MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Gravitation Page 3

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13.

Binding Energy of a Satellite

Definition: The minimum energy that must be provided to a satellite to free it from the gravitational influence of the earth is called its binding energy.

1. Expression for binding energy when body is at rest on the surface of the earth
Total Energy of the body = K.E. + P.E.

K.E. of rotation = 0
P.E. = gravitational potential × mass

 = - GM ´ m R

 = - GMm R

 \ T.E. = - GMm R

B.E.

= Energy necessary to free the body from the gravitation field
=
- (T.E.)

\B.E.

 = GMm R

2. Expression for binding energy when body is orbiting

When a body is revolving around the earth at height 'h' above the surface of the earth

\ Total energy of the satellite = K.E. + P.E.

Now,

K.E. =

 1 mVc2 2

=

 1 m . GM 2 r

=

 GMm 2r

P.E.

= mass ´ gravitational potential

 = m ´ - GM = - GMm r r

\ Total energy, T.E. =

 GMm + - GMm 2r r

\T.E. =

 - GMm 2r

The negative sign indicates that the satellite is bound to the earth by the gravitational force.

Binding energy = Energy necessary to free the satellite from the gravitational field.

 \ Binding energy = - (T.E.) = GMm 2r

 \ B.E. = GMm 2(R + h)

14.

Escape Velocity of a Body from the Surface of the Earth

Definition: The minimum velocity with which a body should be projected from the surface of the earth so that it escapes the gravitational field of the earth is called the escape velocity of the body.

Expression for Escape Velocity (Ve)

When body is at rest on the surface of the earth:

 Binding energy = GMm 2r

For a body projected from the surface of the earth with escape velocity Ve , the kinetic energy of projection is equal to the binding energy.

 \ 1 mVe2 = GMm 2 R

 \ Ve =

Alternate expressions for Ve:

 Ve = = =
1. Ve = Vc where Vc is the critical velocity when body is orbiting very close to the surface of the earth.
 Ve = = =

Remember!
When a body is revolving around the earth at height '
h' above the surface of the earth:

 1 mVe2 = GMm 2 2r

 \ Ve = =

 15. Weightlessness in a Satellite The weight of a body is equal to the gravitational force of attraction. On the earth, the astronaut exerts a force on the surface of the earth. The earth exerts a reaction force on him. Hence, he feels his weight. In an orbiting satellite, the astronaut and the satellite have the same centripetal acceleration. The astronaut does not exert a force on the floor and in return does not experience a reaction force. Hence, he feels weightlessness.

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