Question

Consider a thin uniform annular disc of mass $M=4kg$ having inner and outer radius $3m$ and $4m$ respectively. A point mass $m=2kg$ is taken from point $P$ to infinity so that at infinity it has kinetic energy of $20J$. If work done by external agent in this process is $W=\{20+\frac{16G}{7}(N\sqrt{2}-5\left)\right\}\text{J}$ ($G$ is universal gravitational constant in $SI$ unit). Find the value of $N$.

Answer 1

We know that the work required to take a unit mass from P to infinity =$\Delta V$​, where =$\Delta V$ is the gravitational potential at $P$ due to the disc.

Let we take an elemental ring of length $dx$

Mass of elemental ring
$dm=$$\frac{\left(M2\pi xdx\right)}{7\pi }$

Distance between point $P$ and elemental ring is
$\sqrt{x^2+4^2}$

$\Delta V=\frac{GM2\pi xdx}{7\pi \sqrt{x^2+4^2}}$

$\Delta V=\frac{2GM}{7}\underset{3}{\overset{4}{\int }}\frac{xdx}{\sqrt{x^2+16}}$
After integrating and putting limits we will get:
$\Rightarrow \Delta V=\frac{2GM}{7}(4\sqrt{2}-5)$