Question

A wheel of radius $R$ and mass $M$ is placed at the bottom of a fixed step of height $R$ as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis normal to the plane of the paper passing through the point $Q$. Which of the following options is/are correct?

• A. If the force is applied at point 𝑃 tangentially then $\tau$ decreases continuously as the wheel climbs
• B. If the force is applied normal to the circumference at point 𝑋 then $\tau$ is constant
• C. If the force is applied normal to the circumference at point 𝑃 then $\tau$ is zero
• D. If the force is applied tangentially at point 𝑆 then $\tau \neq 0$ but the wheel never climbs the step

Answer 1

Answer: (B), (C)

As the force acts as shown and considering its direction to be constant, then
$\tau_{\text {net }}= F (2 R \cos \theta)- mg R \cos \theta= R \cos \theta(2 F - mg )$
Which definitely infers that as $\theta$ increases, $\tau_{\text {net }}$ decreases.
* As soon as force is applied as shown, it leads to frictional force between bottom point and horizontal surface, which will produce non-zero torque about point $Q$.