Question

A chain of length $\ell <\frac{\pi R}{2}$ is placed on a smooth surface whose some part is horizontal and some part is on quarter circle of radius $R$ in the vertical plane as shown. Initially the whole part of chain lies on the circular part with one end at topmost point of circular surface. If the mass of chain is $m$ , then work required to pull very slowly the whole chain on horizontal part is –

• A. $\frac{m_{\ell }}{g{R}^2\left[\sin \left(\frac{\ell }{R}\right)\right]}$
• B. $\frac{m_{\ell }}{gR}{}^2\left[\cos \left(\frac{\ell }{R}\right)\right]$
• C. $\frac{m_{\ell }}{g{R}^2}\left[\left(\frac{\ell }{R}\right)-\sin \left(\frac{\ell }{R}\right)\right]$
• D. $\frac{m}{\ell }g{R}^2\left[\left(\frac{\ell }{R}\right)-\cos \left(\frac{\ell }{R}\right)\right]$

Answer 1

Answer: (C)

Taking flat surface as reference,

$d{U}_i=-\left(\frac{m}{\ell }Rd\theta \right)\times g\times R[1-\cos \theta ]$
$d{U}_i=-\frac{mg{R}^2}{\ell }[1-\cos \theta ]d\theta$
$\therefore U_i=-\frac{mg{R}^2}{\ell }\left[\left(\frac{\ell }{R}\right)-\sin \left(\frac{\ell }{R}\right)\right]$
and $U_f=0$
$\therefore W_{\text{ext }}=-\Delta U$