The gravitational force is a conservative force and hence we can define a gravitational potential energy associated with this conservative force field.

**Gravitational
Potential Energy**

The
concept of potential energy and its physical meaning were dealt in unit 4. The
gravitational force is a conservative force and hence we can define a
gravitational potential energy associated with this conservative force field.

Two
masses m_{1} and m_{2} are initially separated by a distance *r*â€². Assuming m_{1} to be fixed
in its position, work must be done on m_{2} to move the distance from *r*â€² to *r* as shown in Figure 6.12(a)

To
move the mass *m*_{2} through
an infinitesimal displacement *d* from * to * + *d* (shown in the
Figure 6.12(b)), work has to be done externally. This infinitesimal work is
given by

The
work is done against the gravitational force, therefore,

Substituting
Equation (6.22) in 6.21, we get

Also
we know,

Thus
the total work done for displacing the particle from *r*â€² to *r* is

This
work done W gives the gravitational potential energy difference of the system
of masses m_{1} and m_{2} when the separation between them are *r* and *r*â€² respectively.

Since
gravitational force is attractive, m_{2} is attracted by m_{1}.Then
m_{2} can move from *r* to *r*â€² without any external work (Figure
6.13). Here work is done by the system spending its internal energy and hence
the work done is said to be negative.

Work
has to be done against gravity to move the object from *r*â€² to r. Therefore work is done on the
body by external force and hence work done is positive.

It
is to be noted that only potential energy difference has physical significance.
Now gravitational potential energy can be discussed by choosing one point as
the reference point.

Let
us choose *râ€™=âˆž* . Then the second term
in the equation (6.28) becomes zero.

Now
we can define gravitational potential energy of a system of two masses m_{1}
and m_{2} separated by a distance r as the amount of work done to bring
the mass m_{2} from infinity to a distance r assuming m_{1} to
be fixed in its position and is written as

It is to be noted that the gravitational potential energy of the system consisting of two masses m1 and m2 separated by a distance r, is the gravitational potential energy difference of the system when the masses are separated by an infinite distance and by distance r. U ( r ) = U ( r ) - U ( âˆž ). Here we choose U ( âˆž )= 0 as the reference point. The gravitational potential energy U( r ) is always negative because when two masses come together slowly from infinity, work is done by the system.

The
unit of gravitational potential energy *U (
r ) *is Joule and it is a scalar quantity.* *The gravitational potential energy depends upon the two masses and
the distance between them.

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11th Physics : UNIT 6 : Gravitation : Gravitational Potential Energy |

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