I'm having trouble with the following questions so if anyone can help it would be greatly appreciated.

1) Have to work out the tension in the cable in the diagram below. Structure supported by steel cable attached to point X, 6.45m above the hinge. Angle between wall and cable is 37 degrees. Weight of ABCD is 950N. Load of 1509.2N lifted at C. Dimensions of AD and DC are 2.5m and BD 1.5m



I know that all forces should be equal to 0 and the sum of the anticlockwise moment=sum of the clockwise moment.

2) Have to work out how much the tension in the support wire changes if the sign is moved to the end section of the support beam. Sign hung by two chains.

For the first problem, you can do it in just one equation if you choose your moment axis cleverly. There are three unknown forces (besides the internal forces in the truss, which you don't care about for this problem), which are the tension in the cable, and the horizontal and vertical forces at the hinge.

Placing the moment axis at the hinge causes the two unknown forces at the hinge to produce moments of zero, so they disappear from the equation. You are left with T*6.5*sin(37) - 950*2.5 - 1509.2*5 = 0. Now it is very easy to solve for T.

For the second problem, once again you use moment around the hinge. Since you only want the change in the tension, you only need to consider how much the moment around the hinge increases when you move the sign out by 1 meter. Let's call the weight of the sign W, so the moment around the hinge is originally W*d, and later it is W*(d+1meter), so the change in moment is the difference of these two, which is just W*1meter. The tension in the cable must increase enough to counteract this extra moment, so the difference between the new and old moments of the cable is T_new*sin(30)*3meters - T_old*sin(30)*3meters = (delta T)*0.5*3meters. Setting this equal to the change in moment from moving the sign gives (delta T)*1.5meters = W*1meter, so delta T = 2/3*W.

Hope this helps!

--Stuart Anderson

Thanks for your help mr homm. I understand what you have done. I've just two more questions that I really need help on.

First one is in relation to question 1 on my first post regarding the the metal lift structure. Need to calculate the magnitude of the reaction force of the wall on the structure at A and determine what direction it is coming from.

2) Piano tuner tightens mild steel wire of diameter 1.7mm so that it its original length of 1.64m is extended by 1.4mm. Have to calulate force exerted on the frame of the piano by the wire.

If someone could help with these final questions I would much appreciate your help.

To find the force reactions in question 1 is easy now that you know the tension. Summing the vertical forces gives A_y + T_y - 950 - 1509.2 = 0, and you know T_y = T*cos(37), so you can solve for A_y. Summing the horizontal forces is even easier: A_x - T_x = 0, and you know T_x = T*sin(37). Now that you have A_x and A_y, the Pythagorean theorem gives you the magnitude of the total reaction at A, and the formula tan(theta) = (A_y / A_x) gives the angle measured counterclockwise from the +x axis.

For the second question, you need to relate force to stress, stress to strain, and strain to length. Force = area*stress, and area is known (pi*r^2, where r is half the diameter of the wire). Stress = elasticity*strain, and elasticity is a property of the material, which is given (mild steel, look up its elastic modulus E in a table). Strain = (change in length)/(original length), which are both given. Therefore, Force = area*elasticity*strain = (pi*(.0017/2)^2)*E*(.0014/1.64).

Hope this helps!

--Stuart Anderson