Question

If mass of the earth is M, radius is R, and gravitational constant is G, then work done to take 1kg mass from earth surface to infinity will be:
A. $ \sqrt{\dfrac{GM}{2R}} $
B. $ \dfrac{GM}{R} $
C. $ \sqrt{\dfrac{2GM}{R}} $
D. $ \dfrac{GM}{2R} $

Answer 1

Hint: First, find out the formula for work done in terms of potential energy. Find out the potential energy of an object on the surface of Earth and at infinity. Using these values calculate the work done for a given body.

Complete step-by-step answer:
The work done to move an object in the gravitational field of the Earth is nothing but the change in the potential energy of the body. Therefore,
Workdone = Change in potential energy = Final potential energy – Initial potential energy
The potential energy of a body on the surface of the Earth is given by
 $ P.E.=-\dfrac{GMm}{R} $
Where G is the universal constant of gravitation.
     R is radius of the Earth
     M is mass of the Earth
     Mass of the body.
For a given body, m=1kg
 $ \therefore P.E.=-\dfrac{GM(1)}{R}=-\dfrac{GM}{R} $
The potential energy of a body at infinity is zero.
Thus, work done to take 1kg mass from earth surface to infinity will be
\[\begin{align}
  & Workdone=0-(-\dfrac{GM}{R}) \\
 & \therefore Workdone=\dfrac{GM}{R} \\
\end{align}\]
Answer is B. $ \dfrac{GM}{R} $

Additional Information:
If h is the height of the body from the surface of the Earth then the potential energy can also be written as
$P.E.=mgh$
Where g is acceleration due to gravity.

Note: The energy possessed by the body due to its position in a gravitational field is called the potential energy (or gravitational potential energy) of that body. The potential energy of the body depends on its distance from the surface of the Earth. At infinity i.e. far away from the gravitational field of the Earth, the potential energy will become zero and the total energy is purely kinetic.