A body of mass m is moving in a circular orbi | vedclass

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(C) The total energy of a body of mass $m$ in a circular orbit of radius $r$ around a planet of mass $M$ is given by $E = -\frac{GMm}{2r}$.Initial tot

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$ - \frac{GMm}{6R}$

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(C) The total energy of a body of mass $m$ in a circular orbit of radius $r$ around a planet of mass $M$ is given by $E = -\frac{GMm}{2r}$.Initial total energy of the body of mass $m$ at radius $R$ is:$E_i = -\frac{GMm}{2R}$After splitting,the body divides into two masses,each of mass $m' = \frac{m}{2}$.The first mass moves in an orbit of radius $r_1 = \frac{R}{2}$,and the second mass moves in an orbit of radius $r_2 = \frac{3R}{2}$.The final total energy $E_f$ is the sum of the energies of the two masses:$E_f = -\frac{GM(m/2)}{2(R/2)} - \frac{GM(m/2)}{2(3R/2)}$$E_f = -\frac{GMm}{2R} - \frac{GMm}{6R}$$E_f = -\frac{3GMm + GMm}{6R} = -\frac{4GMm}{6R} = -\frac{2GMm}{3R}$The difference between the final and initial total energies is:$\Delta E = E_f - E_i = -\frac{2GMm}{3R} - (-\frac{GMm}{2R})$$\Delta E = -\frac{2GMm}{3R} + \frac{GMm}{2R} = \frac{-4GMm + 3GMm}{6R} = -\frac{GMm}{6R}$

List-$I$	List-$II$
$(A)$ Kinetic energy of planet	$(1)$ $-\frac{GMm}{a}$
$(B)$ Gravitational potential energy of Sun-planet system	$(2)$ $\frac{GMm}{2a}$
$(C)$ Total mechanical energy of planet	$(3)$ $\frac{GM}{r}$
$(D)$ Escape energy at the surface of planet for unit mass object	$(4)$ $-\frac{GMm}{2a}$

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