Question

A massless rope is wrapped several times on a disc of mass M and radius R. The other end is tied to a mass m, which at the beginning is at a height of h above the ground shown in the figure. When released the velocity of the mass as it touches the ground is

• A. $\frac{mMI}{2(M+m)}$
• B. $\frac{ml}{M+m}$
• C. $\frac{ml}{2(M+m)}$
• D. $v=\sqrt{\frac{4mgh}{2m+M}}$

Answer 1

The correct option is D $v=\sqrt{\frac{4mgh}{2m+M}}$

By conservation of energy:

Change in $PE=$ Change in $KE$

$mgh=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2\Rightarrow mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\left(\frac{1}{2}M{R}^2\right){\omega }^2\Rightarrow mgh=\frac{1}{2}m{v}^2+\frac{M{v}^2}{4}\Rightarrow v=\sqrt{\frac{4mgh}{2m+M}}$