Question

A body of mass \[m\] is raised to a height \[10R\] from the surface of the earth, \[R\] where is the radius of the earth. The increase in potential energy is (\[G\]=universal gravitational constant, \[M\]=mass of the earth and \[g\]=acceleration due to gravity)
A. \[\dfrac{{GMm}}{{11R}}\]
B. \[\dfrac{{GMm}}{{10R}}\]
C. \[\dfrac{{mgR}}{{11G}}\]
D. \[\dfrac{{10GMm}}{{11R}}\]

Answer 1

Hint: Use the formulae for the potential energy of the body on the surface of the earth and at a particular height from the surface of the earth. Then calculate the change in potential energy of the body.
Formula used:
The potential energy \[U\] of an object on the surface of the Earth is
\[U = - \dfrac{{GMm}}{R}\] …… (1)
Here, \[G\] is the universal gravitational constant, \[M\] is the mass of the earth, \[m\] is the mass of the object and \[R\] is the radius of the earth.
The potential energy \[{U_h}\] of an object at a height \[h\] from the surface of the Earth is
\[{U_h} = - \dfrac{{GMm}}{{R + h}}\] …… (2)
Here, \[G\] is the universal gravitational constant, \[M\] is the mass of the earth, \[m\] is the mass of the object and \[R\] is the radius of the earth.

Complete step by step answer:
A body of mass \[m\] is raised to a height \[10R\] from the surface of the earth.
The potential energy \[U\] of the body on the surface of the earth is
\[U = - \dfrac{{GMm}}{R}\]
Calculate the potential energy \[{U_h}\] of the body at a height \[10R\] from the surface of the earth.
Substitute \[10R\] for \[h\] in equation (2).
\[{U_h} = - \dfrac{{GMm}}{{R + 10R}}\]
\[ \Rightarrow {U_h} = - \dfrac{{GMm}}{{11R}}\]
The change in potential energy \[\Delta U\] of the body is the difference of the potential energy \[{U_h}\] of the body at a height \[10R\] from the surface of the earth and the potential energy \[U\] of the body on the surface of the earth.
\[\Delta U = {U_h} - U\]
Substitute \[ - \dfrac{{GMm}}{{11R}}\] for \[{U_h}\] and \[ - \dfrac{{GMm}}{R}\] for \[U\] in the above equation.
\[\Delta U = \left[ { - \dfrac{{GMm}}{{11R}}} \right] - \left[ { - \dfrac{{GMm}}{R}} \right]\]
\[ \Rightarrow \Delta U = - \dfrac{{GMm}}{{11R}} + \dfrac{{GMm}}{R}\]
\[ \Rightarrow \Delta U = \dfrac{{ - GMmR + 11GMmR}}{{11{R^2}}}\]
\[ \Rightarrow \Delta U = \dfrac{{10GMmR}}{{11{R^2}}}\]
\[ \Rightarrow \Delta U = \dfrac{{10GMm}}{{11R}}\]

Therefore, the change in potential energy of the body is \[\dfrac{{10GMm}}{{11R}}\].

So, the correct answer is “Option D”.

Note:
The gravitational potential energy of the body increases with increase in height from the surface of the Earth as for the body of a fixed mass the gravitational potential energy is inversely proportional to the negative of the height of the body from the earth’s surface.