Consider two configurations of a system of three particles of masses $m$, $2m$and $3m$. The work done by external agent in changing the configuration of the system from figure(i) to figure (ii) is $\frac{{\mathrm{KGm}}^2}{\mathrm{a}}\left[1-\frac{1}{\sqrt{2}}\right]$ .Find the value of K.

Gravitational potential energy in configuration (i) is

${\mathrm{U}}_1=-\left[\frac{\mathrm{Gm}2\mathrm{m}}{\mathrm{a}}+\frac{\mathrm{Gm}3\mathrm{m}}{\mathrm{a}}+\frac{\mathrm{G}2\mathrm{m}3\mathrm{m}}{\sqrt{2}\mathrm{a}}\right]$

${\mathrm{U}}_1=-\frac{{\mathrm{Gm}}^2}{\mathrm{a}}\left[5+\frac{6}{\sqrt{2}}\right]$.......(1)

G.P.E in configuration (ii) is 

${\mathrm{U}}_2=-\left[\frac{\mathrm{Gm}2\mathrm{m}}{\mathrm{a}}+\frac{\mathrm{G}2\mathrm{m}3\mathrm{m}}{\mathrm{a}}+\frac{\mathrm{G}3\mathrm{mm}}{\mathrm{a}}\right]$

${\mathrm{U}}_2=-11\frac{{\mathrm{Gm}}^2}{\mathrm{a}}$........(2)

$\mathrm{\Delta }{\omega }_{\mathrm{g}}=-\mathrm{\Delta u}=-\left[{\mathrm{u}}_2-{\mathrm{u}}_1\right]$

$\mathrm{\Delta }{\omega }_{\mathrm{g}}=-\frac{6{\mathrm{Gm}}^2}{\mathrm{a}}\left[\frac{1}{\sqrt{2}}-1\right]=\frac{6{\mathrm{Gm}}^2}{\mathrm{a}}\left[1-\frac{1}{\sqrt{2}}\right]$

$\therefore {\omega }_{ext}=-{\omega }_{\mathrm{g}}=\frac{-6{\mathrm{Gm}}^2}{\mathrm{a}}\left[1-\frac{1}{\sqrt{2}}\right]$