Question

A solid cylinder of mass $M$ and radius $2R$ is connected to a string through a frictionless yoke and axle. The string runs over a disk shaped pulley of mass $M$ and radius $R$. The mass $M$ is attached to the other end of the string. The cylinder rolls without slipping on the table top. Find the acceleration of the block after the system is released from rest.

• A. $\frac{2g}{5}$
• B. $\frac{g}{2}$
• C. $\frac{g}{3}$
• D. $\frac{g}{5}$

Answer 1

Answer: (C)

$\frac{g}{3}$

Let ${\alpha }_1$ and ${\alpha }_2$ be the angular accelerations of the cylinder and the disc respectively.

As the linear acceleration of the string is constant, so we get
$2R\times {\alpha }_1=R\times {\alpha }_{2}or{\alpha }_2=2\times {\alpha }_1$
Now doing torque balance on the disc we get,
$(T_2-T_1)R=\left(\frac{M{R}^2}{2}\right)\left(2\alpha \right)$
or
$T_2-T_1=Ma$ ...(1)
Also using the FBD of the mass M we have
$Mg-T_1=Ma$ ...(2)
and torque balance about the instantaneous center of rotation of the cylinder we have
$T_2\left(2R\right)=\frac{M(2R{)}^2}{2}\left(\alpha \right)or{T}_2=Ma$ ....(3)
Adding (1),(2) and (3)
$a=\frac{g}{3}$