Question

A chain of length $l$ is placed on a smooth spherical surface of radius r with one of its ends fixed at the top of the surface. Length of chain is assumed to be $l$ < $\frac{\pi r}{2}$. Acceleration of each element of chain when upper end is released is

• A. $\frac{lg}{r}(1-cos\frac{r}{l})$
• B. $\frac{rg}{l}(1-sin\frac{l}{r})$
• C. $\frac{lg}{r}(1-sin\frac{l}{r})$
• D. $\frac{rg}{l}(1-cos\frac{l}{r})$

Answer 1

The correct option is D $\frac{rg}{l}(1-cos\frac{l}{r})$

Let mass per unit length of chain $=\frac{m}{l}$
Consider an element of chain of length $dl$ subtending an angle of $d\theta$ at the centre of spherical surface.
Mass of element $dm=\frac{m}{l}dl=\frac{m}{l}rd\theta$
Force acting on element, $dF=dmgsin\theta$
$=\frac{m}{l}rgsin\theta d\theta$
$\therefore F=\frac{m}{l}rg{\int }_0^{\alpha }sin\theta d\theta$
$=\frac{m}{l}rg(1-cosa)$
$=\frac{m}{l}rg(1-cos\frac{l}{r})$

Then $\alpha =\frac{F}{m}=\frac{rg}{l}(1-cos\frac{l}{r})$