Solveeit Logo
Login

Question

Question: In order to shift a body of mass\[m\] from a circular orbit of radius\[3R\] to a higher orbit of rad...

In order to shift a body of massmm from a circular orbit of radius3R3R to a higher orbit of radius 5R5R around the earth, the work done is
A) 3GMm5R\dfrac{{3GMm}}{{5R}}
B) GMm2R\dfrac{{GMm}}{{2R}}
C) 215 GMmR\dfrac{2}{{15}}\,\dfrac{{GMm}}{R}
D) GMm5R\dfrac{{GMm}}{{5R}}

Explanation

Solution

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=−GMmRU = - \dfrac{{GMm}}{R}
Where U, G, M, m and RU,\,G,\,M,\,m\,and\,Rare 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 orbitR1=3R{R_1} = 3R
Radius of final orbit R2=5R{R_2} = 5R
By using the formula of gravitational potential energy whenR1=3R{R_1} = 3R
Initial gravitational potential energy Ui=−GMmR1=−GMm3R{U_i} = - \dfrac{{GMm}}{{{R_1}}} = - \dfrac{{GMm}}{{3R}}
By using the formula of gravitational potential energy whenR2=5R{R_2} = 5R
Final gravitational potential energy Uf=−GMmR2=−GMm5R{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=Uf−UiW = {U_f} - {U_i}
⇒W=−GMm5R−(−GMm3R)\Rightarrow W = - \dfrac{{GMm}}{{5R}} - \left( { - \dfrac{{GMm}}{{3R}}} \right)
⇒W=−GMm5R+GMm3R\Rightarrow W = - \dfrac{{GMm}}{{5R}} + \dfrac{{GMm}}{{3R}}
⇒W=2GMm15R\Rightarrow W = \dfrac{{2GMm}}{{15R}}

Hence the required work done is equal to 2GMm15R\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.