Question

The potential at the centre of the base of a solid hemisphere of mass $m$ and radius $R$ is

• A. $\frac{-Gm}{2R}$
• B. $\frac{3Gm}{2R}$
• C. $\frac{-3Gm}{2R}$
• D. $\frac{-2Gm}{R}$

Answer 1

The correct option is C $\frac{-3Gm}{2R}$


Consider a hemispherical shell of radius $r$ and thickness $dr$.

So mass $dm$ of elementary part will be

$dm=\frac{m}{\frac{2}{3}\pi R^3}\times \left(2\pi r^2\right)dr=\frac{3m}{R^3}{r}^{2}dr$

Since all the points of this hemispherical shell are at the same distance $r$ from $O$, hence potential at $O$ is

$dV=\frac{-Gdm}{r}=\frac{-G\left(3m{r}^{2}dr\right)}{r{R}^3}$

$\Rightarrow dV=-\frac{3Gm}{R^3}rdr$

integrating both side, we get

$V={\int }_0^R-\frac{3Gm}{R^3}rdr=-\frac{3Gm}{R^3}{\int }_0^{R}rdr$

$\Rightarrow V=\frac{-3}{2}\frac{Gm}{R}$

Hence, option $\left(c\right)$ is correct.