of density "ρ" relative to its upper surface so the upper surface experience a force of water pressure equal to
P1= ρ*g*d
Also assuming that the object itself is "h" meters in heights then the lower part is h +d meters down the surface and experiences a Pressure of
P2 = ρ*g* (d+h)
obviously P2 > P1
so ΔP = P2 - P1 = ρ*g* (d + h - d) = ρ*g*h
and then,
ΔP = ΔF / A = ρ*g*h
so we have
ΔF = ρ*g*h*A = ρ*g*V ; here V is the volume of the object that is submerged in the liquid, g is the gravity constant, and ρ is the density of the liquid.
Also since we have:
ΔF = ρ*g*V and ρ' = m / V so V = m / ρ' we have,
ΔF = ΔF = ρ*g*Vρ*g* (m / ρ') = (ρ * ρ')(mg) = W(object)* (ρ * ρ')
so the buyant force is also equal to the weight of the object times the multiple of the densities of object and the liquid medium. Buoyant force is also equal to the displaced weight of the liquid since again:
ΔF = ρ*g*V = (ρ * V) g = m(liquid) * g = W (liquid) (This derivative is also referred to as the Archimedes Principal)
Pascal's Principal:
F1/A1 =F2/A2 ; this applies in a closed system of connected tubes.
Terminal velocity: [Ultra-Centrifugation]
If an object falls through a viscous medium (such as air) as is the case for a person he or she will eventually attain a terminal velocity:
As you can see the person is pushed down by a force of gravity W = mg
He also experiences a Buoyant force of B = ρ*g*V where V is his volume and ρ is the density of the air.
As he falls he picks up speed and as he goes faster the resistive R air friction increases. Air friction is R = kv where k is a constant unique to an object's shape, and size and v is the velocity of the object.
So since at the beginning his speed is not fast enough he would accelerate since the resistive force is not large enough to compensate for his weight:
R + B < mg there fore a ≠ 0
but as he picks up speed, the R increases until:
R + B = mg ; at that time he is moving at a constant speed before a = 0
so we have:
kv + ρ*g*V = mg
and
v = g ( m - ρV) / k
if the person himself also has a constant density of "ρ'" then we have:
v = g (ρ'V - ρV) / k = gV/k (ρ' -ρ)
This is the magnitude of the terminal velocity for an object of density ρ' in a medium of density ρ and a resistive constant of k.
Surface Tension: S = F / L ; where L is the length over which the force F acts.
Reynold's Number: RN = ρvd / η ; where v is the velocity, and η is the viscosity of the liquid, and d is the radius of the tube through which this fluid flows and ρ is the density of the liquid.
Ideal Gas Law:
PV = nRT ; where P is pressure in atm, V is volume in Litter, n is the number of moles, R is equal to .08021, T is temperature in Kelvin.
Boyle's Law: PV = constant
Charles' Law: V/T = constant
Kinetic Energy of a gas:
KE = 3/2*Kb*T ; where T is the temperature in Kelvin, Kb is Ludwig Boltzmann's constant [kb = 1.38E-23 J/K]
Average speed (rms) of a gas:
u(rms) = √ 3RT / MM ; where MM is the molecular mass in g/mol, T is the temperature in Kelvin, and R is the gas constant equal to 8.31 J/mol * K.
Internal Energy of the monoatomic gas:
U = 3/2 nRT
_____________________________________________
Problem Solving Steps:
This is a very diverse category of discussion. Usually the most important formulas are the Buoyancy force, and Pascal's equation. I recommend reading the question clearly, identifying whether the question is about a liquid, gas or solid and whether about pressure, motion, or liquid motion. From there on, any one of the formulas above is applicable. Of course as usual, read the question carefully, and try to re-evaluate at the end.
_____________________________________________
Sample Problem: [REAL LIFE APPLICATION]
How to identify a pure gold item: Lets say you buy a gold crown of mass "m" that you assume is pure gold but you want to test it. Here is how it is done using buoyancy:
First weigh the item in air (if you could in vacuum to avoid buoyancy of air)
you find that the object weighs W = mg
You then immerse the object in water holding it with a thread connected to a balance.
Obviously since the crown is not moving we have the net force equal to zero so we have:
B + T = W
ρ*g*V + T = mg or T = mg - ρ*g*V
also we know that the buoyant force on the crown is equal to the volume of the water displaced by the crown we have:
B = ρ*g*V(crown) so we can have then:
T = mg - ρ*g*V(crown) and since density of crown is ρ' = m / V
we have:
T = mg - ρ*g* (m/ρ ')
re-writing the equation above gives us:
(Wρ) / (W - T) = ρ'
so if you measure the weight or "W" of the crown in air and then find its weight when completely submerged in water to be T then you can find its density from the formula above. Now if:
If your salesman cheated on you and added some impurity your estimated ρ' would be off from the actual value for gold and if it is close or the same you got a sweet deal.
__________________________________
[End of This Topic]
**********************************************************
_________________
___________
______
*********************************************************
TOPIC 5: ELECTRICITY (DC)
Definitions:
Current: Flow of charged particles due to a potential difference from one point to the next. Charge always moves from a place of high potential to a point of law potential. Current is conventionally taken to be the movement of the positive charge but it is in fact the electrons that move in the metallic Crystal lattice from a place of high negative potential to a place of low negative potential. Currents conventionally move from the positive end of the battery to the negative end (long length in drawing).
Charge: Anything that has an excess or lack of electron.
Circuit: A connection that allows movement of electrons in a predetermined fashion from a source of high potential to a source of low potential.
Resistance: The impediment of current movement or the impediment of amount of charge allowed to pass through any conducting wire.
Conservation of Charge: Charge lost from a source is gained by another charge but the net amount of charge in universe is always conserved. This was discovered by Benjamin Franklin along with his countless other contribution to electricity principals.
Conduction: Charging a neutral object by bringing it into contact with an object of negative or positive charge and therefore inducing the flow of electrons from the neutral object to the positive object or to induce the flow of electrons from the charged negative object to our neutral object. Conduction demands 1)direct contact, and 2) conductivity in both charged and non-charged object. Conduction does not occur in insulators or even semi-conductors under normal conditions.
Induction: To create a net charge in a neutral object without ANY contact between the two objects. The only condition is that the object being charged has to be a conductor. The effector (object causing the induction, could or could not be a conductor). Look at the picture below for samples of Induction. Induction works due to the principle of attraction of opposite charges.
Polarization: This is Induction done in a non-conductor. The result is that charges or ions oriented to have their positive end close to the external negative charge and vice versa.
Capacitors: Two parallel plates separate by a distance among them often filled with a non conducting material or air or vacuum. Capacitors develop Voltage gradually and therefore impede current in a circuit.
Resistors: Component of Circuit that impede the current.
______________________________
Formulas:
Sir Charles Coloumbs Law in vacuum:
F = K Q1 * Q2 / d^2 , where Q1 and Q2 are respectively the charges involved and d is the distance between them and K is a constant equal to 9 * 10^9 N.m^2 /C^2
Note: Like charges repel each other and opposites attract.
Sir Charles Coloumbs Law in a space filled with matter:
F = K Q1 * Q2 / º d^2
where º is the permittivity of the matter.
Electric Field Strength:
E = F / Q where F is the electric force or force created by the electric field and E is the magnitude of the electric field. Q is the charge over which the electric field effects the force F.
Electric field strength when the source is a single point charge:
E = F / Q = K Q Q(source) / d^2 / Q = K Q(source) / d^2 where d is the distance in meters away from the charge source or Q source.
Electric field strength for a constant electric field (like a plate charge):
E = F / Q = V/d where V is the voltage across the plates seperate by a distance of d.
Electric Field lines: Hypothetical spatial lines representing the vector presence of the electric field. Field lines are drawn as if the electric field is acting on a positive charge on space. They also radiate outward from any positive charge particle and inward toward any negative charge.
Equipotential lines: Lines in space where the potential difference is zero. No charge would move from one equipotential line to the next or vice versa.
Electric Flux This is very fundamental concept in electronics and is defined as the number of electric field lines that penetrate any surface A at any time in space. Electric flux is defined by the following formula /sign:
Ô = E . A where E is the electric field strength and A is the area over which the field acts.
Electric Flux for a tilted area:
Ô = E A Cos a where a is the angle between the normal on the page and the electric field lines
or
Ô = E A Sin b where b is the angle between the page and a page that is perpendicular to the electric field.
Johann Carl Friedrich Gauss's Law:
This law applies to any point charge constricted into a close spacial membrane called the Gaussian surface. Take a point charge constricted in a Gaussian membrane formed in shape of a sphere:
**Picture property of http://atom.physics.calpoly.edu. All other images are produced by Ulrich Drude and you can manipulate them or use them in commercial or non commercial settings.
As you can in the picture above, the electric field lines penetrate our hypothetical Gaussian surface that is a sphere. The blue sphere has a radius of "r", and the charge has a charge of Q. OK lets star writing some formulas:
E = K Q / r^2 (the expression of electric field on the surface)
Ô = E A = K Q A / r^2
and
A = 4ër^2 the area of a sphere.
so we have now Ô = E A = 4ëK Q
we also know that permittivity of the free space º = 1 / 4ëK so we have:
Ô = 4ëKQ = Q / º , where º = 9 * 10^(-12) C^2/Nm^2
With the help of integration Gauss's law can be extended to include any point charge in any surface whether symmetrical or not. Ideally however symmetrical surfaces answer better than random ones.
Potential Difference and Potential energy for constant Electric field:
V = E d , and PE = Q V , so
PE = Q E d where Q is the charge E is the electric field to which it is expose, and d is the distance between the plates that created the electric charge.
Potential of a point charge:
V = k Q(source) /r
Potential energy of a point charge:
PE = Q V = k Q(source) Q / r
Capacitance:
C = Q / V where Q is the charge on EACH PLATE, and V is the voltage created by the voltage. C has units of C / V or Faraday (F).
C = א º A / d where º is the permittivity of free space 9 * 10^(-12)
, and where א is the dielectric constant which is unique to each substance.
The above formula can be proven:
V = E d and C = Q / V so we have:
C = Q / E .d = Q/E * 1/d and we also have Ô = E A = Q / º, So we have:
C = E A º / e * 1/d = A º /d
Stored energy in a Capacitor:
W = 1/2 C V^2
Resistance:
R = ρ L / A where L is the length of the resister through which the current moves, A is the cross-sectional area, and ρ is a constant unique to each conductor called the resistivity factor.
__________________
()_________________() <-- A
<----------L-------------->
Gorge Simon Ohm's Law
V = RI
Circuit Laws:
Series: 1) Resistance : Add them up ; 2)Capacitor : 1/C = 1/C1 + 1/C2+ ...
Parallel: 1)Resistance : 1/R = 1/R1 + 1/R2 +... ; 2)Capacitor: add them all up.
Paul Drude's Drift Speed Derivation:
I = n*v*Q*A ,
where A is the area of the wire, n is the number of mobile charge carriers per unit of space volume, and v is the drift velocity or average velocity of the charge carrier.
Temperature Variation of Resistance:
R = R1 (1+ a (T2 - T1) ) where a is a constant and 2 refers to the secondary state of the system while 1 refers to the primary state of the system.
Superconductivity:
Every conductor ideally has certain internal Resistance in it which impedes the movement of electrons. Dutch were the first people to notice Superconductivity in Mercury at low temperature. Thanks to work of J. George Benorz of Germany, and K. Alex Muller of IBM Zurich Lab, temperature for superconductivity has been raised to an almost applicable level.
1)But what is Superconductivity?
It is when a metal or an alloy at a certain critical temperature loses all internal resistivity and allows a current to move about unimpeded.
2)How is it explained?
BCS theory explains superconductivity in the following way:
(Excerpt from http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bcs.html)
Two electrons approach each other in the normal temperature. They collapse and leave off resulting in a random collision. We decrease the temperature until we reach the critical temperate. At this point two electrons come into contact. The electron is a fermion so it has a state of -,+1/2 Quantum spin number. The resultant attraction between opposite spins brings the electrons together and produces a union that is a Boson!
This Boson is called a "cooper pair" and it has an integer spin quantum number and can condensate into a very small space. The Bosonic effect decrease the internal energy and this decreases the resistance resulting in unimpeded flow.
Electrons acting as pairs via lattice interaction? How did they come up with that idea for the BCS theory of superconductivity? The evidence for a small band gap at the Fermi level was a key piece in the puzzle. That evidence comes from the existence of a critical temperature, the existence of a critical magnetic field, and the exponential nature of the heat capacity variation in the Type I superconductors.
Power in DC circuits:
Power = I V = V^2/R = I ^2 R
Gustav Kirchhoff's Principle:
RULE #1: I(input) = I(output)
All the current that is put into a junction shall leave the junction . In other words, conservation of current.
RULE#2: The sum of potential differences across all the elements around any closed-circuit loop must be zero.
Sub-Rule #2: In any circuit first choose a direction of movement in each loop and then use the following to give the potentials signs accordingly:
------------ I ---------->
A______/\/\/\________B if you go from A to B then V = -RI where R is the resistance.
A______/\/\/\________B if you go A to B then V = RI.
<----------- I ------------
A___(+)Battery(-)____B if you go from A to B, then V = - Voltage of Battery.
A___(-)Battery(+)____A if you go from A to B, then V = + Voltage of Battery.
RC Circuits: Any circuit that has a batter or voltage source, conductors, and resistors and capacitors in series with each other.
Note: when u connect the RC circuit it takes some time for the capacitor to charge up so the voltage would increase gradually. The capacitor would eventually have a voltage of Q = CV where V is the voltage of the battery but that would occur very gradually over a long period of time.
so instead the following formula is advised for any circuit such as bellow:
When the switch is closed at "a" the charge on the capacitor starts to build up according to :
Qt = CV (1 - e ^ (-t /RC) ) where t is instantaneous time and Qt is instantaneous charge and e is the Euler's constant.
T = RC this is the expression for what is called a time constant. Lets see what happens when t=T=RC
Qt = CV (1 - e ^ (-RC /RC) ) = CV (1 - e ^ (-1) ) = CV (1 - 1/e ) = .63 CV = 63% CV = 63% Q(final)
so at one times constant the capacitor has 63% of its maximum charge.
NOTE: A capacitor approaches a finite value CV as t approaches infinity so no capacitor would charge up anytime....unless given billions of years till end of the world but it gets very close.
If switch is turned to b or discharge the following Charge would remain on the capacitor at any time:
Qt = CV (e ^ (-t/RC) )
_________________________________________
Problem Solving Steps:
Do more problems that is the only way when the topic is so broad as this.
__________________________________________
Sample Problem:
If an appliance has a voltage of 200 V and its current is 20 ampere, then what is its power?
Power = IV = 4000 Watts
___________________________________________
This is marvellous and must have taken you a long time, Drude. This will make a fine sticky for the top of the homework section and people can be guided to the relevant section. I was going to suggest expanding your projectile piece but not by as much as you have done - great contribution.
Good to hear that mate 
. I am gonna add a few more common sections tonite like electricity, rotational motion, torque, and maybe nuclear/relativistic physics.
. I am gonna add a few more common sections tonite like electricity, rotational motion, torque, and maybe nuclear/relativistic physics.
This is great stuff. You should add a section on fluids and buoyancy.
QUOTE
This is great stuff. You should add a section on fluids and buoyancy.
I added the section on Buoyancy and liquids in the first post. Hope it helps.
I have just sent someone here - sure they will have all questions answered.
I await your electrical section as I have come to an impasse in my deliberations regarding charge transfer. I just know that almost everything I have been taught about electron flow, current and charge itself are fudged and I cannot bring myself to delve deeper in search of truth. Something has been lost between Maxwell, Gauss, Coulomb and Ampere and our 'easy' SI structure does not help look for it.
http://www.ee.surrey.ac.uk/Workshop/advice...s/unit_systems/
is where I make my head hurt when I feel up to it.
I await your electrical section as I have come to an impasse in my deliberations regarding charge transfer. I just know that almost everything I have been taught about electron flow, current and charge itself are fudged and I cannot bring myself to delve deeper in search of truth. Something has been lost between Maxwell, Gauss, Coulomb and Ampere and our 'easy' SI structure does not help look for it.
http://www.ee.surrey.ac.uk/Workshop/advice...s/unit_systems/
is where I make my head hurt when I feel up to it.
Surely, you wont need my help being a professional yourself in this area. I should probably ask for your input if anything but I will add the electrical section as promised. Thank you very much for your input.
This just gets better and better - now when anyone posts a question about anything we can just send them here - no more troublesome homework.
I haven't read the newer sections thoroughly yet but did send some recent poster there to find his answers - hope he isn't overawed. I must remember to point to a specific section in future.
Once again - fine work Sir.
I haven't read the newer sections thoroughly yet but did send some recent poster there to find his answers - hope he isn't overawed. I must remember to point to a specific section in future.
Once again - fine work Sir.
You know you are right, it might a bit hard to pinpoint the Topic headers so I kind of took the liberty to use a lot of blue and make the titles huge hehe, so any body can see the clear boundary from one topic to the next. Also thank for the comment, happy to hear that.
Hopefully today or tomorrow I will add the last sections of quantum theory, relativity and even String Theory since it is such a hot topic (though I dont believe in any of it). In any case, have a good day.
Hopefully today or tomorrow I will add the last sections of quantum theory, relativity and even String Theory since it is such a hot topic (though I dont believe in any of it). In any case, have a good day.
Quantum Physics:
What led to creation of Quantum theory?
_Electromagnetic radiation emitted by a heated object (blackbody effect)
_The emission of electrons by illuminated metals (Photoelectric effect)
_The emission of sharp spectral lines by gas atoms in an electric discharge tube.
_Dual nature of light as both a particle and energy
_Aberrations observed under the assumptions of relativity, and classical physics.
Who were most influential in creation of Quantum Physics?
_Max Planck
_Werner Heisenberg
_Max Born
_Louis De Broglie
_Albert Einstein
_Erwin Schrodinger
_Wolfgang Pauli
_Niel Bohr
_Paul Dirac
_Arthur Holly Compton
_Max von Laue
_Wilhelm Roentgen
_and many more who contributed but did not get recognized.
QUANTUM PHENOMENA:
1)BLACK BODY RADIATION: An object at any temperature is known to emit electromagnetic radiation that is sometimes referred to as thermal radiation. The radiated energy varies at different Temperatues, and different Wavelength:
λmax *T = .2898e(-2) m.K
where λmax is the wavelength at which the intensity of blackbody radiation is most, and T is the temperature in Kelvin.
Attempts to use the classical ideas to explain this quality of radiation intensity versus wavelength failed. Problem was that at long wavelegnths the classical theory was in good agreement with the experimental data, but at short wavelengths however, there exists a very broad difference and contradiction from expected theoretical data and the experimental data. According to the theory the wavelength should approach infiniti as the wavelegnth approaches zero but it is in fact zero! This contradiction is called the Ultraviolet Catasrophe because of the spectrum in which the experimental and classical theoretical data have huge disagreement.
This contradictoin was resolved by Max Planck who is by some called to be the father of the quantum physics. Planck hypothesized that black body radiation was produced by microscopic oscillations, called resonators. Plank argued that these resonators were allowed to have only certain discrete energies "En" given by the equation:
En = n*h*f
where n is a positive integer called the principal quantum number, f is the frequency of the vibration, and h is a constant , now termed Planck's constant in his honor, equal to 6.626e(--34)J*s.
since the energy of each superatomic resonator is given by equation above the energy is said to be "Quantized." Each energy state therefore represents a quantum state.
When planck presented his hypothesis many scientists (including Planck himself!) thought this theory was faulty. The quantized theory of atomic energy has stood the test of time and is an indespensable tool in explaining a number of other critical atomic processes unexplanable by any other theory.
_________________
2)THE PHOTOELECTRIC EFFECT AND THE PARTICLE THEORY OF LIGHT:
In the later part of the 19th century, experiments showed that when light is incident on certain metallic surfaces, electrons are emitted from the surfaces. This was known as the "Photoelectric effect." This first discovery was made by Heinrich Hertz, who was also the first to produce the electromagnetic waves predicated by Maxwell's equations.
Above is a simplified circuit of a photoelectric effect apparatus. As light hits the metal sheet at a high enough frequency it excites the metallic electrons and causes a flow of current in the circuit that is recorded or otherwise depicted by the ammeter. There is a battery connected to the circuit. Whenver the voltage of the selected battery is equal to or less than a value called "Vs"the current stops regardless of the light. This potential is called "The Stopping Potential." Below stopping potential regardless of how much the intensity of the light is, no electron jumps the gap.
Since at the stopping potential no electron can jump the voltage their Potential energy becomes Zero or PE = 0 and since had they been able to jump the gap their Potential energy would have been:
PE = q V = e ΔVs
all that energy is now converted to the kinetic energy. Therefore the KEmax of the photoelectrons is
KEmax = e ΔVs
Several of the features that can not be explained by classical physics:
_Classical theory says that at any frequency some electrons should be emitted which is not what really happens since after a cuttoff frequency (fc) no more electron is emitted.
_Classical theory says the higher the intensity of light the more the electron's kinetic when in fact, in reality there is a limit for the KEmax given by the equation above.
_Classical theory says that the there is no relationship between the KE of the electrons and the intensity of the light, when in fact in reality the higher the intensity the higher the KE up until it reaches a point where KEmax and goes no longer.
_Classical theory says that is takes time for electrons to reach the receiving end but in reality it happens so fast that it is almost unexplainable by classical concepts.
Albert Einstein was the man who solved this dilemma. Einstein expanded Planck's concept of quantization and resonators to electrons and their jump. He said that a tiny packet of light energy or "photon" would be emitted when a quantized oscillator made a jump from an energy state En=nhf to the next lower state, En-1 =(n-1)hf , Conservation of energy would require that the energy of the photon is:
E = En - E(n-1) = nhf - (n-1)hf = (n-n+1)hf = hf
hence he gave birth to the notion of photon having an energy of hf where h is planck's constant and f is the frequency of the electromagnetic wave that is incident on the electron.
When a photon of energy hf collides with an electron all of its energy is transferred to the photoelectron that now moves with a KEmax that is equal to the energy of the photon minus the binding energy of the electron to its neighbouring electron or in the lattic.
That means that:
photon + electron ----> photoelectron
hf + -ø -------> hf - ø
so in a sense the KEmax = hf - ø = photon energy - binding energy
ø is called the "work function" which represents the minimum energy with which the electron is bound to the metal and is on the order of few electron Volts.
KEmax and f have a linear relationship. We mentioned that below a certain cutoff frequency "fc" and fc relates to ø through the following relationship:
fc = ø / h
we also know that v = fc * λc
so we have:
λc = c / fc = c / (ø / h) = hc / ø
wavelenghts bigger than λc or smaller than fc do not give off any photoelectric effect.
Heisenberg's Uncertainty Principal:
Image property of http://www.aip.org/history/heisenberg/imag...g-new-small.gif
If you were to measure the position and speed of a particle at any instanct, you would always be faced with experimental uncertainties in your measurments. According to classical mechanics, no fundamental barrier to an ultimate refinement of the apparatus or experimental procedures exists. Werner Heisenber masterfully proved the false nature of this assumption!
Heisenber published a paper in which he explained that if the position of particle or Δx of an object is measured accurately, and a simultaneous measurment of its linear momentum Δpx is made, then the product of the two can NOT be smaller than h/4ë !
In other words:
h/4ë ≤Δpx * Δx
Another form of the same equation dealing with Energy in a finite time interval is given by:
h/4ë ≤ΔE * Δt
It can be inferred from this relationship that one can not measure Energy content and time during which this measurment is made with utmost certainty.
Wave Property of Objects and Broglie's equation:
Louis De Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties. According to Broglie, electrons just as light have a wave nature also. Using the relativity constant of
p = E /c which is the expression for the momentum of a photon Broglie argued that the expression can be changed into:
p = E /c = h /λ
in other words Broglie argued that a photon's wavelenght can be described through its momentum :
λ = h / p where h is the planck's constant and p is the momentum of the photon. He then argued that this applies to all particles including electrons!
He also used Einstein-Planck relationship E = hf to stay that:
λ = h / E and this was a good expression for the wave-matter nature of objects since momentum is mv a mass related property while Energy was a quantized concept.
Erwin Schrodinger's Wave Function:
Ideas of Broglie was the initiation upon which Schrodinger based his highly specialized theory of wave function that explained the wave nature of matter in space. Schrodinger's wave function are so critically important today that hydrogen atom's behavior at certain conditions is impossible without the "wave functions."
time-independent Schrodinger wave equation in one dimension is as follows:
where ø (x) is the is the wave function, m is mass, ž is Planck's constant divided by 2ë, E is the total energy of the particle, and U(x) is the potential energy function of the particle. The wave function is shown by the sign "ψ."
This is one of those ubber equations that only concerns the professional physicists and is out of the scope of simple physics and so we shall leave it be.
The Compton Effect on X-Rays:
Arthur Holly Compton directed an x-ray beam of wavelength λ1 toward a block of graphite. He found out that the scattered x-rays held a slightly longer wavelenght of λ. He proved that the change of wavelength called "the Compton shift" to be:
Δλ = λ - λ1 = h /me*c (1 - cosa)
where me is the mass of electron, and "a" is the angle between the directions of the scattered and incident photon. The quantity "h/me*c) is called the Compton wavelength and has a constant value of h /me*c = .00243nm.
Compton derived the equation above by combining the assumption of conservation of momentum for a two dimentional glancing collision along with relativistic conservation of momentum and energy.
FUN APPLICATION OF QUANTUM THEORY:
MILITARY:
A helicopter wants to drop its load onto a target according to Heisenberg's principal the missing distance is at least:
Δx = ( h /2ëm)^1/2 * (H/2g)^1/4
where, H is the initial height of the helicotper, m is the mass of each package, h is planck's constant and g is the gravitational pull. In star war the form of the equation above changes into:
Δx = ( h /2*ë*m)^1/2 * (H*d^2/2G*Mp)^1/4
where h is planck's constant, m is the mass of the projectile, Mp is the mass of the the closet planet and d is the distance from the centre of that planet and the spaceship.
SPORTS!
You can use De Broglie equation to predict the place of landing of a ping pong ball should you consider its wave nature! I wont get into it cuz the equation would just go crazy but ponit made simple, quantum physics is very applicable in everyday situations.
What led to creation of Quantum theory?
_Electromagnetic radiation emitted by a heated object (blackbody effect)
_The emission of electrons by illuminated metals (Photoelectric effect)
_The emission of sharp spectral lines by gas atoms in an electric discharge tube.
_Dual nature of light as both a particle and energy
_Aberrations observed under the assumptions of relativity, and classical physics.
Who were most influential in creation of Quantum Physics?
_Max Planck
_Werner Heisenberg
_Max Born
_Louis De Broglie
_Albert Einstein
_Erwin Schrodinger
_Wolfgang Pauli
_Niel Bohr
_Paul Dirac
_Arthur Holly Compton
_Max von Laue
_Wilhelm Roentgen
_and many more who contributed but did not get recognized.
QUANTUM PHENOMENA:
1)BLACK BODY RADIATION: An object at any temperature is known to emit electromagnetic radiation that is sometimes referred to as thermal radiation. The radiated energy varies at different Temperatues, and different Wavelength:
λmax *T = .2898e(-2) m.K
where λmax is the wavelength at which the intensity of blackbody radiation is most, and T is the temperature in Kelvin.
Attempts to use the classical ideas to explain this quality of radiation intensity versus wavelength failed. Problem was that at long wavelegnths the classical theory was in good agreement with the experimental data, but at short wavelengths however, there exists a very broad difference and contradiction from expected theoretical data and the experimental data. According to the theory the wavelength should approach infiniti as the wavelegnth approaches zero but it is in fact zero! This contradiction is called the Ultraviolet Catasrophe because of the spectrum in which the experimental and classical theoretical data have huge disagreement.
This contradictoin was resolved by Max Planck who is by some called to be the father of the quantum physics. Planck hypothesized that black body radiation was produced by microscopic oscillations, called resonators. Plank argued that these resonators were allowed to have only certain discrete energies "En" given by the equation:
En = n*h*f
where n is a positive integer called the principal quantum number, f is the frequency of the vibration, and h is a constant , now termed Planck's constant in his honor, equal to 6.626e(--34)J*s.
since the energy of each superatomic resonator is given by equation above the energy is said to be "Quantized." Each energy state therefore represents a quantum state.
When planck presented his hypothesis many scientists (including Planck himself!) thought this theory was faulty. The quantized theory of atomic energy has stood the test of time and is an indespensable tool in explaining a number of other critical atomic processes unexplanable by any other theory.
_________________
2)THE PHOTOELECTRIC EFFECT AND THE PARTICLE THEORY OF LIGHT:
In the later part of the 19th century, experiments showed that when light is incident on certain metallic surfaces, electrons are emitted from the surfaces. This was known as the "Photoelectric effect." This first discovery was made by Heinrich Hertz, who was also the first to produce the electromagnetic waves predicated by Maxwell's equations.
Above is a simplified circuit of a photoelectric effect apparatus. As light hits the metal sheet at a high enough frequency it excites the metallic electrons and causes a flow of current in the circuit that is recorded or otherwise depicted by the ammeter. There is a battery connected to the circuit. Whenver the voltage of the selected battery is equal to or less than a value called "Vs"the current stops regardless of the light. This potential is called "The Stopping Potential." Below stopping potential regardless of how much the intensity of the light is, no electron jumps the gap.
Since at the stopping potential no electron can jump the voltage their Potential energy becomes Zero or PE = 0 and since had they been able to jump the gap their Potential energy would have been:
PE = q V = e ΔVs
all that energy is now converted to the kinetic energy. Therefore the KEmax of the photoelectrons is
KEmax = e ΔVs
Several of the features that can not be explained by classical physics:
_Classical theory says that at any frequency some electrons should be emitted which is not what really happens since after a cuttoff frequency (fc) no more electron is emitted.
_Classical theory says the higher the intensity of light the more the electron's kinetic when in fact, in reality there is a limit for the KEmax given by the equation above.
_Classical theory says that the there is no relationship between the KE of the electrons and the intensity of the light, when in fact in reality the higher the intensity the higher the KE up until it reaches a point where KEmax and goes no longer.
_Classical theory says that is takes time for electrons to reach the receiving end but in reality it happens so fast that it is almost unexplainable by classical concepts.
Albert Einstein was the man who solved this dilemma. Einstein expanded Planck's concept of quantization and resonators to electrons and their jump. He said that a tiny packet of light energy or "photon" would be emitted when a quantized oscillator made a jump from an energy state En=nhf to the next lower state, En-1 =(n-1)hf , Conservation of energy would require that the energy of the photon is:
E = En - E(n-1) = nhf - (n-1)hf = (n-n+1)hf = hf
hence he gave birth to the notion of photon having an energy of hf where h is planck's constant and f is the frequency of the electromagnetic wave that is incident on the electron.
When a photon of energy hf collides with an electron all of its energy is transferred to the photoelectron that now moves with a KEmax that is equal to the energy of the photon minus the binding energy of the electron to its neighbouring electron or in the lattic.
That means that:
photon + electron ----> photoelectron
hf + -ø -------> hf - ø
so in a sense the KEmax = hf - ø = photon energy - binding energy
ø is called the "work function" which represents the minimum energy with which the electron is bound to the metal and is on the order of few electron Volts.
KEmax and f have a linear relationship. We mentioned that below a certain cutoff frequency "fc" and fc relates to ø through the following relationship:
fc = ø / h
we also know that v = fc * λc
so we have:
λc = c / fc = c / (ø / h) = hc / ø
wavelenghts bigger than λc or smaller than fc do not give off any photoelectric effect.
Heisenberg's Uncertainty Principal:
Image property of http://www.aip.org/history/heisenberg/imag...g-new-small.gif
If you were to measure the position and speed of a particle at any instanct, you would always be faced with experimental uncertainties in your measurments. According to classical mechanics, no fundamental barrier to an ultimate refinement of the apparatus or experimental procedures exists. Werner Heisenber masterfully proved the false nature of this assumption!
Heisenber published a paper in which he explained that if the position of particle or Δx of an object is measured accurately, and a simultaneous measurment of its linear momentum Δpx is made, then the product of the two can NOT be smaller than h/4ë !
In other words:
h/4ë ≤Δpx * Δx
Another form of the same equation dealing with Energy in a finite time interval is given by:
h/4ë ≤ΔE * Δt
It can be inferred from this relationship that one can not measure Energy content and time during which this measurment is made with utmost certainty.
Wave Property of Objects and Broglie's equation:
Louis De Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties. According to Broglie, electrons just as light have a wave nature also. Using the relativity constant of
p = E /c which is the expression for the momentum of a photon Broglie argued that the expression can be changed into:
p = E /c = h /λ
in other words Broglie argued that a photon's wavelenght can be described through its momentum :
λ = h / p where h is the planck's constant and p is the momentum of the photon. He then argued that this applies to all particles including electrons!
He also used Einstein-Planck relationship E = hf to stay that:
λ = h / E and this was a good expression for the wave-matter nature of objects since momentum is mv a mass related property while Energy was a quantized concept.
Erwin Schrodinger's Wave Function:
Ideas of Broglie was the initiation upon which Schrodinger based his highly specialized theory of wave function that explained the wave nature of matter in space. Schrodinger's wave function are so critically important today that hydrogen atom's behavior at certain conditions is impossible without the "wave functions."
time-independent Schrodinger wave equation in one dimension is as follows:
where ø (x) is the is the wave function, m is mass, ž is Planck's constant divided by 2ë, E is the total energy of the particle, and U(x) is the potential energy function of the particle. The wave function is shown by the sign "ψ."
This is one of those ubber equations that only concerns the professional physicists and is out of the scope of simple physics and so we shall leave it be.
The Compton Effect on X-Rays:
Arthur Holly Compton directed an x-ray beam of wavelength λ1 toward a block of graphite. He found out that the scattered x-rays held a slightly longer wavelenght of λ. He proved that the change of wavelength called "the Compton shift" to be:
Δλ = λ - λ1 = h /me*c (1 - cosa)
where me is the mass of electron, and "a" is the angle between the directions of the scattered and incident photon. The quantity "h/me*c) is called the Compton wavelength and has a constant value of h /me*c = .00243nm.
Compton derived the equation above by combining the assumption of conservation of momentum for a two dimentional glancing collision along with relativistic conservation of momentum and energy.
FUN APPLICATION OF QUANTUM THEORY:
MILITARY:
A helicopter wants to drop its load onto a target according to Heisenberg's principal the missing distance is at least:
Δx = ( h /2ëm)^1/2 * (H/2g)^1/4
where, H is the initial height of the helicotper, m is the mass of each package, h is planck's constant and g is the gravitational pull. In star war the form of the equation above changes into:
Δx = ( h /2*ë*m)^1/2 * (H*d^2/2G*Mp)^1/4
where h is planck's constant, m is the mass of the projectile, Mp is the mass of the the closet planet and d is the distance from the centre of that planet and the spaceship.
SPORTS!
You can use De Broglie equation to predict the place of landing of a ping pong ball should you consider its wave nature! I wont get into it cuz the equation would just go crazy but ponit made simple, quantum physics is very applicable in everyday situations.
The Bohr Theory of Hydrogen Atom:
This is an attache as requested by a private message. Since the topic is basically universal, I might as well put it here.
At the begining of the 20th century there was a great deal of confusion as to why the classical physics had not been able to explain or account for the observed characteristic atomic spectra. The question that had many wondering was, why did the atomic spectra contained line of light at only certain frequencies, and in return only absorved light of the same frequny (or wavelength). The dilemma was well understood. The earlier works by Max Planck had offered a great frame work for developing new perspective work into the field, but the person who eventually managed to marry classical notions, with the new Quantum outlook was none other than Niel Bohr.
Bohr postulated a primitive, yet precise working model for a simple atom (H) in the following diagram:
Borh postulated that atom are made up of a nucleus with electrons going around it in near perfect circle. He then made the following assumptions:
1)Using previously proven Plank's quantized concept of energy, Bohr claimed that when an electron moves from a far shell to a shell closer to the nucleus, it releases a "quantized" amount of energy which is the different of energy between the primary and the secondary shells:
E (lost) = | Ei - Ef |
this difference then leads to an electromagnetic radiation with a frequency of f and accodring to Max Planck's E = hf , he concluded that:
E (lost) = | Ei - Ef | = hf
2)Bohr then made a ridiculously out of blue assumption which interestingly was quite factual. He stated that reason why the atomic spectra line is segmented is because electrons are in two kinds of orbital: the stable orbital and the unstable orbitals. Stable orbitals do not allow the hydrogen electorns to emit any sort of energy and only unstable orbitals allow escape of energy. He then argued that the size of the allowed electron orbitals is determined by a relationship with the atom's electron's orbital angular momentum! He then state that the electrons in the stable orbitals must have an agular momentum of L , which is equal to n? , where ? = h/2ï , where h is planck's constant.
3)Bohr hypothesized that the orbit of an electron around the nucleus can be thought of as a classical case of rotational motion.
With that in mind lets get calculating:
we know the electron rotates around the nucleus so:
F centripetal = m * v^2/r
we also know that the angular momentum of electron is L = Iw = m*r^2*w and we also know that w *r = v , so we have
L = m * r * v
where m is the mass of electron, r is the radius of the orbit in which the electron is located in, and v is the velocity of its motion.
we then know that the Bohr condition (2) is that L = n?
so we have:
m * r * v =n?
we also know that it is a form of rotation motion and the centripetal force is the force of attraction between the electron and the proton in hydrogen so we have:
F (centripetal) = - k e^2/r^2 where e and -e are the charges of proton and electron respectively.
so
F (centripetal) = - k e^2/r^2 = m v^2/ r
so,
k e^2/r^2 = m v^2/ r
simplifying the above expression we get:
k e^2/2r = 1/2m v^2 = KE (electron)
Also the potential energy between the electron and the proton is:
PE = V.q=- k e^2 / r
Assuming that the atom is stable and not moving the mechanical energy of the atom is :
ME = KE + PE = 1/2 m v^2 + k e^2 / r
Using the value for 1/2mv^2 from the previous equation we get:
ME = k e^2/2r - k e^2 / r = (ke^2 - 2ke^2 )/2r = -ke^2/2r
this in short means:
ME = - k e^2 / r , where e is the charge of an electron, r is the radius of its orbit, and k is the coloumbic constant .
Hold this conclusion for now and lets go back to m * r * v =n? , so we have:
v = n?/mr
and so if we square both sides we have:
v^2 = (n^2 * ?^2)/(m^2 * r^2)
also using k e^2/2r = 1/2m v^2 , we get:
v^2 = k e^2/rm
combine the two expressions for v^2 and we have:
v^2 = k e^2/rm = (n^2 * ?^2)/(m^2 * r^2)
and this leads us to be able to solve for r to be:
r = (n^2 * ?^2) / (m * k * e^2)
The above is the Bohr's claim that atomic radii is quantaized!!
If we however replace the value found for Bohr's radius into the expression we found for ME (Mechanical energy) = - k e^2 / r , we get:
ME = - {(m * k^2 * e^4) / (2* ?^2)} {1/n^2}
Since this mechanical energy is quantized as well as the radius it represented it is possible to actually represent it with En as opposed to ME to refer to its value being specificaly dependent on the principal qunatum number:
so
En = - {(m * k^2 * e^4) / (2* ?^2)} {1/n^2}
But then Bohr also hypothesized based on the data from Max Planck's experimentation with quantized energy that if an electron goes from level i to level f in the atomic shells it would release an energy of :
E (lost) = | Ei - Ef | = hf
since now we can define what Ei and Ef are based on En (with n being the principal quantum number for each of the shells at question) the expression above becomes:
E (lost) = Ei - Ef = hf = [ - {(m * k^2 * e^4) / (2* ?^2)} {1/n(i)^2} ] + [{(m * k^2 * e^4) / (2* ?^2)} {1/n(f)^2}]
where n(i) and n(f) represent the quantum numbers of the initial and the final shell through which the electron moves.
That gives us:
f = E(lost) / h = { (m * k^2 *e^4) / (4ï*?^3) } { 1/n(f)^2 - 1/n(i)^2 }
and that can mean:
1/λ = { (m * k^2 *e^4) / (4ïc*?^3) } { 1/n(f)^2 - 1/n(i)^2 }
the value { (m * k^2 *e^4) / (4ïc*?^3) } is called the Rydberg Constant which experimentation proved to be accurate beyond doubt. Borh was now recognized as a well rounded theorteical physicist.
-ulrich D. (In response to Sofia's request)
This is an attache as requested by a private message. Since the topic is basically universal, I might as well put it here.
At the begining of the 20th century there was a great deal of confusion as to why the classical physics had not been able to explain or account for the observed characteristic atomic spectra. The question that had many wondering was, why did the atomic spectra contained line of light at only certain frequencies, and in return only absorved light of the same frequny (or wavelength). The dilemma was well understood. The earlier works by Max Planck had offered a great frame work for developing new perspective work into the field, but the person who eventually managed to marry classical notions, with the new Quantum outlook was none other than Niel Bohr.
Bohr postulated a primitive, yet precise working model for a simple atom (H) in the following diagram:
Borh postulated that atom are made up of a nucleus with electrons going around it in near perfect circle. He then made the following assumptions:
1)Using previously proven Plank's quantized concept of energy, Bohr claimed that when an electron moves from a far shell to a shell closer to the nucleus, it releases a "quantized" amount of energy which is the different of energy between the primary and the secondary shells:
E (lost) = | Ei - Ef |
this difference then leads to an electromagnetic radiation with a frequency of f and accodring to Max Planck's E = hf , he concluded that:
E (lost) = | Ei - Ef | = hf
2)Bohr then made a ridiculously out of blue assumption which interestingly was quite factual. He stated that reason why the atomic spectra line is segmented is because electrons are in two kinds of orbital: the stable orbital and the unstable orbitals. Stable orbitals do not allow the hydrogen electorns to emit any sort of energy and only unstable orbitals allow escape of energy. He then argued that the size of the allowed electron orbitals is determined by a relationship with the atom's electron's orbital angular momentum! He then state that the electrons in the stable orbitals must have an agular momentum of L , which is equal to n? , where ? = h/2ï , where h is planck's constant.
3)Bohr hypothesized that the orbit of an electron around the nucleus can be thought of as a classical case of rotational motion.
With that in mind lets get calculating:
we know the electron rotates around the nucleus so:
F centripetal = m * v^2/r
we also know that the angular momentum of electron is L = Iw = m*r^2*w and we also know that w *r = v , so we have
L = m * r * v
where m is the mass of electron, r is the radius of the orbit in which the electron is located in, and v is the velocity of its motion.
we then know that the Bohr condition (2) is that L = n?
so we have:
m * r * v =n?
we also know that it is a form of rotation motion and the centripetal force is the force of attraction between the electron and the proton in hydrogen so we have:
F (centripetal) = - k e^2/r^2 where e and -e are the charges of proton and electron respectively.
so
F (centripetal) = - k e^2/r^2 = m v^2/ r
so,
k e^2/r^2 = m v^2/ r
simplifying the above expression we get:
k e^2/2r = 1/2m v^2 = KE (electron)
Also the potential energy between the electron and the proton is:
PE = V.q=- k e^2 / r
Assuming that the atom is stable and not moving the mechanical energy of the atom is :
ME = KE + PE = 1/2 m v^2 + k e^2 / r
Using the value for 1/2mv^2 from the previous equation we get:
ME = k e^2/2r - k e^2 / r = (ke^2 - 2ke^2 )/2r = -ke^2/2r
this in short means:
ME = - k e^2 / r , where e is the charge of an electron, r is the radius of its orbit, and k is the coloumbic constant .
Hold this conclusion for now and lets go back to m * r * v =n? , so we have:
v = n?/mr
and so if we square both sides we have:
v^2 = (n^2 * ?^2)/(m^2 * r^2)
also using k e^2/2r = 1/2m v^2 , we get:
v^2 = k e^2/rm
combine the two expressions for v^2 and we have:
v^2 = k e^2/rm = (n^2 * ?^2)/(m^2 * r^2)
and this leads us to be able to solve for r to be:
r = (n^2 * ?^2) / (m * k * e^2)
The above is the Bohr's claim that atomic radii is quantaized!!
If we however replace the value found for Bohr's radius into the expression we found for ME (Mechanical energy) = - k e^2 / r , we get:
ME = - {(m * k^2 * e^4) / (2* ?^2)} {1/n^2}
Since this mechanical energy is quantized as well as the radius it represented it is possible to actually represent it with En as opposed to ME to refer to its value being specificaly dependent on the principal qunatum number:
so
En = - {(m * k^2 * e^4) / (2* ?^2)} {1/n^2}
But then Bohr also hypothesized based on the data from Max Planck's experimentation with quantized energy that if an electron goes from level i to level f in the atomic shells it would release an energy of :
E (lost) = | Ei - Ef | = hf
since now we can define what Ei and Ef are based on En (with n being the principal quantum number for each of the shells at question) the expression above becomes:
E (lost) = Ei - Ef = hf = [ - {(m * k^2 * e^4) / (2* ?^2)} {1/n(i)^2} ] + [{(m * k^2 * e^4) / (2* ?^2)} {1/n(f)^2}]
where n(i) and n(f) represent the quantum numbers of the initial and the final shell through which the electron moves.
That gives us:
f = E(lost) / h = { (m * k^2 *e^4) / (4ï*?^3) } { 1/n(f)^2 - 1/n(i)^2 }
and that can mean:
1/λ = { (m * k^2 *e^4) / (4ïc*?^3) } { 1/n(f)^2 - 1/n(i)^2 }
the value { (m * k^2 *e^4) / (4ïc*?^3) } is called the Rydberg Constant which experimentation proved to be accurate beyond doubt. Borh was now recognized as a well rounded theorteical physicist.
QUOTE
Rydberg constant = { (m * k^2 *e^4) / (4ïc*?^3) }
-ulrich D. (In response to Sofia's request)
This is great!!!
QUOTE
_Classical theory says that the there is no relationship between the KE of the electrons and the intensity of the light, when in fact in reality the higher the intensity the higher the KE up until it reaches a point where KEmax and goes no longer.
This is incorrect. You contradict yourself here with previous statements. Classical theory DOES say that there is a relationship between the kinetic energy of the electrons and the intensity of the light (the higher the intensity of light, the greater the KE of the ejected electrons). Quantum theory says that changing the intensity only alters the number of incident photons, thus only changes the current, not the stopping potential (hence KE) of the ejected electrons.
Rod.
Awesome thread Drude!!!!
On Magnets and their mathematics
The innovations of Andre Marie Ampere and Hans Christian Oersted
Until 1820, the only magnetism known was that of iron magnets and of "lodestones", natural magnets of iron-rich ore. It was believed that the inside of the Earth was magnetized in the same fashion, and scientists were greatly puzzled when they found that the direction of the compass needle at any place slowly shifted, decade by decade, suggesting a slow variation of the Earth's magnetic field.
Edmond Halley's Theories
How can an iron magnet produce such changes? Edmond Halley (of comet fame) ingeniously proposed that the Earth contained a number of spherical shells, one inside the other, each magnetized differently, each slowly rotating in relation to the others.
Hans Christian Oersted - Electromagnetism Experiments
Hans Christian Oersted was a professor of science at Copenhagen University. In 1820 he arranged in his home a science demonstration to friends and students. He planned to demonstrate the heating of a wire by an electric current, and also to carry out demonstrations of magnetism, for which he provided a compass needle mounted on a wooden stand.
While performing his electric demonstration, Hans Christian Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but in the months that followed worked hard trying to make sense out of the new phenomenon.
However, Hans Christian Oersted could not explain why. The needle was neither attracted to the wire nor repelled from it. Instead, it tended to stand at right angles. In the end he published his findings without any explanation.
Andre Marie Ampere and Electromagnetism
Andre Marie Ampere in France felt that if a current in a wire exerted a magnetic force on a compass needle, two such wires also should interact magnetically. In a series of ingenious experiments Andre Marie Ampere showed that this interaction was simple and fundamental - parallel (straight) currents attract, anti-parallel currents repel. The force between two long straight parallel currents was inversely proportional to the distance between them and proportional to the intensity of the current flowing in each.
There thus existed two kinds of forces associated with electricity--electric and magnetic. In 1864, James Clerk Maxwell demonstrated a subtle connection between the two types of force, unexpectedly involving the velocity of light. From this connection sprang the idea that light was an electric phenomenon, the discovery of radio waves, the theory of relativity and a great deal of present-day physics.
The earth is a magnet. The magnetic north pole is the direction in which all compasses point. The earth’s force field can be observed by the needle deflection of the compass. A compass is an example of a free-floating magnet.
If a small bar magnet is dipped into iron filings the filings are attracted to its ends. The magnetic force comes from two points called the poles of a magnet.
The Earth exerts a force forces on the poles of a magnet.If a bar magnet is suspended it swings around until it lies roughly north-south. This effect is used to name the two poles of a magnet. These are called the north and south poles of the magnets.
A Law which also is true in real life to a certain extent
Like poles repel and unlike poles attract
Induced magnetism: This is when magnetic materials are magnetized because of nearby magnetic materials.
Hard magnetic materials: Steel and alloys called alcomax and Magnadur are difficult to magnetize but do not readily lose their magnetism.They are used fro permanent magnets.
Soft magnetic materials such as iron and Mumetal are relatively easy to magnetize but their magnetism is only temporary. These are used in electromagnets.
Where magnetism comes from:
In an atom tiny electrical particles called electrons move around a central nucleus. Each electron has a magnetic effect as it spins and orbits the nucleus.In an unmagnetized material, the atomic magnets point in random directions. But as they are magnetized most of these atomic magnets move in line to form the magnetism that we observe. Hammering and heating throw these atomic magnets out of line and thus the material becomes demagnetized.
*Magnetic field lines run from the N-pole to the S-pole. The field direction shown usually by arrowheads is defined as the direction in which the force on a N-pole would act.
*The magnetic field is strongest where the field lines are closest together
* Earth's geographic North pole is about 1200+km from its true North pole
If an electric current is passed through a wire a weak magnetic field is produced.The field has the following features: * the magnetic field lines are circles * the field is strongest close to the wire * increasing the current increases the strength of the field.Right-hand grip rule:
The direction of the magnetic field produced by a current can be determined by gripping the wire with your right hand so that your thumb points in the same direction as the field lines.
*A long coil is a solenoid
An Electromagnet
An electromagnet starts with a battery (or some other source of power) and a wire. What a battery produces is electrons.
If you look at a battery, say at a normal D-cell from a flashlight, you can see that there are two ends, one marked plus (+) and the other marked minus (-). Electrons collect at the negative end of the battery, and, if you let them, they will flow to the positive end. The way you "let them" flow is with a wire. If you attach a wire directly between the positive and negative terminals of a D-cell, three things will happen:
Electrons will flow from the negative side of the battery to the positive side as fast as they can.
The battery will drain fairly quickly (in a matter of several minutes). For that reason, it is generally not a good idea to connect the two terminals of a battery to one another directly. Normally, you connect some kind of load in the middle of the wire so the electrons can do useful work. The load might be a motor, a light bulb, a radio or whatever.
A small magnetic field is generated in the wire. It is this small magnetic field that is the basis of an electromagnet.
Flemings Left hand rule:
When a wire carrying an electric current is moved in a magnetic field of a magnet the magnetic field induced by the wire reacts with the magnetic field of the magnet causing the wire to move outwards. Fleming's left hand rule helps you to predict the movement.
First finger - direction of magnetic field (N-S)
SeCond finger - direction of current (positive to negative)
ThuMb - movements of the wire
The mathematics aspect:
Magnetic quantities in the SI
coercivity Hc core factor Σl/A
effective area Ae effective length le
effective permeability μe flux linkage λ
induced voltage e inductance L inductance factor Al initial permeability μi
intensity of magnetization I magnetic field strength H
magnetic flux Φ magnetic flux density B
magnetic mass susceptibility χρ magnetic moment m magnetic polarization J magnetic susceptibility χ magnetization M magnetomotive force Fm
permeability μ permeability of vacuum μ0 relative permeability μr reluctance Rm
remnance Br
You may find it easier to obtain an intuitive grasp of the relationships between magnetic quantities by thinking in terms of 'magnetic circuits' with flux flowing round a core in a fashion analogous to current flowing round an electric circuit.
Analogy:
Magnetic quantity, Electric quantity
magnetomotive force , electromotive force (voltage)
magnetic field strength ,electric field strength
permeability ,conductivity
magnetic flux ,current
magnetic flux density , current density
reluctance , resistance
Inductance:
Inductance in the SI
Quantity name Inductance
Quantity symbol L
Unit name henry
Unit symbol H
Base units kg m2 s-2 A-2
Any length of wire has inductance. Inductance is a measure of a coil's ability to store energy in the form of a magnetic field. It is defined as the rate of change of flux with current -
L = N × d Φ / d I Equation TMO
If the core material's permeability is considered constant then the relation between flux and current is linear and so:
L = N × Φ / I Equation TMI
By Substitution of Equation TMM and Rowland's Law -
L = N2 / Rm Equation TMA
You can relate inductance directly to the energy represented by the surrounding magnetic field -
L = 2 W / I2 Equation TME
Where W is the field energy in joules.
In practice, where a high permeability core is used, inductance is usually determined from the Al value specified by the manufacturer for the core -
L = 10-9 Al × N2 Equation TMK
Core Factor
Core Factor in the SI
Quantity name core factor or
geometric core constant
Quantity symbol Σl/A
Unit name per metre
Unit symbols m-1
The idea of core factor is, apart from adding to the jargon :-( , to encapsulate in one figure the contribution to core reluctance made by the size and shape of the core. It is usually quoted in the data sheet but it is calculated as -
Σl/A = le / Ae m-1
Core factors are often specified in millimetres-1. You should then multiply by 1000 before using them in the formula for reluctance.
Effective Area
Effective Area in the SI
Quantity name effective Area
Quantity symbol Ae
Unit name square metre
Unit symbols m2
The 'effective area' of a core represents the cross sectional area of one of its limbs. Usually this corresponds closely to the physical dimensions of the core but because flux may not be distributed completely evenly the manufacturer will specify a value for Ae which reflects this.
The need for the core area arises when you want to relate the flux density in the core (limited by the material type) to the total flux it carries -
Ae = Φ / B the area could be determined approximately as the product of the core height and the difference between the major and minor radii -
Magnetic Field Strength
Magnetic Field Strength in the SI
Quantity name magnetic field strength
alias magnetic field intensity
alias the auxiliary field
alias magnetizing force
Quantity symbol H
Unit name ampere per metre
Unit symbols A m-1
Whenever current flows it is always accompanied by a magnetic field. Scientists talk of the field as being due to 'moving electric charges' - a reasonable description of electrons flowing along a wire.The strength, or intensity, of this field surrounding a straight wire is given by
H = I / (2 π r) Equation TML
where r, the distance from the wire, is small in comparison with the length of the wire. The situation for short wires is described by the Biot-Savart equation.
By the way, don't confuse the speed of the charges (such as electrons) with the speed of a signal travelling down the wire they are in. Think of the signal as being the boundary between those electrons that have started to move and those that have yet to get going. The boundary might move close to the speed of light (3x108 m s-1) whilst the electrons themselves drift (on average) something near to 0.1 mm s-1. The electrons would be outpaced by a snail - even if it wasn't in a hurry.
You may object that magnetic fields are also produced by permanent magnets (like compass needles, door catches and fridge note holders) where no current flow is evident. It turns out that even here it is electrons moving in orbit around nuclei or spinning on their own axis which are responsible for the magnetic field.
Magnetic field strength is analogous to electric field strength. Where an electric field is set up between two plates separated by a distance, d, and having an electromotive force, e, between them the electric field is given by -
ε = e / d V m-1 Equation TMG
Similarly, magnetic field strength is -
H = Fm / le Equation TMH
The analogy with electric field strength is mathematical and not physical. An electric field has a clearly defined physical meaning: simply the force exerted on a 'test charge' divided by the amount of charge. Magnetic field strength cannot be measured in the same way because there is no 'magnetic monopole' equivalent to a test charge.
Do not confuse magnetic field strength with flux density, B. This is closely related to field strength but depends also on the material within the field. The strict definition of H is
H = B / μ0 - M Sommerfeld Field Equation
This formula applies generally, even if the materials within the field have non-uniform permeability or a permanent magnetic moment. It is rarely used in coil design because it is usually possible to simplify the calculation by assuming that within the field the permeability can be regarded as uniform. With that assumption we say instead that
H = B / μ Equation TMU
Flux also emerges from a permanent magnet even when there are no wires about to impose a field.
A field strength of about 2000 A m-1 is about the limit for cores made from iron powder.
Electron energy:
gain in kinetic energy = eV
for an electron moving through a voltage = charge * voltage
Faraday's law of electromagnetic induction:
The EMF induced in a conductor is propotional to the rate at which magnetic field lines are cut by the conductor.
Following reverses direction of the induced EMF and current:
Moving the wire in the opposite direction
Turning the magnet round so that the field direction is reversed
Hope that helps...
The innovations of Andre Marie Ampere and Hans Christian Oersted
Until 1820, the only magnetism known was that of iron magnets and of "lodestones", natural magnets of iron-rich ore. It was believed that the inside of the Earth was magnetized in the same fashion, and scientists were greatly puzzled when they found that the direction of the compass needle at any place slowly shifted, decade by decade, suggesting a slow variation of the Earth's magnetic field.
Edmond Halley's Theories
How can an iron magnet produce such changes? Edmond Halley (of comet fame) ingeniously proposed that the Earth contained a number of spherical shells, one inside the other, each magnetized differently, each slowly rotating in relation to the others.
Hans Christian Oersted - Electromagnetism Experiments
Hans Christian Oersted was a professor of science at Copenhagen University. In 1820 he arranged in his home a science demonstration to friends and students. He planned to demonstrate the heating of a wire by an electric current, and also to carry out demonstrations of magnetism, for which he provided a compass needle mounted on a wooden stand.
While performing his electric demonstration, Hans Christian Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but in the months that followed worked hard trying to make sense out of the new phenomenon.
However, Hans Christian Oersted could not explain why. The needle was neither attracted to the wire nor repelled from it. Instead, it tended to stand at right angles. In the end he published his findings without any explanation.
Andre Marie Ampere and Electromagnetism
Andre Marie Ampere in France felt that if a current in a wire exerted a magnetic force on a compass needle, two such wires also should interact magnetically. In a series of ingenious experiments Andre Marie Ampere showed that this interaction was simple and fundamental - parallel (straight) currents attract, anti-parallel currents repel. The force between two long straight parallel currents was inversely proportional to the distance between them and proportional to the intensity of the current flowing in each.
There thus existed two kinds of forces associated with electricity--electric and magnetic. In 1864, James Clerk Maxwell demonstrated a subtle connection between the two types of force, unexpectedly involving the velocity of light. From this connection sprang the idea that light was an electric phenomenon, the discovery of radio waves, the theory of relativity and a great deal of present-day physics.
The earth is a magnet. The magnetic north pole is the direction in which all compasses point. The earth’s force field can be observed by the needle deflection of the compass. A compass is an example of a free-floating magnet.
If a small bar magnet is dipped into iron filings the filings are attracted to its ends. The magnetic force comes from two points called the poles of a magnet.
The Earth exerts a force forces on the poles of a magnet.If a bar magnet is suspended it swings around until it lies roughly north-south. This effect is used to name the two poles of a magnet. These are called the north and south poles of the magnets.
A Law which also is true in real life to a certain extent
Like poles repel and unlike poles attract
Induced magnetism: This is when magnetic materials are magnetized because of nearby magnetic materials.
Hard magnetic materials: Steel and alloys called alcomax and Magnadur are difficult to magnetize but do not readily lose their magnetism.They are used fro permanent magnets.
Soft magnetic materials such as iron and Mumetal are relatively easy to magnetize but their magnetism is only temporary. These are used in electromagnets.
Where magnetism comes from:
In an atom tiny electrical particles called electrons move around a central nucleus. Each electron has a magnetic effect as it spins and orbits the nucleus.In an unmagnetized material, the atomic magnets point in random directions. But as they are magnetized most of these atomic magnets move in line to form the magnetism that we observe. Hammering and heating throw these atomic magnets out of line and thus the material becomes demagnetized.
*Magnetic field lines run from the N-pole to the S-pole. The field direction shown usually by arrowheads is defined as the direction in which the force on a N-pole would act.
*The magnetic field is strongest where the field lines are closest together
* Earth's geographic North pole is about 1200+km from its true North pole
If an electric current is passed through a wire a weak magnetic field is produced.The field has the following features: * the magnetic field lines are circles * the field is strongest close to the wire * increasing the current increases the strength of the field.Right-hand grip rule:
The direction of the magnetic field produced by a current can be determined by gripping the wire with your right hand so that your thumb points in the same direction as the field lines.
*A long coil is a solenoid
An Electromagnet
An electromagnet starts with a battery (or some other source of power) and a wire. What a battery produces is electrons.
If you look at a battery, say at a normal D-cell from a flashlight, you can see that there are two ends, one marked plus (+) and the other marked minus (-). Electrons collect at the negative end of the battery, and, if you let them, they will flow to the positive end. The way you "let them" flow is with a wire. If you attach a wire directly between the positive and negative terminals of a D-cell, three things will happen:
Electrons will flow from the negative side of the battery to the positive side as fast as they can.
The battery will drain fairly quickly (in a matter of several minutes). For that reason, it is generally not a good idea to connect the two terminals of a battery to one another directly. Normally, you connect some kind of load in the middle of the wire so the electrons can do useful work. The load might be a motor, a light bulb, a radio or whatever.
A small magnetic field is generated in the wire. It is this small magnetic field that is the basis of an electromagnet.
Flemings Left hand rule:
When a wire carrying an electric current is moved in a magnetic field of a magnet the magnetic field induced by the wire reacts with the magnetic field of the magnet causing the wire to move outwards. Fleming's left hand rule helps you to predict the movement.
First finger - direction of magnetic field (N-S)
SeCond finger - direction of current (positive to negative)
ThuMb - movements of the wire
The mathematics aspect:
Magnetic quantities in the SI
coercivity Hc core factor Σl/A
effective area Ae effective length le
effective permeability μe flux linkage λ
induced voltage e inductance L inductance factor Al initial permeability μi
intensity of magnetization I magnetic field strength H
magnetic flux Φ magnetic flux density B
magnetic mass susceptibility χρ magnetic moment m magnetic polarization J magnetic susceptibility χ magnetization M magnetomotive force Fm
permeability μ permeability of vacuum μ0 relative permeability μr reluctance Rm
remnance Br
You may find it easier to obtain an intuitive grasp of the relationships between magnetic quantities by thinking in terms of 'magnetic circuits' with flux flowing round a core in a fashion analogous to current flowing round an electric circuit.
Analogy:
Magnetic quantity, Electric quantity
magnetomotive force , electromotive force (voltage)
magnetic field strength ,electric field strength
permeability ,conductivity
magnetic flux ,current
magnetic flux density , current density
reluctance , resistance
Inductance:
Inductance in the SI
Quantity name Inductance
Quantity symbol L
Unit name henry
Unit symbol H
Base units kg m2 s-2 A-2
Any length of wire has inductance. Inductance is a measure of a coil's ability to store energy in the form of a magnetic field. It is defined as the rate of change of flux with current -
L = N × d Φ / d I Equation TMO
If the core material's permeability is considered constant then the relation between flux and current is linear and so:
L = N × Φ / I Equation TMI
By Substitution of Equation TMM and Rowland's Law -
L = N2 / Rm Equation TMA
You can relate inductance directly to the energy represented by the surrounding magnetic field -
L = 2 W / I2 Equation TME
Where W is the field energy in joules.
In practice, where a high permeability core is used, inductance is usually determined from the Al value specified by the manufacturer for the core -
L = 10-9 Al × N2 Equation TMK
Core Factor
Core Factor in the SI
Quantity name core factor or
geometric core constant
Quantity symbol Σl/A
Unit name per metre
Unit symbols m-1
The idea of core factor is, apart from adding to the jargon :-( , to encapsulate in one figure the contribution to core reluctance made by the size and shape of the core. It is usually quoted in the data sheet but it is calculated as -
Σl/A = le / Ae m-1
Core factors are often specified in millimetres-1. You should then multiply by 1000 before using them in the formula for reluctance.
Effective Area
Effective Area in the SI
Quantity name effective Area
Quantity symbol Ae
Unit name square metre
Unit symbols m2
The 'effective area' of a core represents the cross sectional area of one of its limbs. Usually this corresponds closely to the physical dimensions of the core but because flux may not be distributed completely evenly the manufacturer will specify a value for Ae which reflects this.
The need for the core area arises when you want to relate the flux density in the core (limited by the material type) to the total flux it carries -
Ae = Φ / B the area could be determined approximately as the product of the core height and the difference between the major and minor radii -
Magnetic Field Strength
Magnetic Field Strength in the SI
Quantity name magnetic field strength
alias magnetic field intensity
alias the auxiliary field
alias magnetizing force
Quantity symbol H
Unit name ampere per metre
Unit symbols A m-1
Whenever current flows it is always accompanied by a magnetic field. Scientists talk of the field as being due to 'moving electric charges' - a reasonable description of electrons flowing along a wire.The strength, or intensity, of this field surrounding a straight wire is given by
H = I / (2 π r) Equation TML
where r, the distance from the wire, is small in comparison with the length of the wire. The situation for short wires is described by the Biot-Savart equation.
By the way, don't confuse the speed of the charges (such as electrons) with the speed of a signal travelling down the wire they are in. Think of the signal as being the boundary between those electrons that have started to move and those that have yet to get going. The boundary might move close to the speed of light (3x108 m s-1) whilst the electrons themselves drift (on average) something near to 0.1 mm s-1. The electrons would be outpaced by a snail - even if it wasn't in a hurry.
You may object that magnetic fields are also produced by permanent magnets (like compass needles, door catches and fridge note holders) where no current flow is evident. It turns out that even here it is electrons moving in orbit around nuclei or spinning on their own axis which are responsible for the magnetic field.
Magnetic field strength is analogous to electric field strength. Where an electric field is set up between two plates separated by a distance, d, and having an electromotive force, e, between them the electric field is given by -
ε = e / d V m-1 Equation TMG
Similarly, magnetic field strength is -
H = Fm / le Equation TMH
The analogy with electric field strength is mathematical and not physical. An electric field has a clearly defined physical meaning: simply the force exerted on a 'test charge' divided by the amount of charge. Magnetic field strength cannot be measured in the same way because there is no 'magnetic monopole' equivalent to a test charge.
Do not confuse magnetic field strength with flux density, B. This is closely related to field strength but depends also on the material within the field. The strict definition of H is
H = B / μ0 - M Sommerfeld Field Equation
This formula applies generally, even if the materials within the field have non-uniform permeability or a permanent magnetic moment. It is rarely used in coil design because it is usually possible to simplify the calculation by assuming that within the field the permeability can be regarded as uniform. With that assumption we say instead that
H = B / μ Equation TMU
Flux also emerges from a permanent magnet even when there are no wires about to impose a field.
A field strength of about 2000 A m-1 is about the limit for cores made from iron powder.
Electron energy:
gain in kinetic energy = eV
for an electron moving through a voltage = charge * voltage
Faraday's law of electromagnetic induction:
The EMF induced in a conductor is propotional to the rate at which magnetic field lines are cut by the conductor.
Following reverses direction of the induced EMF and current:
Moving the wire in the opposite direction
Turning the magnet round so that the field direction is reversed
Hope that helps...
Bravo Fysics! I owe you 2 more positive feedbacks for I really wanted to get into magnetism but was always lax about it.
I think it is not a bad idea to add this section also to this threat. Please note any errors and inform me to correct them.
Today I feel energetic so I have decided to completely and fully explain the pendulum motion for you in a non-deal system with air friction and loss of kinetic energy:
PENDULUM & HARMONIC MOTION
As Moseley explained in an IDEAL system it does not. But of course in your case there is air friction and so air friction would hinder the motion making the periods longer. Air resistance of coures is often F = kv^2 and k depends on size and shape and even mass of the pendulum bob.
Lets try to derive the pendulum period in a system that is non-ideal
the force affecting the system are the force of gravity mg on the pendulum bob, and of course the tension in the pendulum cable, and air friction. Now since the motion of the pendulum is centripetal the resistive force is tangent to the motion of the pendulum in the direction oppositie to its motion.
If you look at this picture you see that:
Harmonic motion is any motion in which a force is restoring the return to the initial position. F = kx is the magical formula which we wish to establish for a pendulum. We know by looking at the picture that the force the restores motion is the force of the gravity component of mgSinA. However in this special case the air resistance is impeding the motion of the ball. In this case the restoring force is the force of gravity component mgsinA minus the force of friction or F[r].
the net restoring force therefore is F = mgsinA - F[r]
obviously for the system to move F[r]<<mgSinA
since we know that any harmonic motion is F = kx
we have
mgsinA - F[r] = kx
but what is x? In the diagram above the semicircular curb is the distance that the ball has to travel and we call it x. The angle A is obviously connected to this length by x = l *A where A is in radian and l is the length of the massless thread that is holding the ball.
so we have:
mgSinA - F[r] = k (l*A)
so k = { mgSinA - F[r] } / l A
we also konw that in harmonic motion the period of the motion is very tiny so we can almost easily assume that SinA=TanA ~ A so we have
k = mgA - F[r] / l A
Now normally F[r] is zero in an ideal system so we have k = mg/l and when we put that into the harmonic motion equation we get
T = 2pi * square root (m/k) = 2 pi * square root ( l / g)
but in our case F[r] is not zero so we have
k = ( mgA - F[r] ) / l A
but lets use the wisdom of those scientists who thought about these things before us. We know that George Strokes in 1845 found that the magnitude of resistive force on a very small spherical object falling through fluid (including air or water or etc) is given by
F[r]= 6* pi* r * v *
where r is the radius of the sphere, v is the velocity of the motion of the ball, and is actually viscocity of the fluid in which the pendulum is working.
so now replacing in F[r] we have:
k = ( mgA - F[r] ) / l A and F[r]= 6* pi* r * v *
so
k = (mgA - 6* pi* r * v *)/ lA
Do remember that we are assuming that the pendulum bob is perfect sphere so we are actually dealing with a special case pendulum which is well very common composed of a very light thread connected to a spherical bob of a considerable mass and round.
We also konw that the motion of the pendulum bob is circular so we have v = wR
where w is omega or the angular velocity and r is the radius of the motion which in this case is actually l + r , since the mass of the ball can be assumed to be in the center of the sphere and so the distance from this point and the pivot is l (or the length of the wire holding the bob) plus the radius of the circle.
so we have:
v = w ( l + r ) so then K is
k = (mgA - 6* pi* r * w * (l+r) *)/ lA
Simplifying the above formula and we have:
k = ()mg - 6*pi*r*w*(l+r)*1/A*n)/ l [ instead of the pic "n" now represents viscocity]
in a very realistic case we can assume that l + r >> r or that we are making a super dense ball, or simply by assuming that that the centre of mass of a sphere is an infinitesimally tiny point in the centre of the sphere so we have:
l + r ~ r
then our K is
k = (mg/l) - 6*pi*r*w*1/A*n)
now that we have found our K lets see what is the formula for the motion of a pendulum:
T = 2 pi * square root ( m /k) IS the general formula for the period of any harmonic motion. so we have:
Today I feel energetic so I have decided to completely and fully explain the pendulum motion for you in a non-deal system with air friction and loss of kinetic energy:
PENDULUM & HARMONIC MOTION
As Moseley explained in an IDEAL system it does not. But of course in your case there is air friction and so air friction would hinder the motion making the periods longer. Air resistance of coures is often F = kv^2 and k depends on size and shape and even mass of the pendulum bob.
Lets try to derive the pendulum period in a system that is non-ideal
the force affecting the system are the force of gravity mg on the pendulum bob, and of course the tension in the pendulum cable, and air friction. Now since the motion of the pendulum is centripetal the resistive force is tangent to the motion of the pendulum in the direction oppositie to its motion.
If you look at this picture you see that:
Harmonic motion is any motion in which a force is restoring the return to the initial position. F = kx is the magical formula which we wish to establish for a pendulum. We know by looking at the picture that the force the restores motion is the force of the gravity component of mgSinA. However in this special case the air resistance is impeding the motion of the ball. In this case the restoring force is the force of gravity component mgsinA minus the force of friction or F[r].
the net restoring force therefore is F = mgsinA - F[r]
obviously for the system to move F[r]<<mgSinA
since we know that any harmonic motion is F = kx
we have
mgsinA - F[r] = kx
but what is x? In the diagram above the semicircular curb is the distance that the ball has to travel and we call it x. The angle A is obviously connected to this length by x = l *A where A is in radian and l is the length of the massless thread that is holding the ball.
so we have:
mgSinA - F[r] = k (l*A)
so k = { mgSinA - F[r] } / l A
we also konw that in harmonic motion the period of the motion is very tiny so we can almost easily assume that SinA=TanA ~ A so we have
k = mgA - F[r] / l A
Now normally F[r] is zero in an ideal system so we have k = mg/l and when we put that into the harmonic motion equation we get
T = 2pi * square root (m/k) = 2 pi * square root ( l / g)
but in our case F[r] is not zero so we have
k = ( mgA - F[r] ) / l A
but lets use the wisdom of those scientists who thought about these things before us. We know that George Strokes in 1845 found that the magnitude of resistive force on a very small spherical object falling through fluid (including air or water or etc) is given by
F[r]= 6* pi* r * v *
where r is the radius of the sphere, v is the velocity of the motion of the ball, and is actually viscocity of the fluid in which the pendulum is working.
so now replacing in F[r] we have:
k = ( mgA - F[r] ) / l A and F[r]= 6* pi* r * v *
so
k = (mgA - 6* pi* r * v *)/ lA
Do remember that we are assuming that the pendulum bob is perfect sphere so we are actually dealing with a special case pendulum which is well very common composed of a very light thread connected to a spherical bob of a considerable mass and round.
We also konw that the motion of the pendulum bob is circular so we have v = wR
where w is omega or the angular velocity and r is the radius of the motion which in this case is actually l + r , since the mass of the ball can be assumed to be in the center of the sphere and so the distance from this point and the pivot is l (or the length of the wire holding the bob) plus the radius of the circle.
so we have:
v = w ( l + r ) so then K is
k = (mgA - 6* pi* r * w * (l+r) *)/ lA
Simplifying the above formula and we have:
k = ()mg - 6*pi*r*w*(l+r)*1/A*n)/ l [ instead of the pic "n" now represents viscocity]
in a very realistic case we can assume that l + r >> r or that we are making a super dense ball, or simply by assuming that that the centre of mass of a sphere is an infinitesimally tiny point in the centre of the sphere so we have:
l + r ~ r
then our K is
k = (mg/l) - 6*pi*r*w*1/A*n)
now that we have found our K lets see what is the formula for the motion of a pendulum:
T = 2 pi * square root ( m /k) IS the general formula for the period of any harmonic motion. so we have:
T = 2 pi square root ( m / { (mg/l) - 6*pi*r*w*1/A*n} )
and this is the general formula for any actual pendulum period and as you can see mass seems to play a role.
______________________________
George Strokes
http://www.cord.edu/faculty/ulnessd/legacy...onja/stokes.htm
http://en.wikipedia.org/wiki/Stokes'_law
Tossbag.
I think it is not a bad idea to add this section also to this threat. Please note any errors and inform me to correct them.
QUOTE
I don't know quite what you are after but the mass of the pendulum bob does not affect the period of oscillation.
T = 2*pi*sqrt(l/g)
T = 2*pi*sqrt(l/g)
Today I feel energetic so I have decided to completely and fully explain the pendulum motion for you in a non-deal system with air friction and loss of kinetic energy:
PENDULUM & HARMONIC MOTION
As Moseley explained in an IDEAL system it does not. But of course in your case there is air friction and so air friction would hinder the motion making the periods longer. Air resistance of coures is often F = kv^2 and k depends on size and shape and even mass of the pendulum bob.
Lets try to derive the pendulum period in a system that is non-ideal
the force affecting the system are the force of gravity mg on the pendulum bob, and of course the tension in the pendulum cable, and air friction. Now since the motion of the pendulum is centripetal the resistive force is tangent to the motion of the pendulum in the direction oppositie to its motion.
If you look at this picture you see that:
Harmonic motion is any motion in which a force is restoring the return to the initial position. F = kx is the magical formula which we wish to establish for a pendulum. We know by looking at the picture that the force the restores motion is the force of the gravity component of mgSinA. However in this special case the air resistance is impeding the motion of the ball. In this case the restoring force is the force of gravity component mgsinA minus the force of friction or F[r].
the net restoring force therefore is F = mgsinA - F[r]
obviously for the system to move F[r]<<mgSinA
since we know that any harmonic motion is F = kx
we have
mgsinA - F[r] = kx
but what is x? In the diagram above the semicircular curb is the distance that the ball has to travel and we call it x. The angle A is obviously connected to this length by x = l *A where A is in radian and l is the length of the massless thread that is holding the ball.
so we have:
mgSinA - F[r] = k (l*A)
so k = { mgSinA - F[r] } / l A
we also konw that in harmonic motion the period of the motion is very tiny so we can almost easily assume that SinA=TanA ~ A so we have
k = mgA - F[r] / l A
Now normally F[r] is zero in an ideal system so we have k = mg/l and when we put that into the harmonic motion equation we get
T = 2pi * square root (m/k) = 2 pi * square root ( l / g)
but in our case F[r] is not zero so we have
k = ( mgA - F[r] ) / l A
but lets use the wisdom of those scientists who thought about these things before us. We know that George Strokes in 1845 found that the magnitude of resistive force on a very small spherical object falling through fluid (including air or water or etc) is given by
F[r]= 6* pi* r * v *
where r is the radius of the sphere, v is the velocity of the motion of the ball, and
is actually viscocity of the fluid in which the pendulum is working.so now replacing in F[r] we have:
k = ( mgA - F[r] ) / l A and F[r]= 6* pi* r * v *
so
k = (mgA - 6* pi* r * v *
)/ lADo remember that we are assuming that the pendulum bob is perfect sphere so we are actually dealing with a special case pendulum which is well very common composed of a very light thread connected to a spherical bob of a considerable mass and round.
We also konw that the motion of the pendulum bob is circular so we have v = wR
where w is omega or the angular velocity and r is the radius of the motion which in this case is actually l + r , since the mass of the ball can be assumed to be in the center of the sphere and so the distance from this point and the pivot is l (or the length of the wire holding the bob) plus the radius of the circle.
so we have:
v = w ( l + r ) so then K is
k = (mgA - 6* pi* r * w * (l+r) *
)/ lASimplifying the above formula and we have:
k = ()mg - 6*pi*r*w*(l+r)*1/A*n)/ l [ instead of the pic "n" now represents viscocity]
in a very realistic case we can assume that l + r >> r or that we are making a super dense ball, or simply by assuming that that the centre of mass of a sphere is an infinitesimally tiny point in the centre of the sphere so we have:
l + r ~ r
then our K is
k = (mg/l) - 6*pi*r*w*1/A*n)
now that we have found our K lets see what is the formula for the motion of a pendulum:
T = 2 pi * square root ( m /k) IS the general formula for the period of any harmonic motion. so we have:
QUOTE (->
| QUOTE |
| I don't know quite what you are after but the mass of the pendulum bob does not affect the period of oscillation. T = 2*pi*sqrt(l/g) |
Today I feel energetic so I have decided to completely and fully explain the pendulum motion for you in a non-deal system with air friction and loss of kinetic energy:
PENDULUM & HARMONIC MOTION
As Moseley explained in an IDEAL system it does not. But of course in your case there is air friction and so air friction would hinder the motion making the periods longer. Air resistance of coures is often F = kv^2 and k depends on size and shape and even mass of the pendulum bob.
Lets try to derive the pendulum period in a system that is non-ideal
the force affecting the system are the force of gravity mg on the pendulum bob, and of course the tension in the pendulum cable, and air friction. Now since the motion of the pendulum is centripetal the resistive force is tangent to the motion of the pendulum in the direction oppositie to its motion.
If you look at this picture you see that:
Harmonic motion is any motion in which a force is restoring the return to the initial position. F = kx is the magical formula which we wish to establish for a pendulum. We know by looking at the picture that the force the restores motion is the force of the gravity component of mgSinA. However in this special case the air resistance is impeding the motion of the ball. In this case the restoring force is the force of gravity component mgsinA minus the force of friction or F[r].
the net restoring force therefore is F = mgsinA - F[r]
obviously for the system to move F[r]<<mgSinA
since we know that any harmonic motion is F = kx
we have
mgsinA - F[r] = kx
but what is x? In the diagram above the semicircular curb is the distance that the ball has to travel and we call it x. The angle A is obviously connected to this length by x = l *A where A is in radian and l is the length of the massless thread that is holding the ball.
so we have:
mgSinA - F[r] = k (l*A)
so k = { mgSinA - F[r] } / l A
we also konw that in harmonic motion the period of the motion is very tiny so we can almost easily assume that SinA=TanA ~ A so we have
k = mgA - F[r] / l A
Now normally F[r] is zero in an ideal system so we have k = mg/l and when we put that into the harmonic motion equation we get
T = 2pi * square root (m/k) = 2 pi * square root ( l / g)
but in our case F[r] is not zero so we have
k = ( mgA - F[r] ) / l A
but lets use the wisdom of those scientists who thought about these things before us. We know that George Strokes in 1845 found that the magnitude of resistive force on a very small spherical object falling through fluid (including air or water or etc) is given by
F[r]= 6* pi* r * v *
where r is the radius of the sphere, v is the velocity of the motion of the ball, and
is actually viscocity of the fluid in which the pendulum is working.so now replacing in F[r] we have:
k = ( mgA - F[r] ) / l A and F[r]= 6* pi* r * v *
so
k = (mgA - 6* pi* r * v *
)/ lADo remember that we are assuming that the pendulum bob is perfect sphere so we are actually dealing with a special case pendulum which is well very common composed of a very light thread connected to a spherical bob of a considerable mass and round.
We also konw that the motion of the pendulum bob is circular so we have v = wR
where w is omega or the angular velocity and r is the radius of the motion which in this case is actually l + r , since the mass of the ball can be assumed to be in the center of the sphere and so the distance from this point and the pivot is l (or the length of the wire holding the bob) plus the radius of the circle.
so we have:
v = w ( l + r ) so then K is
k = (mgA - 6* pi* r * w * (l+r) *
)/ lASimplifying the above formula and we have:
k = ()mg - 6*pi*r*w*(l+r)*1/A*n)/ l [ instead of the pic "n" now represents viscocity]
in a very realistic case we can assume that l + r >> r or that we are making a super dense ball, or simply by assuming that that the centre of mass of a sphere is an infinitesimally tiny point in the centre of the sphere so we have:
l + r ~ r
then our K is
k = (mg/l) - 6*pi*r*w*1/A*n)
now that we have found our K lets see what is the formula for the motion of a pendulum:
T = 2 pi * square root ( m /k) IS the general formula for the period of any harmonic motion. so we have:
T = 2 pi square root ( m / { (mg/l) - 6*pi*r*w*1/A*n} )
and this is the general formula for any actual pendulum period and as you can see mass seems to play a role.
______________________________
George Strokes
http://www.cord.edu/faculty/ulnessd/legacy...onja/stokes.htm
http://en.wikipedia.org/wiki/Stokes'_law
NA
At the insistence of some friends here are some of the msot common organic equations. if you pursue a chemical degree these would be handy.
R-OH + PCl3 --> R-Cl + HOCl2
R-OH + SOCl2 ---> R-Cl
R-OH + HX -----> R-X
R-Cl + CN¯ ----> R-CN , R-CN + H3O (H2O, H) ----> RCOOH, RCOOH + PCl3 ------>RCOCl
RCOCl + NH3 -----> RCONH2 + NaOBr ----> RNH2
R-Cl + NH3 ------>RNH2
R-COOH + LiAlH4 ----> R-CH2OH
RCOR' + NaBH4 ------>RCOHR', RCHO + NaBH4 ----->R-CH2OH
R-Cl + Br¯ -----> RBr + Cl¯ , RI + Cl¯ -----> NR (in SN2 only the stronger nucleophile replaces the leaving group)
R-COOR' + LiAlH4 ------> RCH2OH + R'OH
R-Cl + Mg ----> RMgCl , RMgCl + H2O(or EtOH, or any weak acid) ----> RH
R-X + Li + Cu ----> organic complex . Organic complex + R'-X(primary halide) ---> R-R'
R-X + Zn/H2SO4 ------> R-H
CH2==CHCH3 + H2/Pt or Pd or Ni ---------> CH3CH2CH3
CH2==CHCH3 + HCl(dry) --------> extensive polymerization ( for i.e. 4-methyl-2-pentene, 4,6-dimethyl-2-heptene and etc...)
CH2==CHCH3 + HCl(wet; H2O) -------> CH3CHOHCH3
R-H + X2(Cl2, I2, Br2) -------> R-X
E1, E2 reactions
RCHXCH3 + KOH(alcoholic) ------> RC==CH2 (anti-elimination and prefers tertiary to secondary and secondary to primary hydrogens)
RCHXCH2X + Zn/Pt --------------> RC==CH2
RCHOHCH3 + H2SO4/heat ---------> RC==CH2 (Be cautious of carbocation movement through 1,2 hydride or R shift. Once you figure out the most stable cation then look for the most substituted combination to create a double bond).
R―C≡CR' + H2/Pt -----> Cis- RCH==CHR'
R―C≡CR' + Na/NH3 -----------> Trans - RCH==CHR'
RCH=CH2 + X2 -----> RCHXCH2X
RCH=CH2 + HX -----> (Markovnicov addition of hydrogen) RCHXCH3
RCH=CH2 + aqueous acid (H2SO4, H2O) ------> RCHOHCH3
RCH=CH2 + H2SO4 ------> RCH(OSO3)CH3 (A very rare reaction. It often does not happen)
RCH=CH2 + B2H6(or unstable BH3) + H2O2 + OH¯ ------> (Anti-Markovnicov) RCH2CH2OH
RCH=CH2 + cold. diluted KMnO4 --------> (Important reaction; syn addition of glycol) RCHOHCH2OH
RCH=CH2 + Cl2 ----> Halonium ring; Halonium ring + H2O(Br¯ ) ----> RCHOH(or Br)CH3
RCH=CH2 + RCO3H + aqueous acid(H2SO4, H2O) -----> RCHOHCH2OH (anti addition of alcohol)
RCH=CH2 + O3 + Zn + H2O ----> RCH=O + O=CH2
RCH=CH2 + CHCl3 + strong base (ie. NaOH) ----> RCH2CH2CCl3
RCH=CH2 + HBr + peroxide ----------> RCH2CH2Br (Peroxide forces an antiMarkovnicov addition)
RCH=CH2 + HBr (no peroxide) --------> RCHBrCH3
R―CH2―CH=CH2 + Cl2 + Δ (stands for heat) [or uv light] -----> R―CHCl―CH=CH2 ↔ RCH=CH―CH2Cl
CH―CH2―CHCl + AgNO3 -----> CH―CH2―CH(+) (Minor product) ↔ (Major) CH―CH(+)―CH3
C6H5CH3 + uv or high heat + Cl2(Br2) ------> C6H5CH2―Cl or Br
C6H5CH3 + hot KMnO4(OH¯) -----> C6H5―COO¯ + weak acid ----> C6H5COOH
RC≡CH + HX ----> RCX=CH2 , RCX=CH2 + HX ----> RCX2CH3
RC≡CH + NaNH2 ( or just NH2¯) ----> RC≡C¯ , RC≡C¯ + R'X (Primary) -----> RC≡CR' + X¯
RC≡CH + Ag ------> RC≡CAg (precipitates)
RC≡CH (think of it as a weak acid) + R'MgX (very strong based) -------> RC≡CMgX + R'H
RC≡CMgX + R'COR'' ------> RC≡C―CR'OH―R''
CH2=CHCH=CH2 + HI -----> 1-Iodo and 3-Iodo (Consult your teacher about allylic additions.)
CH=CHCH=CH2 + CH2=CH2 ------> (Diel-Alder reaction) Cyclohexene
RMgX + RCOR'(H) ------> R―COH®―R'
RMgX + RCOOR' -----> R―COH®―R + R'OMgX
R-CH2―OH + KMnO4 (Na2Cr2O7) ------> R-COOH (upon treatment with permanganate or chromate a primary alcohol is converted into a carboyxlic acid)
R-CHR'―OH + KMnO4 (Na2Cr2O7)-------> R-CO-R' (upon treatment with permanganate or chromate a secondary alchol turns into a ketone)
R-CR'(R'')―OH + KMnO4 (Na2Cr2O7) ------> NR (tertiary alcohols can not be oxidized)
R-CH2―OH + PCC (Pyridine.HCl.CrO3)-----> R-CH=O (the only reaction that converts a primary alcohol into an aldehyde with considerable yield)
HC≡CCH3 + Hg(H2SO4) ------> H3CCOCH3 (Markovnicov)
R-CO-R' + Et-OH(ethanol) + H2SO4 ↔ R-COH(EtO)R' (Hemiacetal) + EtOH --->R-C(EtO)2-R' (Acetal)
R―CO―R' + Zn/H(+) (or Hg in H2SO4) -------------> R―CH2―R' (very useful Clemensen reduction)
R―CO―R + N2H4/OH¯ --------> R―CH2―R' (Wolff-Kishner reduction similar to Clemensen but only in Base)
HOOC(CH2)nCOOH + BaO + heat ---------> .....(CH2)―CO―(CH2)―(CH2)―..... (Cyclic ketone)
R-CH=O + PCl5 ----------> R-CHCl2
R-CH=O + H2/Ni -----> R-CH-OH
R-CH==HC-R' + H2/Ni -----> R-CH2CH2R'
R-CH=O + Ag(NH3)2(+) [Tollen's reagent]----------> R-COOH + Ag(mirro) [a test for aldehydes as well as a way to synthesize carboxylic acids]
R-CH2-COCl + LiAlH(t-BuO)3 --------> R-CH2-CH=O (replacement of Cl with H)
R-CH2―CH=O + CN¯ -------->R-CH2―CHOH(CN)
R-CH2―CH=O + H2NNHAr -------> R-CH2―CH=NNHAr
Ar―CH=O + OH¯ + H2O -----> Ar―COO¯ + Ar―CH2OH (Cannizzaro rection; certainly important)
Some variations of Cannizzaro reaction:
Ar―CD=O + OH¯ + H2O -----> Ar―COO¯ + Ar―CD2OH (D is deutorium and not Hydrogen)
Ar―CH=O + OD¯ + D2O-----> Ar―COO¯ + Ar―CH2OD
Ar―CH=O + OH¯ + HCOH (formaldehyde) ----- > (Consult your instructor on this one; you will better understand it if you have to ask it rather than juts memorize it.)
―CO―CO― + HIO4 -----> ―COOH + HOOC―
―CO―CHOH― + HIO4 ----> ―COOH + OHC―
―CHOH―CHOH― + HIO4 ----> ―CHO + OHC―
Ar―H + R―COCl + AlCl4( or FeCl3, or BF3, or any lewis.brÖnsted acid) -----> Ar―CO―R , For an extensive explanation for electrophilic reactions please click here.
Iodoform Test: (mechanism steps):
Overall view of the reaction: ―CO―CH3 + I2/OH¯ (NaOI¯) ------> ―COO¯ + HCI3
Mechanism of iodoform test: Step1. ―CO―CH3 + OH¯ ------> ―(C―O¯)==CH2 + H2O
Step2. ―(C―O¯)==CH2 + I―I ------> ―CO―CI3 (first it is one Iodide that replaces one hydrogen producing ―CH2I, then upon second treatment with I2 you get ―CHI2, and eventually you end up with ―CI3, as it is in shown in step 2)
Step3. ―CO―CI3 + OH¯ -----> ―(C―O¯)OH―CI3 (Unstable tetrahedral complex formed as electrorich OH attackes the carbonyl group and leads to migration the carbonyl pi electron to the oxygen. At this point we have a carbon with OH , CI3 and a negative O attached to it; this complex is a highly energetic complex and therefore is not favored at all and soon resumes its carbonyl characteristic in step 4, by leaving CI3 as a leaving group, and forming a carboxylic acid)
Step4. ―(C―O¯)OH―CI3 --------> ―COOH + ¯CI3
Step 5. ―COOH + ¯CI3 --(acid-base rxn)--> ―COO¯ + HCI3
Once you have turned the initial ―CO―CH3 (called a methyl ketone) into a ―COO (carboxylic acid), you can just add a little water and weak acid to add a hydrogen to ―COO and make it a ―COOH.
CH2=CH―CH3 + NBS (N-bromoSuccinimide) ----> CH2=CH―CH2―Br + CH2=CH―CH2―Br (C is isotopically labeled. As you can see although NBS adds a Bromide to the allylic carbon, resonance alternates the double bond with the halide, so if you have two terminal carbons that are not equivalent for example in this case we isotopically labeled them] we will have two products).
―C=C― + Br2/CCl4 -----> (anti addition of halogen to a double bond) ―CHBr―CHBr―
―CHBr―CHBr― + OH¯ ----(E2 rxn)--> ―CH==CBr― + NH2¯ ------>―C≡C― (Note: if you have a double bond and you wish to transform it into a triple bond just treat it with Br2/CCl4 to get the reagent above and then do the reaction above and you got yourself a triple bond!)
―CHI― + HOAc (Acetic acid) ----> ―CH(+)― (whenever you see an acetic acid and a halide and no other reaction comes into your mind, then you are most probably dealing with a carbocation formation and should first try to locate the most stable carbocation by the use of 1,2shif and then probably develop a double bond using Saytzeff rule.)
One way to produce a specific ketone:↓
RCH2―OH + Na2Cr2O7 -------> R―COOH , RCOOH + SOCl2 ------> RCOCl
R'X + Li + CuX -------> R'2CuLi
RCOCl + R'2CuLi ----> R―CO―R'
The other way to produce the same thing:↓
R―CHO + R'MgX --------> R―CHOH―R'
R―CHOH―R' + Na2Cr2O7 -----> R―CO―R'
Yet another way to produce the same thing:↓
RMgX + R'―CHO -------> R―CHOH―R'
R―CHOH―R' + K2Cr2O7 -----> R―CO―R'
Carbene Chemistry reaction:
CH2N2 + uv --------> CH2(singlet) , CH2(singlet) + Inert gasses ----> CH2(tripplet)
Cis-2butene + CH2(singlet) -------> (ONLY one product) cis-1,2,dimethylcyclopropane
Cis-2butene + CH2(tripplet) ------->(3 products) cis-1,2,dimethylcyclopropane + Trans-1,2dimethylcyclopropane + enantiomer of Trans-1,2dimethylcyclopropane
HCCl3 + K(tBuO) -----> :CCl3¯ , :CCl3¯ ----> CCl2 + Cl¯
:CCl2 (singlet) + CH2=CH2 ---> 1,1,dichlorocyclopropane
CH2=CH2CH3 + O2 -----> CH2=CH2COOH
RCH2OH + SF4 ----> RCH2F
RCH=CH2 + Hg(HOAc)2 + H2O(or ROH) + NaBH4 -----> RCOH(H)―CH3 or RCOR(H)―CH3
R―OH + Na -----> R―O¯ Na(+) (Alkoxide sodium salt; you could also use Li)
CH3―CH2―Br + HS¯ -----> CH3―CH2―SH, CH3―CH2―SH + KMnO4 ---> CH3―CH2―SO3H
Reactions with Ethoxide or other potent alkoxide: Whenever you see EtO¯ or other strong basic alkoxide(like tertiary butyl oxide) you should always and I mean always, remember that there is either a substitution of SN2 type or there is an elimination of E2 type. It is often hard however to realize which one occurs even if one know which one occurs how can he/she find the answer? If these are the questions you have then do not worry. Here is a rule(or a collection of rules for lack of a better word) that explains it.
1-If the starting material has a halogen, find the carbon on which the halogen is located. Once you found that carbon, then determine if it is primary, secondary or tertiary. For primary carbons like the example below the major (and only) reaction is substitution of X with EtO. The product is the same as reagent but the X(halide) is replaced by EtO and since this is a SN2 reaction, there is also inversion of the structure. Inversion of structure makes sense only when we have a chiral carbon otherwise, we don't have to care at all. So:
R―C(D)H―X + EtO¯ ----> EtO―C(H)(D)―R (as you can see the structure is inverted since R has moved from left to write)
2-If the starting material has a halogen, and the carbon that hold that halogen is secondary then two reactions occur with almost the same speed and therefore we get two products: One product is the result of SN2 replacement which gives us an inverted chiral carbon and halide is replaced by EtO; the other product is result of an E2 elimination since EtO¯ is also a strong base as well a good nucleophile. This means that the product of elimination is a double bond formed between the carbon holding the halogen and any adjacent carbon that has priority according to saytzeff rule.So:
R―C(R')H―X + EtO¯ ---------> EtO―C(R')H―R + R==C(R')H (it could also make a double bond with R'; you see you should know the saytzeff rule that says if you have an option between two hydrogen for elimination choose the tertiary hydrogen over the secondary and secondary over the primary; I however strongly recommend that you go to your instructor and see his explanation and realize it).
3-If the starting material has a halogen, and the carbon that holds the halogen is a tertiary then no SN2 substitution can occur and the major (and the only) reaction is E2 elimination.
R―C(R')R''―X + EtO¯ -------> R==C(R')R'' (double bond could form with either R, R' or R'' depending on what saytzeff rule dictates)
Ether Chemistry:
Crown Ether + Inorganic salts (or ions) -------> Crown-salt complex (important application in drug and petroleum industry)
2ROH + H2SO4 -----> ROR
ROH + Na ----> RO¯ , RO¯ + R'X (a primary halide) ------> ROR' (Williamson ether synthesis)
ROR' + HI -------> RI + R'I + some alcohols
Any epoxide + H2O(ROH) ----> RO―C(..)―C(..)―OH (Note: whenever you are dealing with an epoxide that is protonated you attack the carbon which is more substituted but when you are in a basic solution always attack the carbon that is less substituted).
R2C=O + Mg + HCl ------> R2C(OH)―C(OH)R2
R―CH2―COOH + NaOH(CaO, heat) -----> R―CH3
R―CH2―COOH + Ag + Br2 ----> RCH2Br (Hundsdiecker reaction)
R―CH2MgX + CO2 + H2SO4 ----> R―CH2―COOH
RCN + H2/Ni -----> R―CH2―NH2
R―CH2OH + KMnO4 ----> R―COOH
Ar―R + KMnO4 + OH¯ + Δ-----> Ar―COO¯
Ar―COO¯ + dil. acid ------> Ar―COOH
Ar―R + Na2Cr2O7 + H(+) ------> Ar―COOH
Ar―CH3 + Cl2 + uv ----------> Ar―CH2―Cl , Ar―CH2―Cl + Mg + CO2 -----> Ar―CH2―COO¯
Ar―CH2―COO¯ + dil. acid -------> Ar―CH2―COOH
R―C≡N + H2 ----> R―CH2―NH2
R―C≡N + H2O + acid/base -----> R―COOH , Ar―C≡N + H2O + acid/base -----> Ar―COOH
Ar―CH2―X + CN¯ -----> Ar―CH2―CN , Ar―CH2―CN + aq. Base (in reflux) -----> ArCH2COOH
Ar―COOH + Base¯ ------> Ar―COO¯ + Base:H (A simple acid-base reaction)
Ar―COOH + PCl3/PCl5/SOCl2 ------> Ar―COCl
Ar―H + HNO3 + H2SO4 ----> Ar―NO2 (Nitration)
Ar―H + H2SO4 + Heat or cold -----> Ar―SO3H
Ar―H + FeCl3 + Cl2 -------> Ar―Cl
Ar―H + FeBr3 + Br2 -------> Ar―Br
Ar―H + AlCl3 + R―X -----> Ar―R (Rearrangement occurs)
Ar―H + AlCl3(or FeCl3) + RCOCl -----> Ar―OCR
Ar―H + R―OH + BF3 ------> Ar―R
Ar'―CH2Cl + NH3 -----> Ar'―CH2―NH2
Ar―H + Ag + HIO3/HNO3 -----> Ar―I (do not use this to make iodoarenes; You can instead use the diazonium method).
Ar―CH=CH2 + KMnO4 (cold) -----> Ar―CH(OH)―CH2―OH
Ar―CH=CH2 + KMnO4 (hot) ------> ArCOO¯---dil acid----> ArCOOH
Ar―CH=CH2 + Cl2/CCl4 --------> Ar―CH(Cl)―CH2Cl
Ar―CH3 + HNO3 + H2SO4 ------> NO2―Ar―CH3 (ortho or para)
Ar―CH3 + H2SO4 --------> SO3H―Ar―CH3 (ortho or para)
Ar―CH3 + FeCl3 + Cl2 ------> Cl―Ar―CH3 (ortho or para)
Ar―CO―CH3 + I2/OH¯ -------> Ar―COO¯
R―CH2OH + KMnO4 -----> R―COOH
Cyclohexane + HNO3 ------>HOOCCH2CH2CH2CH2COOH (Adipic acid)
BrCH2CH2I + CN¯ -------> CN―CH2CH2―CN ; CN―CH2CH2―CN + H3O (aq. acid) ----> HOOCCH2CH2COOH
CH3―C(CH3)2―I (tertiary iodide) + CN¯ ------> CH3―C(CH3)==CH2 (E2) Note: Do not use Cyanide to creat carboxylic acids in tertiary positions. Elimination competes with nucleophilic addition.
Ar―COOH + LiAlH4 ----> Ar―CH2―OH
R―CH2COOH + X2 + P -------> R―CHXCOOH (HVZ reaction)
R―CH2COOH + NaOH + CaO + Heat or hot radiation --------> R―CH3
R―CH2COOH + Ag(+) + Br2 -----> R―CH2Br , R―CH2Br + Mg + H2O ----> R―CH3
R―CH2COOH + NH3 ------> R―CH2CO―NH2 + OBr¯ (or Br2/NaOH) ----> RCH2NH2 (Hoffman degradation)
R―CH2COOH + MnO + heat from a reactor or industrial heat -----> RCH2―CO―CH2R
R―CH2COOH + PCl3 ----> R―CH2COCl
R―CH2COR' + Zn/H(+) -----> R―CH2CH2R'
CH3―COOH + Cl2/P -----> Cl―CH2COOH, Cl―CH2COOH + NaOH ------> Cl―CH2COO¯Na(+), Cl―CH2COO¯Na(+) + CN¯ ----> CN―CH2COO¯Na(+), CN―CH2COO¯Na(+) + H2O/strong acid -----> HOOC―CH2―COOH (NaOH is used to avoid acid-base reaction with Cyanide).
CH2―COOH + CH3―NH2 ------> CH3―COO¯......(+) NH3―CH3 (Acid-base reaction)
R―COOH + R'OH + H2SO4 -------> R―COOR' (Fisher Acid catalyzed esterification)
R―COOH + NaOH ------> RCOO¯Na(+) ; RCOO¯Na(+) + R'I (primary) -----> R―COOR'
RCOOR' + aq. H2SO4 ------> RCOOH + R'OH (Acid catalyzed hydrolysis)
R―COCl + R'OH + Pyridine --------> R―COOR' (Acid chlorides are the best candidate for ester production because they are great leaving groups and do not often require a catalyst; use of pyridine however facilitates the reactions and increase the yield to an acceptable extend).
__________________________________________________________________
1)Esters
Preparation:
Carboxylic acids -----> esters
RCOOH + R'OH + H2SO4 + heat -------> RCOOR' (Fisher Acid-catalyzed esterification)
RCOOH + NaOH ----> RCOONa ; RCOONa + R'I (Primary) ----> RCOOR'
Acid chlorides -------> esters
RCOCl + R'OH + Pyridine or traces of base -----> RCOOR'
RCOCl + AlH(t-OBu)3 -----> RCHO ; RCHO + KMnO4 + base ----> RCOOH ; RCOOH + R'OH + H2SO4 + heat -----> RCOOR'
Acid Anhydrides --------> Esters
RCOOOCR + R'OH -------> RCOOR'
Reactions:
-Hydrolysis and Saponification:
CH3COOCH3 + H3O(+) -----> CH3COOH + CH3OH
CH3COOCH3 + NaOH + heat -----> CH3COO¯Na(+) + CH3OH
-Reduction:
RCOOR' + LiAlH4 ----->RCH2―OH + R'OH
RCOOR' + R'MgX (from R-X + Mg) ----> R―COH(R')―R' + R'OH (It is a tertiary alcohol since the R'MgX, replaces the OR' group).
-Amide formation: (could be also utilized in a Hoffman degredation)
RCOOR' + NH3 ---->RCONH2 ; RCONH2 + OBr¯ ------> RNH2
2)Acid Chlorides
Preparation:
CH3COOH + PCl3 (or SOCl2, P(OH)3, POCl3, or PCl5) ------> CH3COCl
Reactions:
CH3COCl + H2O -----> CH3COOH
CH3COCl + ROH -----> CHRCOOR
CH3COCl + NH3 -----> CH3CONH2
CH3COCl + LiAlH4 + water -----> CH3CH2―OH
CH3COCl + AlCl3 (FeCl3, or any lewis acid) + Ar―H -----> Ar―COCH3
CH3COCl + H2/Pd + BaSO4 + S (I am not entirely sure of this one...I saw this in a chemistry magazine...please consult your instructor about this) ------> CH3CHO (an aldehyde)
3)Amides
Preparation:
CH3COOH + NH3 -----> CH3COONH4 (An acid-base reaction); CH3COONH4 + Industrial heat (do not use this to make amides; remember in the synthesis questions you are asked to prepare stuff in laboratory conditions and temperatures ranging in industrial diagram are totally off your chart) -----> CH3CONH2
CH3COCl + NH3 -------> CH3CONH2 (good way to make amides)
(CH3CO)2O + NH3 -------> CH3CONH2
CH3COOCH3 + NH3 -----> CH3CONH2 (The other product is omitted for simplicity)
Reactions:
CH3CONH2 + OH¯ ------>CH3COO¯ + NH3
CH3CONH2 + OBr¯ (or I2/OH¯) ------> CH3NH2 (Famous Hoffman Degradation reaction)
CH3CONH2 + P4H10 ------> CH3CN
4)Acid Anhydrides
Preparation:
CH3COO¯ + RCOCl ------> RCOOOCCH3
Reactions:
(CH3CO)2O + H2O -----> 2CH3COOH
(CH3CO)2O + ROH -----> CH3COOH + CH3COOR
(CH3CO)2O + NH3 ---->CH3CONH3 + CH3COONH4
R-OH + PCl3 --> R-Cl + HOCl2
R-OH + SOCl2 ---> R-Cl
R-OH + HX -----> R-X
R-Cl + CN¯ ----> R-CN , R-CN + H3O (H2O, H) ----> RCOOH, RCOOH + PCl3 ------>RCOCl
RCOCl + NH3 -----> RCONH2 + NaOBr ----> RNH2
R-Cl + NH3 ------>RNH2
R-COOH + LiAlH4 ----> R-CH2OH
RCOR' + NaBH4 ------>RCOHR', RCHO + NaBH4 ----->R-CH2OH
R-Cl + Br¯ -----> RBr + Cl¯ , RI + Cl¯ -----> NR (in SN2 only the stronger nucleophile replaces the leaving group)
R-COOR' + LiAlH4 ------> RCH2OH + R'OH
R-Cl + Mg ----> RMgCl , RMgCl + H2O(or EtOH, or any weak acid) ----> RH
R-X + Li + Cu ----> organic complex . Organic complex + R'-X(primary halide) ---> R-R'
R-X + Zn/H2SO4 ------> R-H
CH2==CHCH3 + H2/Pt or Pd or Ni ---------> CH3CH2CH3
CH2==CHCH3 + HCl(dry) --------> extensive polymerization ( for i.e. 4-methyl-2-pentene, 4,6-dimethyl-2-heptene and etc...)
CH2==CHCH3 + HCl(wet; H2O) -------> CH3CHOHCH3
R-H + X2(Cl2, I2, Br2) -------> R-X
E1, E2 reactions
RCHXCH3 + KOH(alcoholic) ------> RC==CH2 (anti-elimination and prefers tertiary to secondary and secondary to primary hydrogens)
RCHXCH2X + Zn/Pt --------------> RC==CH2
RCHOHCH3 + H2SO4/heat ---------> RC==CH2 (Be cautious of carbocation movement through 1,2 hydride or R shift. Once you figure out the most stable cation then look for the most substituted combination to create a double bond).
R―C≡CR' + H2/Pt -----> Cis- RCH==CHR'
R―C≡CR' + Na/NH3 -----------> Trans - RCH==CHR'
RCH=CH2 + X2 -----> RCHXCH2X
RCH=CH2 + HX -----> (Markovnicov addition of hydrogen) RCHXCH3
RCH=CH2 + aqueous acid (H2SO4, H2O) ------> RCHOHCH3
RCH=CH2 + H2SO4 ------> RCH(OSO3)CH3 (A very rare reaction. It often does not happen)
RCH=CH2 + B2H6(or unstable BH3) + H2O2 + OH¯ ------> (Anti-Markovnicov) RCH2CH2OH
RCH=CH2 + cold. diluted KMnO4 --------> (Important reaction; syn addition of glycol) RCHOHCH2OH
RCH=CH2 + Cl2 ----> Halonium ring; Halonium ring + H2O(Br¯ ) ----> RCHOH(or Br)CH3
RCH=CH2 + RCO3H + aqueous acid(H2SO4, H2O) -----> RCHOHCH2OH (anti addition of alcohol)
RCH=CH2 + O3 + Zn + H2O ----> RCH=O + O=CH2
RCH=CH2 + CHCl3 + strong base (ie. NaOH) ----> RCH2CH2CCl3
RCH=CH2 + HBr + peroxide ----------> RCH2CH2Br (Peroxide forces an antiMarkovnicov addition)
RCH=CH2 + HBr (no peroxide) --------> RCHBrCH3
R―CH2―CH=CH2 + Cl2 + Δ (stands for heat) [or uv light] -----> R―CHCl―CH=CH2 ↔ RCH=CH―CH2Cl
CH―CH2―CHCl + AgNO3 -----> CH―CH2―CH(+) (Minor product) ↔ (Major) CH―CH(+)―CH3
C6H5CH3 + uv or high heat + Cl2(Br2) ------> C6H5CH2―Cl or Br
C6H5CH3 + hot KMnO4(OH¯) -----> C6H5―COO¯ + weak acid ----> C6H5COOH
RC≡CH + HX ----> RCX=CH2 , RCX=CH2 + HX ----> RCX2CH3
RC≡CH + NaNH2 ( or just NH2¯) ----> RC≡C¯ , RC≡C¯ + R'X (Primary) -----> RC≡CR' + X¯
RC≡CH + Ag ------> RC≡CAg (precipitates)
RC≡CH (think of it as a weak acid) + R'MgX (very strong based) -------> RC≡CMgX + R'H
RC≡CMgX + R'COR'' ------> RC≡C―CR'OH―R''
CH2=CHCH=CH2 + HI -----> 1-Iodo and 3-Iodo (Consult your teacher about allylic additions.)
CH=CHCH=CH2 + CH2=CH2 ------> (Diel-Alder reaction) Cyclohexene
RMgX + RCOR'(H) ------> R―COH®―R'
RMgX + RCOOR' -----> R―COH®―R + R'OMgX
R-CH2―OH + KMnO4 (Na2Cr2O7) ------> R-COOH (upon treatment with permanganate or chromate a primary alcohol is converted into a carboyxlic acid)
R-CHR'―OH + KMnO4 (Na2Cr2O7)-------> R-CO-R' (upon treatment with permanganate or chromate a secondary alchol turns into a ketone)
R-CR'(R'')―OH + KMnO4 (Na2Cr2O7) ------> NR (tertiary alcohols can not be oxidized)
R-CH2―OH + PCC (Pyridine.HCl.CrO3)-----> R-CH=O (the only reaction that converts a primary alcohol into an aldehyde with considerable yield)
HC≡CCH3 + Hg(H2SO4) ------> H3CCOCH3 (Markovnicov)
R-CO-R' + Et-OH(ethanol) + H2SO4 ↔ R-COH(EtO)R' (Hemiacetal) + EtOH --->R-C(EtO)2-R' (Acetal)
R―CO―R' + Zn/H(+) (or Hg in H2SO4) -------------> R―CH2―R' (very useful Clemensen reduction)
R―CO―R + N2H4/OH¯ --------> R―CH2―R' (Wolff-Kishner reduction similar to Clemensen but only in Base)
HOOC(CH2)nCOOH + BaO + heat ---------> .....(CH2)―CO―(CH2)―(CH2)―..... (Cyclic ketone)
R-CH=O + PCl5 ----------> R-CHCl2
R-CH=O + H2/Ni -----> R-CH-OH
R-CH==HC-R' + H2/Ni -----> R-CH2CH2R'
R-CH=O + Ag(NH3)2(+) [Tollen's reagent]----------> R-COOH + Ag(mirro) [a test for aldehydes as well as a way to synthesize carboxylic acids]
R-CH2-COCl + LiAlH(t-BuO)3 --------> R-CH2-CH=O (replacement of Cl with H)
R-CH2―CH=O + CN¯ -------->R-CH2―CHOH(CN)
R-CH2―CH=O + H2NNHAr -------> R-CH2―CH=NNHAr
Ar―CH=O + OH¯ + H2O -----> Ar―COO¯ + Ar―CH2OH (Cannizzaro rection; certainly important)
Some variations of Cannizzaro reaction:
Ar―CD=O + OH¯ + H2O -----> Ar―COO¯ + Ar―CD2OH (D is deutorium and not Hydrogen)
Ar―CH=O + OD¯ + D2O-----> Ar―COO¯ + Ar―CH2OD
Ar―CH=O + OH¯ + HCOH (formaldehyde) ----- > (Consult your instructor on this one; you will better understand it if you have to ask it rather than juts memorize it.)
―CO―CO― + HIO4 -----> ―COOH + HOOC―
―CO―CHOH― + HIO4 ----> ―COOH + OHC―
―CHOH―CHOH― + HIO4 ----> ―CHO + OHC―
Ar―H + R―COCl + AlCl4( or FeCl3, or BF3, or any lewis.brÖnsted acid) -----> Ar―CO―R , For an extensive explanation for electrophilic reactions please click here.
Iodoform Test: (mechanism steps):
Overall view of the reaction: ―CO―CH3 + I2/OH¯ (NaOI¯) ------> ―COO¯ + HCI3
Mechanism of iodoform test: Step1. ―CO―CH3 + OH¯ ------> ―(C―O¯)==CH2 + H2O
Step2. ―(C―O¯)==CH2 + I―I ------> ―CO―CI3 (first it is one Iodide that replaces one hydrogen producing ―CH2I, then upon second treatment with I2 you get ―CHI2, and eventually you end up with ―CI3, as it is in shown in step 2)
Step3. ―CO―CI3 + OH¯ -----> ―(C―O¯)OH―CI3 (Unstable tetrahedral complex formed as electrorich OH attackes the carbonyl group and leads to migration the carbonyl pi electron to the oxygen. At this point we have a carbon with OH , CI3 and a negative O attached to it; this complex is a highly energetic complex and therefore is not favored at all and soon resumes its carbonyl characteristic in step 4, by leaving CI3 as a leaving group, and forming a carboxylic acid)
Step4. ―(C―O¯)OH―CI3 --------> ―COOH + ¯CI3
Step 5. ―COOH + ¯CI3 --(acid-base rxn)--> ―COO¯ + HCI3
Once you have turned the initial ―CO―CH3 (called a methyl ketone) into a ―COO (carboxylic acid), you can just add a little water and weak acid to add a hydrogen to ―COO and make it a ―COOH.
CH2=CH―CH3 + NBS (N-bromoSuccinimide) ----> CH2=CH―CH2―Br + CH2=CH―CH2―Br (C is isotopically labeled. As you can see although NBS adds a Bromide to the allylic carbon, resonance alternates the double bond with the halide, so if you have two terminal carbons that are not equivalent for example in this case we isotopically labeled them] we will have two products).
―C=C― + Br2/CCl4 -----> (anti addition of halogen to a double bond) ―CHBr―CHBr―
―CHBr―CHBr― + OH¯ ----(E2 rxn)--> ―CH==CBr― + NH2¯ ------>―C≡C― (Note: if you have a double bond and you wish to transform it into a triple bond just treat it with Br2/CCl4 to get the reagent above and then do the reaction above and you got yourself a triple bond!)
―CHI― + HOAc (Acetic acid) ----> ―CH(+)― (whenever you see an acetic acid and a halide and no other reaction comes into your mind, then you are most probably dealing with a carbocation formation and should first try to locate the most stable carbocation by the use of 1,2shif and then probably develop a double bond using Saytzeff rule.)
One way to produce a specific ketone:↓
RCH2―OH + Na2Cr2O7 -------> R―COOH , RCOOH + SOCl2 ------> RCOCl
R'X + Li + CuX -------> R'2CuLi
RCOCl + R'2CuLi ----> R―CO―R'
The other way to produce the same thing:↓
R―CHO + R'MgX --------> R―CHOH―R'
R―CHOH―R' + Na2Cr2O7 -----> R―CO―R'
Yet another way to produce the same thing:↓
RMgX + R'―CHO -------> R―CHOH―R'
R―CHOH―R' + K2Cr2O7 -----> R―CO―R'
Carbene Chemistry reaction:
CH2N2 + uv --------> CH2(singlet) , CH2(singlet) + Inert gasses ----> CH2(tripplet)
Cis-2butene + CH2(singlet) -------> (ONLY one product) cis-1,2,dimethylcyclopropane
Cis-2butene + CH2(tripplet) ------->(3 products) cis-1,2,dimethylcyclopropane + Trans-1,2dimethylcyclopropane + enantiomer of Trans-1,2dimethylcyclopropane
HCCl3 + K(tBuO) -----> :CCl3¯ , :CCl3¯ ----> CCl2 + Cl¯
:CCl2 (singlet) + CH2=CH2 ---> 1,1,dichlorocyclopropane
CH2=CH2CH3 + O2 -----> CH2=CH2COOH
RCH2OH + SF4 ----> RCH2F
RCH=CH2 + Hg(HOAc)2 + H2O(or ROH) + NaBH4 -----> RCOH(H)―CH3 or RCOR(H)―CH3
R―OH + Na -----> R―O¯ Na(+) (Alkoxide sodium salt; you could also use Li)
CH3―CH2―Br + HS¯ -----> CH3―CH2―SH, CH3―CH2―SH + KMnO4 ---> CH3―CH2―SO3H
Reactions with Ethoxide or other potent alkoxide: Whenever you see EtO¯ or other strong basic alkoxide(like tertiary butyl oxide) you should always and I mean always, remember that there is either a substitution of SN2 type or there is an elimination of E2 type. It is often hard however to realize which one occurs even if one know which one occurs how can he/she find the answer? If these are the questions you have then do not worry. Here is a rule(or a collection of rules for lack of a better word) that explains it.
1-If the starting material has a halogen, find the carbon on which the halogen is located. Once you found that carbon, then determine if it is primary, secondary or tertiary. For primary carbons like the example below the major (and only) reaction is substitution of X with EtO. The product is the same as reagent but the X(halide) is replaced by EtO and since this is a SN2 reaction, there is also inversion of the structure. Inversion of structure makes sense only when we have a chiral carbon otherwise, we don't have to care at all. So:
R―C(D)H―X + EtO¯ ----> EtO―C(H)(D)―R (as you can see the structure is inverted since R has moved from left to write)
2-If the starting material has a halogen, and the carbon that hold that halogen is secondary then two reactions occur with almost the same speed and therefore we get two products: One product is the result of SN2 replacement which gives us an inverted chiral carbon and halide is replaced by EtO; the other product is result of an E2 elimination since EtO¯ is also a strong base as well a good nucleophile. This means that the product of elimination is a double bond formed between the carbon holding the halogen and any adjacent carbon that has priority according to saytzeff rule.So:
R―C(R')H―X + EtO¯ ---------> EtO―C(R')H―R + R==C(R')H (it could also make a double bond with R'; you see you should know the saytzeff rule that says if you have an option between two hydrogen for elimination choose the tertiary hydrogen over the secondary and secondary over the primary; I however strongly recommend that you go to your instructor and see his explanation and realize it).
3-If the starting material has a halogen, and the carbon that holds the halogen is a tertiary then no SN2 substitution can occur and the major (and the only) reaction is E2 elimination.
R―C(R')R''―X + EtO¯ -------> R==C(R')R'' (double bond could form with either R, R' or R'' depending on what saytzeff rule dictates)
Ether Chemistry:
Crown Ether + Inorganic salts (or ions) -------> Crown-salt complex (important application in drug and petroleum industry)
2ROH + H2SO4 -----> ROR
ROH + Na ----> RO¯ , RO¯ + R'X (a primary halide) ------> ROR' (Williamson ether synthesis)
ROR' + HI -------> RI + R'I + some alcohols
Any epoxide + H2O(ROH) ----> RO―C(..)―C(..)―OH (Note: whenever you are dealing with an epoxide that is protonated you attack the carbon which is more substituted but when you are in a basic solution always attack the carbon that is less substituted).
R2C=O + Mg + HCl ------> R2C(OH)―C(OH)R2
R―CH2―COOH + NaOH(CaO, heat) -----> R―CH3
R―CH2―COOH + Ag + Br2 ----> RCH2Br (Hundsdiecker reaction)
R―CH2MgX + CO2 + H2SO4 ----> R―CH2―COOH
RCN + H2/Ni -----> R―CH2―NH2
R―CH2OH + KMnO4 ----> R―COOH
Ar―R + KMnO4 + OH¯ + Δ-----> Ar―COO¯
Ar―COO¯ + dil. acid ------> Ar―COOH
Ar―R + Na2Cr2O7 + H(+) ------> Ar―COOH
Ar―CH3 + Cl2 + uv ----------> Ar―CH2―Cl , Ar―CH2―Cl + Mg + CO2 -----> Ar―CH2―COO¯
Ar―CH2―COO¯ + dil. acid -------> Ar―CH2―COOH
R―C≡N + H2 ----> R―CH2―NH2
R―C≡N + H2O + acid/base -----> R―COOH , Ar―C≡N + H2O + acid/base -----> Ar―COOH
Ar―CH2―X + CN¯ -----> Ar―CH2―CN , Ar―CH2―CN + aq. Base (in reflux) -----> ArCH2COOH
Ar―COOH + Base¯ ------> Ar―COO¯ + Base:H (A simple acid-base reaction)
Ar―COOH + PCl3/PCl5/SOCl2 ------> Ar―COCl
Ar―H + HNO3 + H2SO4 ----> Ar―NO2 (Nitration)
Ar―H + H2SO4 + Heat or cold -----> Ar―SO3H
Ar―H + FeCl3 + Cl2 -------> Ar―Cl
Ar―H + FeBr3 + Br2 -------> Ar―Br
Ar―H + AlCl3 + R―X -----> Ar―R (Rearrangement occurs)
Ar―H + AlCl3(or FeCl3) + RCOCl -----> Ar―OCR
Ar―H + R―OH + BF3 ------> Ar―R
Ar'―CH2Cl + NH3 -----> Ar'―CH2―NH2
Ar―H + Ag + HIO3/HNO3 -----> Ar―I (do not use this to make iodoarenes; You can instead use the diazonium method).
Ar―CH=CH2 + KMnO4 (cold) -----> Ar―CH(OH)―CH2―OH
Ar―CH=CH2 + KMnO4 (hot) ------> ArCOO¯---dil acid----> ArCOOH
Ar―CH=CH2 + Cl2/CCl4 --------> Ar―CH(Cl)―CH2Cl
Ar―CH3 + HNO3 + H2SO4 ------> NO2―Ar―CH3 (ortho or para)
Ar―CH3 + H2SO4 --------> SO3H―Ar―CH3 (ortho or para)
Ar―CH3 + FeCl3 + Cl2 ------> Cl―Ar―CH3 (ortho or para)
Ar―CO―CH3 + I2/OH¯ -------> Ar―COO¯
R―CH2OH + KMnO4 -----> R―COOH
Cyclohexane + HNO3 ------>HOOCCH2CH2CH2CH2COOH (Adipic acid)
BrCH2CH2I + CN¯ -------> CN―CH2CH2―CN ; CN―CH2CH2―CN + H3O (aq. acid) ----> HOOCCH2CH2COOH
CH3―C(CH3)2―I (tertiary iodide) + CN¯ ------> CH3―C(CH3)==CH2 (E2) Note: Do not use Cyanide to creat carboxylic acids in tertiary positions. Elimination competes with nucleophilic addition.
Ar―COOH + LiAlH4 ----> Ar―CH2―OH
R―CH2COOH + X2 + P -------> R―CHXCOOH (HVZ reaction)
R―CH2COOH + NaOH + CaO + Heat or hot radiation --------> R―CH3
R―CH2COOH + Ag(+) + Br2 -----> R―CH2Br , R―CH2Br + Mg + H2O ----> R―CH3
R―CH2COOH + NH3 ------> R―CH2CO―NH2 + OBr¯ (or Br2/NaOH) ----> RCH2NH2 (Hoffman degradation)
R―CH2COOH + MnO + heat from a reactor or industrial heat -----> RCH2―CO―CH2R
R―CH2COOH + PCl3 ----> R―CH2COCl
R―CH2COR' + Zn/H(+) -----> R―CH2CH2R'
CH3―COOH + Cl2/P -----> Cl―CH2COOH, Cl―CH2COOH + NaOH ------> Cl―CH2COO¯Na(+), Cl―CH2COO¯Na(+) + CN¯ ----> CN―CH2COO¯Na(+), CN―CH2COO¯Na(+) + H2O/strong acid -----> HOOC―CH2―COOH (NaOH is used to avoid acid-base reaction with Cyanide).
CH2―COOH + CH3―NH2 ------> CH3―COO¯......(+) NH3―CH3 (Acid-base reaction)
R―COOH + R'OH + H2SO4 -------> R―COOR' (Fisher Acid catalyzed esterification)
R―COOH + NaOH ------> RCOO¯Na(+) ; RCOO¯Na(+) + R'I (primary) -----> R―COOR'
RCOOR' + aq. H2SO4 ------> RCOOH + R'OH (Acid catalyzed hydrolysis)
R―COCl + R'OH + Pyridine --------> R―COOR' (Acid chlorides are the best candidate for ester production because they are great leaving groups and do not often require a catalyst; use of pyridine however facilitates the reactions and increase the yield to an acceptable extend).
__________________________________________________________________
1)Esters
Preparation:
Carboxylic acids -----> esters
RCOOH + R'OH + H2SO4 + heat -------> RCOOR' (Fisher Acid-catalyzed esterification)
RCOOH + NaOH ----> RCOONa ; RCOONa + R'I (Primary) ----> RCOOR'
Acid chlorides -------> esters
RCOCl + R'OH + Pyridine or traces of base -----> RCOOR'
RCOCl + AlH(t-OBu)3 -----> RCHO ; RCHO + KMnO4 + base ----> RCOOH ; RCOOH + R'OH + H2SO4 + heat -----> RCOOR'
Acid Anhydrides --------> Esters
RCOOOCR + R'OH -------> RCOOR'
Reactions:
-Hydrolysis and Saponification:
CH3COOCH3 + H3O(+) -----> CH3COOH + CH3OH
CH3COOCH3 + NaOH + heat -----> CH3COO¯Na(+) + CH3OH
-Reduction:
RCOOR' + LiAlH4 ----->RCH2―OH + R'OH
RCOOR' + R'MgX (from R-X + Mg) ----> R―COH(R')―R' + R'OH (It is a tertiary alcohol since the R'MgX, replaces the OR' group).
-Amide formation: (could be also utilized in a Hoffman degredation)
RCOOR' + NH3 ---->RCONH2 ; RCONH2 + OBr¯ ------> RNH2
2)Acid Chlorides
Preparation:
CH3COOH + PCl3 (or SOCl2, P(OH)3, POCl3, or PCl5) ------> CH3COCl
Reactions:
CH3COCl + H2O -----> CH3COOH
CH3COCl + ROH -----> CHRCOOR
CH3COCl + NH3 -----> CH3CONH2
CH3COCl + LiAlH4 + water -----> CH3CH2―OH
CH3COCl + AlCl3 (FeCl3, or any lewis acid) + Ar―H -----> Ar―COCH3
CH3COCl + H2/Pd + BaSO4 + S (I am not entirely sure of this one...I saw this in a chemistry magazine...please consult your instructor about this) ------> CH3CHO (an aldehyde)
3)Amides
Preparation:
CH3COOH + NH3 -----> CH3COONH4 (An acid-base reaction); CH3COONH4 + Industrial heat (do not use this to make amides; remember in the synthesis questions you are asked to prepare stuff in laboratory conditions and temperatures ranging in industrial diagram are totally off your chart) -----> CH3CONH2
CH3COCl + NH3 -------> CH3CONH2 (good way to make amides)
(CH3CO)2O + NH3 -------> CH3CONH2
CH3COOCH3 + NH3 -----> CH3CONH2 (The other product is omitted for simplicity)
Reactions:
CH3CONH2 + OH¯ ------>CH3COO¯ + NH3
CH3CONH2 + OBr¯ (or I2/OH¯) ------> CH3NH2 (Famous Hoffman Degradation reaction)
CH3CONH2 + P4H10 ------> CH3CN
4)Acid Anhydrides
Preparation:
CH3COO¯ + RCOCl ------> RCOOOCCH3
Reactions:
(CH3CO)2O + H2O -----> 2CH3COOH
(CH3CO)2O + ROH -----> CH3COOH + CH3COOR
(CH3CO)2O + NH3 ---->CH3CONH3 + CH3COONH4
This is not a post for you to post about your trucks!!!
Leave this topic and start a seperate in the off topic. This thread has been pinned to contain only contribution to the educational aspect of the physics.
Drude
My god! What happend to physorg! exotic flies? mafia?
Where is the sense of scientific curiosity, and professional capacity? Those posts have absolutely no scientific value and I am appauled that the old sceintific echelon in charge of this form has scooped so low as to give non-scientific title advocating a malice, socially perverted group such as "mafia." This is not a children's website.
Leave this topic and start a seperate in the off topic. This thread has been pinned to contain only contribution to the educational aspect of the physics.
Drude
My god! What happend to physorg! exotic flies? mafia?
Where is the sense of scientific curiosity, and professional capacity? Those posts have absolutely no scientific value and I am appauled that the old sceintific echelon in charge of this form has scooped so low as to give non-scientific title advocating a malice, socially perverted group such as "mafia." This is not a children's website.
QUOTE (Drude+Aug 14 2007, 02:39 AM)
socially perverted group such as "mafia."
I'm a Mafia member, and I find these posts great. We Mafiosos just don't like crackpots, and that's it.
- Gehn
I'm a Mafia member, and I find these posts great. We Mafiosos just don't like crackpots, and that's it.
- Gehn
Awesome thread!!!
www.thecyberprofessor.com
www.thecyberprofessor.com
Interesting stuff here. Good job, Drude and others.
A couple of suggestions:
It would be very nice if somebody did a good tutorial on mechanical wave motion.
Also, I think it would help the forums if someone posted something on special relativity.
It would be very nice if somebody did a good tutorial on mechanical wave motion.
Also, I think it would help the forums if someone posted something on special relativity.
That is a great idea, Grasshopper. Why dont you give a try? I will do the relativity this Friday once I am done with my clinical sites. But it would be even better if you give it a try. Also there are some serious physicists here like mr. hom, maybe he should give it a try as well =].
QUOTE (Drude+Feb 4 2008, 12:03 AM)
That is a great idea, Grasshopper. Why dont you give a try? I will do the relativity this Friday once I am done with my clinical sites. But it would be even better if you give it a try. Also there are some serious physicists here like mr. hom, maybe he should give it a try as well =].
Hey, believe me, I'd love to. The problem is that... I'm currently learning about waves...
Of course, it'd certainly help me to try to do that. If no one else gets around to it in the next week or so, if you want, I'll regurgitate the little bit I've learned, but of course, that could be risky.
Behold! The (a?) Linear Wave Equation! (as it applies to 2nd semester physics)
∂^2y/∂x^2 = 1/(v^2) ∂^2y/∂t^2
But seriously, again, thanks for the work you and the others have done in this thread.
Hey, believe me, I'd love to. The problem is that... I'm currently learning about waves...
Of course, it'd certainly help me to try to do that. If no one else gets around to it in the next week or so, if you want, I'll regurgitate the little bit I've learned, but of course, that could be risky.
Behold! The (a?) Linear Wave Equation! (as it applies to 2nd semester physics)
∂^2y/∂x^2 = 1/(v^2) ∂^2y/∂t^2
But seriously, again, thanks for the work you and the others have done in this thread.
Tossbag.
As always; I win, you lose - fucksack. - same goes for that real funny twat, Dollerz.
QUOTE (Dollerz+Oct 4 2008, 03:45 PM)
£ 0.27
Boring a$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$munch.
Boring a$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$munch.
QUOTE (Dollerz+Oct 4 2008, 06:33 PM)
I'm a totally dull unimaginative cunt-lip fungal infestation.
Agreed.
Agreed.
Please add constructively to this forum and if you do add material make sure they convey something to the rest.
PhysOrg scientific forums are totally dedicated to science, physics, and technology. Besides topical forums such as nanotechnology, quantum physics, silicon and III-V technology, applied physics, materials, space and others, you can also join our news and publications discussions. We also provide an off-topic forum category. If you need specific help on a scientific problem or have a question related to physics or technology, visit the PhysOrg Forums. Here you’ll find experts from various fields online every day.
To quit out of "lo-fi" mode and return to the regular forums, please click here.