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.

$[\frac{2 (M_{1} + M_{2})}{3 m_{1} + M_{2}}] g$

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$[\frac{M_{1} + M_{2}}{3 m_{1} + 2 M_{2}}] g$

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$[\frac{2 (M_{1} + M_{2})}{3 m_{1} + 2 M_{2}}] g$

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$[\frac{2 (M_{1} + M_{2})}{2 m_{1} + 3 M_{2}}] g$

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Solution

The correct option is C
$[\frac{2 (M_{1} + M_{2})}{3 m_{1} + 2 M_{2}}] g$

Step I: Pulley: One rotational acceleration $α$ .1

Cylinder: One rotational acceleration $α_{2}$ . and, linear acceleration $a_{2}$
Step II: Pulley: $τ_{c} = I_{c} α_{1}$
TR= $(\frac{M_{1} R^{2}}{2}) \propto_{1} \Rightarrow \propto_{1} = \frac{2 T}{M_{1} R}$ ...(i)
Cyliner: $τ_{c} = I_{c} α_{2} \Rightarrow T R = (\frac{M_{2} R^{2}}{2}) α_{2}$
$α_{2} = \frac{2 T}{M_{2} R}$ ...(ii) and $M_{2} g - T = M_{2} a_{2}$ ...(iii)

Step III: -How string length between pulley's increases by $a_{2}$ and also increased by $α_{1} R + α_{2} R$

Step IV:- After solving equation (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.