Gravitational Potential
The gravitational potential at a point
is the potential energy associated with a unit mass due to its position in the
gravitational field of another body.
The gravitational
potential at a point in the gravitational field of a body is defined as the
amount of work done in bringing a body of unit mass from infinity to that
point. Gravitational potential,
V = 
= 
The gravitational potential is a scalar
quantity. Its SI unit is J kg−1 and cgs
units is erg g−1.
The dimensional formula of
gravitational potential is 
Gravitational Potential at a Point
Due to the Earth:
The work done in bringing a body of mass 
from infinity to a point at distance r from the centre
of the earth is
W = 
Hence the gravitational potential due to the earth at distance r from its centre
is
V = 
= 
At the surface of the earth, r
= R, therefore
= 
Relation between Gravitational Potential Energy and
Gravitational Potential:
From the above equations, we find that
U = 
= 
Gravitational potential energy = Gravitational potential × mass
Problems:
1. At a point above the surface of the
earth, the gravitational potential is 
and
the acceleration due to gravity is 
. Assuming the mean radius of the earth
to be 6400 km. Calculate the height of this point above the earth’s surface.
Solution:
Let
be the distance of the given point from the
centre of the earth. Then
Gravitational
potential,
V
= 
= −
5.12 × 107 J kg-1 ------
(i)
Acceleration
due to gravity,
g
= 
=
6.4 ms−2 ------
(ii)
Dividing
(i) by (ii),
r
= 
=
8 × 
=
8000 km
Height
of the point from the earth’s surface
=
8000 – 6400
=
1600 km
2. The radius of the earth is 6.37 × 
its mean density is 5.5 × 
and G
= 6.66 × 
. Determine the gravitational potential
on the surface of the earth.
Solution:
Here,
R = 6.37 × ,
= 5.5 × ,
G
= 6.66 × 
Mass
of the earth,
M =
Volume × density
=
Gravitational
potential on the earth’s surface
V
=
= 
= 
= 
= −
6.22 × 107 J Kg−1
3. The gravitational
field intensity at a point 10,000 km from the centre of the earth is 
.
Calculate the gravitational potential at that point.
Solution:
Gravitational field intensity, E = 
Gravitational potential, V = 
∴ 
= R
V = 
× R
= −4.8 × 10,000 × 103
= −4.8 × 107 J Kg−1
4. The radius of the earth is 
and
the acceleration due to gravity at its surface is g. Calculate the work required in raising a body of mass m to a height h from the surface of the earth.
Solution:
Let
M be the mass of the earth. Then
Workdone = change in P.E.
=
=
=
=
=
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