Question

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.

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Step I: Pulley: One rotational acceleration $\alpha$.1

Cylinder: One rotational acceleration ${\alpha }_2$. and, linear acceleration $a_2$
Step II: Pulley: ${\tau }_c=I_c{\alpha }_1$
TR=$\left(\frac{M_1{R}^2}{2}\right){\propto }_1\Rightarrow {\propto }_1=\frac{2T}{M_{1}R}$ ...(i)
Cyliner: ${\tau }_c=I_c{\alpha }_2\Rightarrow TR=\left(\frac{M_2{R}^2}{2}\right){\alpha }_2$
${\alpha }_2=\frac{2T}{M_{2}R}$ ...(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 ${\alpha }_{1}R+{\alpha }_{2}R$

Step IV:- After solving equation (i), (ii), (iii) and (iv), we get
$a_2=\left[\frac{2(M_1+M_2)}{3m_1+2M_2}\right]g$