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spark
Could anyone tell me INTUITIVELY why moment of inertia for point mass is mr^2, please?
I mean,it could have been 'mr' in order to portray the fact that the rotational inertia depends upon the distance from the object and the mass of the object.

Surely,this definition cannot only be because the derivations come out like this--what is the intuitive reason for moment of inertia to be mr^2
AlphaNumeric
Then the units wouldn't match, as mr and mr^2 have different units.

Look at linear momentum and mass, you have F = ma, p = mv, KE = 1/2 mv^2

For rotational systems you have T = m theta'', where T is torque and theta is the angle of rotation. When you integrate this up as you would F=ma you get the angular versions of the linear momentum expressions, including KE = (1/2) I (theta')^2. From this definition of I you can work out how you find more general expressions for I in terms of the layout of an object.

http://en.wikipedia.org/wiki/Moment_of_inertia
Guest_Limon
It is mR² because the tangential velocity of a comet increases as it approaches the Sun. The slow tangential velocity at apogee times the large radius equals the large tangential velocity times the small radius at perigee. Angular velocity in radians per second is equal to tangential velocity / radius. Multiplying by R² times radians (tangential velocity / radius) gets us back to tangential velocity time radius (two R’s drop out). This is called angular momentum conservation, and it only works when the rotating object is under gravitational acceleration from the point around which it is rotating.
AlphaNumeric
QUOTE (Guest_Limon+Aug 2 2009, 12:34 AM)
This is called angular momentum conservation, and it only works when the rotating object is under gravitational acceleration from the point around which it is rotating.

Angular momentum conservation works when you have rotational symmetry, ala Noether's theorem linking symmetries and conserved quantities. Are you still trying to proclaim angular momentum or whatever it is isn't conserved, Limon? Or have you finally realised that you're wrong?
Enthalpy
In case you like to compare a moment of inertia with a mass:

Angular speed is then the equivalent of linear speed
Torque is the equivalent of force

And then, the first R converts the angular speed to a linear speed, whose time derivative combined with mass is a force, and the second R makes a torque from this force.

If the mass of the object you spin is spread over a volume, you'll have to accumulate mR^2 over this volume.
s0cratus
Inertia and Spin. / My opinion./
1.
Aristotle.
Every object needs force/power/energy for its moving .
If no force, no moving.
2.
Newton.
Of course great Aristotle is right saying that there is no movement
without forces . I respect him very much and I won’t make a
fool myself quarrelling with him. However I can say more
and explain Aristotle’s opinion by the formula F= ma.
It means, the force of moving object depends on acceleration
which it gives to this object’s mass.
But here I have two opportunities /possibilities.
a)
The acceleration appears as a result of outside influence.
One body (moving body) interacts with another body (moving or resting).
b)
But if I have only one, single body moving in the straight line
and it doesn’t interact with another body it means that this body
also must have an acceleration. In this situation I don’t know
how the acceleration appears, I don’t know if it is inner
acceleration of body, I know nothing about this acceleration.
But this kind of acceleration must exist and I will name it
“inertia”.
3.
Mach.
Newton doesn’t know the reason of inertia, but maybe inertia depends
on all stars, on all the matter in the Universe.
4.
Planck.
Newton’s inertia is very strange, and Mach’s idea too. But if I
take our Universe like a ‘ black body ’ then I can suggest
that must be some very small particle (quant) which can move
‘ inertial’ with constant speed c=1 over a period of time. I will write
this ‘ inertial ‘ moving of quanta by formula: h=Et. But really,
it is hard for me to believe that I am right.
5.
Einstein.
Of course Planck is right. But I don’t like the way he reached the
result. He says nothing concrete about the particle and the reason
of this acceleration’s beginning. I will take another road. If I use the
Boltzmann resting particle (R/N=k ) and give him Wien’s displacement
constant (b), as an acceleration, then the particle will have the Planck’s
impulse but now the formula is h=kb. Planck’s formulas and my own
are equal, as they explain behavior of quant (light quanta) from
different point of view.
6.
Goudsmit – Uhlenbeck.
It is all well.
But we can see different kinds of movings in the real Nature.
And look at Planck’s formula h=Et. It includes time (t).
And time, by its nature, is a limited parameter. It means that this
particle cannot go straight at all time with constant speed c=1.
This kind of moving must be temporary and can change.
So, another possibility is that the particle can spin around itself
and we will write this kind of moving by formula h=h/ 2pi.
7.
L. de Broglie and Heisenberg.
These two spins of particle are very important parameters, so we
will try to explain all phenomena in the Nature using only these
parameters.
…………………….
But, unfortunately, they both didn’t have success.
Why did they fall?
Because to use only spin parameters is not enough. The spin
parameters belong to the particle who/ which have/has also another
parameters: speed © and volume (a) and all together they can create
particle which we call electron: e^2=ahc. Now using electron and
Boltzmann particle (R/N=k ) is possible to explain the beginning
of star formation (gravitation) and later all another phenomena in Nature.
===============.
P.S.
Inertia is a mysterious phenomena in Nature.
There isn’t any physical ‘inertia unity’ in Nature.
There isn’t any physical apparatus of measuring ‘the unity of inertia’.
The object moves and doesn’t spend energy at all.
Isn’t it strange ?
The Physics is very ill with ‘inertia’ for a long time.
Maybe therefore someone wrote to me:
“An old professor of mine used to say
that anyone who can answer that question
what inertia is , would win a Nobel Prize. “
I don’t say about myself, I say about this article:
“This article waits for its Nobel Prize.”
============ . .
Best wishes.
Israel Sadovnik. / Socratus.

http://www.worldnpa.org/php2/index.php?tab...Display&id=1372
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