Consider the Earth and mass system, with r, the distance between the mass m and the Earthâ€™s centre.

**Gravitational
potential energy near the surface of the Earth**

It
is already discussed in chapter 4 that when an object of mass m is raised to a
height h, the potential energy stored in the object is mgh (Figure 6.14). This
can be derived using the general expression for gravitational potential energy.

Consider
the Earth and mass system, with *r*,
the distance between the mass m and the Earthâ€™s centre. Then the gravitational
potential energy,

Here
r = R_{e}+h, where R_{e} is the radius of the Earth. h is the
height above the Earthâ€™s surface

If
h << R_{e}, equation (6.31) can be modified as

By
using Binomial expansion and neglecting the higher order terms, we get

We
know that, for a mass m on the Earthâ€™s surface,

Substituting
equation (6.34) in (6.33) we get,

It
is clear that the first term in the above expression is independent of the
height h. For example, if the object is taken from height h_{1} to h_{2,}then
the potential energy at h_{1} is

The
potential energy difference between h_{1} and h_{2} is

The
term *mgR _{e}* in equations
(6.36) and (6.37) plays no role in the result. Hence in the equation (6.35) the
first term can be omitted or taken to zero. Thus it can be stated that The
gravitational potential energy stored in the particle of mass m at a height h
from the surface of the Earth is

It
is to be noted that mgh is the work done on the particle when we take the mass
m from the surface of the Earth to a height h. This work done is stored as a
gravitational potential energy in the mass m. Even though mgh is gravitational
potential energy of the system (Earth and mass m), we can take mgh as the
gravitational potential energy of the mass m since Earth is stationary when the
mass moves to height h.

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11th Physics : UNIT 6 : Gravitation : Gravitational potential energy near the surface of the Earth |

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