Question

The work done in shifting a particle of mass $m$ from the centre of earth to the surface of the earth is

Answer 1

Given,The mass of the particle be $m$,

Initilly it is on the centre of the earth, so

The work done in moving a particle from center of earth to its surface$W={\int }_0^{R}Fdr$

$={\int }_0^R(-\frac{G\left(\rho \frac{4}{3}\pi r^3\right)m}{r^2})dr$

$={\int }_0^R(-\frac{4G\rho \pi mr}{3})dr$

$=\frac{GMm}{2R},$we know that $F=\frac{GMm}{R^2}=mg$

$=\frac{1}{2}mgR$

(Since $g=\frac{GM}{R^2}$)

So, The work done in shifting a particle of mass from the centre of the earth to the surface of the earth is $=\frac{mgR}{2}.$