Question

In order to shift a body of mass\[m\] from a circular orbit of radius\[3R\] to a higher orbit of radius \[5R\] around the earth, the work done is
A) \[\dfrac{{3GMm}}{{5R}}\]
B) \[\dfrac{{GMm}}{{2R}}\]
C) \[\dfrac{2}{{15}}\,\dfrac{{GMm}}{R}\]
D) \[\dfrac{{GMm}}{{5R}}\]

Answer 1

Hint: In this question we will proceed with given the definition of the gravitational potential energy of a body and then by using the formula we will find initial and final gravitational potential energy and the difference of them will give us the required work done.

Formula Used:
Gravitational potential energy
The gravitational potential energy at any point is defined as the work done in bringing a body from infinity to the point in the gravitational field.
\[U = - \dfrac{{GMm}}{R}\]
Where \[U,\,G,\,M,\,m\,and\,R\]are gravitational potential energy, gravitational constant, mass of earth, mass of the body and radius of earth respectively.

Complete step by step answer:
Given,
Radius of initial orbit\[{R_1} = 3R\]
Radius of final orbit \[{R_2} = 5R\]
By using the formula of gravitational potential energy when\[{R_1} = 3R\]
Initial gravitational potential energy \[{U_i} = - \dfrac{{GMm}}{{{R_1}}} = - \dfrac{{GMm}}{{3R}}\]
By using the formula of gravitational potential energy when\[{R_2} = 5R\]
Final gravitational potential energy \[{U_f} = - \dfrac{{GMm}}{{{R_2}}} = - \dfrac{{GMm}}{{5R}}\]
here work done is equal to the change in gravitational potential energy of a body revolving in initial orbit and final orbit given in question as follows
\[W = {U_f} - {U_i}\]
\[\Rightarrow W = - \dfrac{{GMm}}{{5R}} - \left( { - \dfrac{{GMm}}{{3R}}} \right)\]
\[\Rightarrow W = - \dfrac{{GMm}}{{5R}} + \dfrac{{GMm}}{{3R}}\]
\[\Rightarrow W = \dfrac{{2GMm}}{{15R}}\]

Hence the required work done is equal to \[\dfrac{{2GMm}}{{15R}}\]. Hence, the option (C) is correct.

Note:
Before answering this question we will not miss describing the needed definition on which the question is based. We must have the knowledge about the energy and work done as well. It is important to note that the sign should be used carefully to avoid mistakes.