Figure P822 l on W 15 Ti on W SolutionThe wheel is iniatiall

Solution

The wheel is iniatially rotating in the clockwise direction. Now for the wheel to slow down, the net torque on the wheel must be in direction opposite to its motion. That is, we must have a net torque trying to rotate the wheel in the anti-clockwise direction.

Let us consider the inner radius to be R1 and the outer radius to be R2. As per the question, we need to determine the maximum value of the ratio R1 / R2 such that the wheel has a net torque trying to rotate it anti-clockwise.

We will form the equations for each of the tensional forces acting on the wheel and then resolve to determine the required ratio.

For the left hand-side tension , we have: Torque(L) = T x R2

For the right hand-side tension, we have: Torque(R) = 1.5 T x R1

Now for the net torque to be anti-clockwise, we must have Torque(L) > Torque(R)

That is, T x R2 > 1.5 T x R1

or R1 / R2 < 2/3

Therefore the ratio of the two radii must be smaller than 2/3 for the wheel to slow down.