Question

In the figure shown, plank is being pulled to the right with a constant speed v. If the cylinder does not slip then :

• A. the speed of the centre of mass of the cylinder is $2v$.
• B. the speed of the centre of mass of the cylinder is zero.
• C. the angular velocity of the cylinder is $v/R$.
• D. the angular velocity of the cylinder is zero.

Answer 1

Answer: (B), (C)

the speed of the centre of mass of the cylinder is zero.
the angular velocity of the cylinder is $v/R$.

Let velocity of COM of cylinder be $u$ towards right, and angular speed about COM be $\omega$ in anti-clockwise direction. so velocity of top point of cylinder is $u-\omega R$ and velocity of botttom point of cylinder is $u+\omega R$.

As there is no slipping at both points, so $u-\omega R=-v-\left(1\right)$ which is velocity of rope towards left, and $u+\omega R=v-\left(2\right)$ which is velocity of plank towards right.

From $\left(1\right)and\left(2\right)$ $u=0$ and $\omega =v/R$

i.e., velocity of COM of cylinder is $0$ and angular speed about COM is $\omega =v/R$