Question

If g is the acceleration due to gravity on the surface of earth, find the gain in potential energy of an object of mass m raised from the surface of earth to a height equal to the radius R of the earth.
A) \[\dfrac{1}{2}mgR\]
B) \[ \dfrac{1}{4}mgR\]
C) \[\dfrac{1}{3}mgR\]
D) \[\dfrac{2}{1}mgR\]

Answer 1

Hint: The velocity of an object changes when the objects fall free from the space. The change in velocity changes the acceleration. This acceleration is known as acceleration due to gravity. The acceleration due to gravity is denoted by ‘g’.

Formula used: \[U = - \dfrac{{GMm}}{R}\]
where, U=potential energy at height equal to R
G=gravitational constant
M=mass of earth
m=mass of body
R=radius of earth

Complete step-by-step answer:
Let us consider the Potential energy of the object at the surface of earth is
\[{U_1} = - \dfrac{{GMm}}{R}\]
We can consider the Potential energy of the object when it is taken to a height equal to the radius of earth is
\[{U_2} = - \dfrac{{GMm}}{{R + R}}\]
We are taking the R twice because the height of the object is equal to the radius.
Hence, the gain in potential energy is:
\[{U_2} - {U_1} = - \dfrac{{GMm}}{{R + R}} - \left( {\dfrac{{ - GMm}}{R}} \right)\]
We all know the Gravitation law,
\[g = \dfrac{{GM}}{{{R^2}}}\]
 Now let us equate the gain potential energy and gravitational energy,
From the given equations,
\[{U_2} - {U_1} = \dfrac{{g{R^2}m}}{R}\].
\[{U_2} - {U_1} = \dfrac{{mgR}}{2}\].
Therefore, from the above equation we found the value of the gain potential energy.

Hence, the correct answer is option (A).

Note: The force at which the earth attracts the body is known as the Gravity. The acceleration that is gained by the object under the gravity force is known as acceleration due to gravity. It is the work of gravity to hold all the planetary objects in the sky. Gravity is considered the weakest force that prevails on the surface of the earth.
The gravity was precisely explained by Einstein in the general theory of relativity.
In the general theory of relativity, Einstein described gravity as a consequence of spacetime curvature and not as a force. He also explained the Blackhole which is the perfect example for the spacetime curvature.
The elementary particles of the gravity force are the Gravitons. For the electromagnetic field, in quantum the associated field is photons. Same as for the gravitational force the associated quantum field is gravitons.