Question

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

• A. $\frac{1}{2}MRA{\omega }^2$
• B. $MRA{\omega }^2$
• C. $2MRA{\omega }^2$
• D. $MRA{\omega }^2\times \cos \omega t$

Answer 1

The correct option is A $\frac{1}{2}MRA{\omega }^2$


Motion of platform $ABCD$ in $X-$ direction is described by:
$x=A\cos \omega t$
$\therefore v=\frac{dx}{dt}=-A\omega \sin \omega t$
and $a=\frac{d^{2}x}{d{t}^2}=-A{\omega }^2\cos \omega t$
$\therefore a_{\text{max}}=A{\omega }^2$
${\tau }_{\text{max}}=I{\alpha }_{\text{max}}=I\frac{a_{\text{max}}}{R}$
$=\left(\frac{1}{2}M{R}^2\right)\frac{A{\omega }^2}{R}=\frac{1}{2}MRA{\omega }^2$