Question

A cylinder is rolling without sliding over two horizontal planks (surfaces) 1 and 2. If the velocities of the surfaces $A$ and $B$ are $-v\hat{i}$ and $2v\hat{i}$ respectively. Then,

• A. position of instantaneous axis of rotation is $\frac{2R}{3}$
• B. position of instantaneous axis of rotation is $\frac{4R}{3}$
• C. angular velocity of the cylinder is $5v/2R$
• D. angular velocity of the cylinder is $3v/2R$

Answer 1

Answer: (A), (D)

The correct options are
A position of instantaneous axis of rotation is $\frac{2R}{3}$
D angular velocity of the cylinder is $3v/2R$

Let $v_1$ be the velocity of COM of cylinder and $w$ be its angular velocity.

Let C be the position of instantaneous axis of rotation.

$2v=w\left(CB\right)⟹2v=(2R-x)w$ ...........(a)

$v=xw$ ............(b)

$2\left(wx\right)=(2R-x)w⟹x=\frac{2R}{3}$

Also there is no slipping at A and B.

Thus $2v=v_1+Rw$ and $v=Rw-v_1$

Eliminating $v_1$, we get $w=\frac{3v}{2R}$