Question

A particle of mass m was transferred from the centre of the base of a uniform hemisphere of mass M and radius R to infinity. How much work would be performed in this process by the gravitational force exerted on the particle by the hemisphere?

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Consider an elementary mass at point P with polar coordinates, $\theta$ and $\varphi$ with respect to the center of the base.

Elementary volume = $r^{2}sin\theta drd\theta d\varphi$

Elementary mass = $r^{2}sin\theta drd\theta d\varphi \rho$

Where density $\rho =\frac{3M}{2\pi R^3}$

Elementary potential at O $dV=-\frac{G{r}^{2}sin\theta drd\theta d\varphi \rho }{r}$

$\therefore$ Total potential = = $\int dV=-G\rho {\int }_0^{R}rdr{\int }_0^{\frac{\pi }{2}}sin\theta d\theta {\int }_0^{2\pi }d\varphi$

= $-G\rho =-G\rho \left(\frac{R^2}{2}\right)\left(1\right)\left(2\pi \right)$

= $-G\frac{3M}{2\pi R^3}\pi R^2=-\frac{3GM}{2R}$

work done = $-\frac{3GMm}{2R}=\frac{2GM}{l}\left[\frac{1}{a-\frac{l}{2}}\right]=\frac{4GM}{l(2a-1)}$

Force of attraction on m = $\frac{4GMm}{l(2a-l)}$.