This pulley has two rims of differing radii connected at the

This pulley has two rims of differing radii, connected at their common center. The pulley is free to rotate around the fixed axis shown and is rotating in the clockwise direction. Describe in words what will happen in the time after the moment depicted here. Explain/show your reasoning with words and/or equations. If the tension in the right string remains as shown in the drawing, what tension should the left string exert so that the angular velocity of the pulley would no longer change? Explain/show your reasoning with words and/or equations. If the tension in the left string remains as shown in the drawing, what minimum tension should the right string exert so that the pulley would go faster in the clock-wise direction? Explain/show your reasoning with words and/or equations.

Solution

We know that for a system with force acting at a point, the torque about an axis is given as product of force and perpendicular distance from the axis [basically cross product of the vectors].

a.) Now, in the above situation, we will denote the objects as object1 [Left hand side object] and object2 [Right hand side object].

Now for object1, the net torque is exerts on the pulley about the axis = T x 2R = 2TR

net torque due to object2 = 1.5T x R = 1.5TR

Hence the net torque is 0.5TR in the anti-clockwise direction, hence the system will slow down in this case

b.) As the angular velocity is to be maintained, net torque needs to be zero. Now, tension in right string is same

hence torque due to right string = 1.5TR

For net torque to be zero, torque due to left string should also be equal to 1.5TR

That is, Tension x 2R = 1.5TR

This implies, Tension = 0.75T

c.) Now for the pulley to accelerate in the clock wise direction,

net torque due to right side string should be greater than that due to left side string

Torque due to left side string = 2TR

Now Tension in right string x R > 2TR

Hence tension in right string > 2T

Therefore minimum possible for the pulley to accelerate in the clock-wise direction, it should be at least 2T