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<πr/2. Acceleration of each element of chain when upper end is released is

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

Answer 1

Answer: (B)

$\frac{rg}{l}\left( 1 - \cos\frac{l}{r} \right)$

Answer 2

Consider a small element dl making an angle dθ

dm = $\frac{m}{l}dl = \frac{m}{l}rd\theta$

force acting on the element

dF = (dm)g sin θ = mrg/l sin θ dθ

F = $\frac{mrg}{l}\int_{0}^{a}{\sin\theta d\theta}$

Where, total angle α = 1/r

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

or a = $\frac{rg}{l}\left( 1 - \cos\frac{l}{r} \right)$