Question

A solid cylinder of mass M and radius R is resting on a horizontal platform (which is parallel to the x-y plane) with its axis fixed along Y-axis and free to rotate about its axis. The platform is given a motion in the X-direction given by $x=Acos\left(\omega t\right)$. There is no slipping between the cylinder and the platform. The maximum torque acting on the cylinder during its motion is:

• A. $\frac{M{\omega }^{2}AR}{3}$
• B. $\frac{M{\omega }^{2}AR}{2}$
• C. $\frac{2}{3}\times M{\omega }^{2}AR$
• D. The situation is not possible.

Answer 1

The correct option is B $\frac{M{\omega }^{2}AR}{2}$

The cylinder is having a fixed axis, so it can perform rotational motion only.
$fR=\frac{M{R}^2}{2}\alpha \Rightarrow f=\frac{MR}{2}\alpha$

For no slipping,
$R\alpha =$ acceleration of the platform
$\frac{2f}{M}={\omega }^{2}Acos\omega t$
For the maximum torque, $f$ would be maximum and $f_{max}=M{\omega }^{2}A/2$

So the maximum torque, ${\tau }_{max}=f_{max}\times R=\frac{M{\omega }^{2}AR}{2}$