An extensible string is wound over a rough pulley of mass $M_{1}$ and radius $R$ and a cylinder of mass $M_{2}$ and radius $R$ such that as the cylinder rolls down, the string unwounds over the pulley as well the cylinder. Find the acceleration of cylinder $M_{2}$

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Solution

Through this illustration, we will learn the application of torque equation and constraint relation in more complex case here. We have changed the block with cylinder $M_{2}$ . Pulley $M_{1}$ will have only pure rotation while cylinder $M_{2}$ will have rotation and translation combined. Let us analyse step by step in the same way as the previous illustration.
Step I: Analyse the motions of the pulley and the cylinder Pulley. One rotational acceleration $α_{1}$ (clockwise) cylinder. One rotational acceleration $α_{2}$ (cloclwise) and a linear acceleration $a_{2}$ (downward)
Step II: Equation of motion for $M_{2}$
$M_{2} g - T = M_{2} a_{2} . . . (i)$
Step III: Torque equation for pulley: $τ_{C} = I_{C} α$
$T R = (\frac{M_{1} R^{2}}{2}) α_{1} \Rightarrow α_{1} = \frac{2 T}{M_{1} R} . . . (i i)$
Torque equation for the cylinder (About centre of mass of cylinder) $τ_{C} = I_{C} α$
$\Rightarrow T R = (\frac{M_{2} R^{2}}{2}) α_{2} \Rightarrow α_{2} = \frac{2 T}{M_{2} R} . . . (i i i)$
Step IV: The acceleration of $P$ and $Q$ should be equal as both are connected with the same inextensible string.
Acceleration of $P, a_{M} = α_{1} R . . . . . (i v)$
Acceleration of $Q, a_{N} = a_{2} - α_{2} R . . . . (v)$ (Downward)
Hence, constraint relation: $α_{1} R = a_{2} - α_{2} R . . . (v i)$
Step V: Solving equations
After solving Eqs. (i), (ii), (iii) and (iv) we get
$a_{2} = [\frac{2 (M_{1} + M_{2})}{{3 M}_{1} + {2 M}_{2}}] g$

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An extensible string is woung over a rough pulley of mass $m_{1}$ and radius R; and a cylinder of mass $m_{2}$ and radius R such that as the cylinder rolls down the string un-wounds over the pulley as well the cylinder. Find the linear acceleration of the cylinder.