1.
Gravitation
Important Information
1. Newton’s Law of Gravitation
The magnitude of the gravitational force of attraction between two particles of
masses m1 and m2 separated by a distance r is given by
1 2
2
Gmm
F
r
Where G is called universal gravitational constant.
G = 6.67 × 10 - 11 Nm2/Kg2
2. The magnitude of the acceleration due to gravity (g) at the surface of the earth is
given by
2
GM
g
R
Where M is the mass of the Earth of radius R
3. The magnitude of the acceleration due to gravity (g1) at a point at a distance r
(r >R) from the centre of the earth is given by
1
2 2
GM GM
g
r R h
Where h = altitude or the height of the point from the surface of the earth.
1 2 2 g R R
g r R h
4. The critical or orbital velocity (VC) of a satellite, (moving in a circular orbit of a
radius r = R + h), is given by
2
1
C
GM GM gR
V g R h
r R h R h
If the satellite is orbiting very close to the surface of the earth, i.e. if h<<R, then
C
GM
V gR
r
The radius of the orbit
2
c
GM
R h
V

2.
5. The period (T) of a satellite revolving round the earth in a circular orbit of radius
r = R + h is given by
3
C
1
1
R h 2 R h
T 2 or T
GM V
R h
Intermsof g ,T 2
g
6. The binding energy of a body of mass m, when it is at rest on the surface of the
earth is
GMm
B.E
R
7. For a satellite, performing a U.C.M., around the earth.
8. The escape velocity of a body projected from the surface of the earth,
E
2GM
V 2gR
R
9. Kepler’s three laws of planetary motion
i. Each planet revolves around the sun in an elliptical orbit, with the sun at one
of the foci of the ellipse.
ii. The straight line joining the sun and the planet or the radius vector, sweeps
out equal areas in equal intervals of time.
iii. The squares of the periodic times (T2) of the planets about the sun are
proportional to the cubes of the semimajor axis (a) of the elliptical orbits i.e.,
T2 ∝ a3.

3.
10. Period of a satellite
3 2
2 3
2 3
2 2
2
2 3 2 3
R h 4
T 2 T R h
GM GM
R h
GM gR T
gR
T R h or T r

This is Kepler’s law.
11. Gravitational potential energy of a body of mass (m) is
GMm
R
12. An astronaut or anybody inside a satellite feels weightless as there in no reaction
of the satellite upon the astronaut.
13. For a body to escape from the earth’s gravitational influence, K.E. of projection =
Binding energy
14. Gravitational field intensity is the gravitational force per unit mass.
15. Variation of g:
a. With altitude: At a height h,
2 2
h 2
R h
g g g 1
R h R
Where R is the radius of the earth.
But if h << R ( h is very small as compared to R) then
h
2h
g g 1
R
b. With depth:
d
d
g g 1
R
Where d is the depth of the body below the surface of the earth. At the
centre of the earth, gd = 0.
Thus the value of g decreases with increase in height as well as depth.

4.
c. With latitude: (effect of rotation of the earth)
2 2
2
equator
pole
g' g R cos ,where is the latitude
At the equator 0 g g R
At thepoles 90 g g
.
This is the maximum value of g.
d. The equatorial radius (RE) > the polar radius (RP)
P E
1
g g g
R

16. For a communication or a geosynchronous satellite, T = 24 hours. It moves in the
equatorial plane from west to east with a velocity of about 3.1 km/s.
The height of the communication satellite is about 36000 km.
17. Useful Points:
i. Gravitational force is a conservative force. It acts as an action-reaction pair.
The ratio of the gravitational force to the electrostatic force between two
electrons is of the order of 10 - 43.
ii. The period of a satellite orbiting very close to the surface of the earth is
about 84 minutes.
iii. The escape velocity of a body from the surface of the earth is about 11.2
km/s, while for the moon escape velocity is about 2.38 km/s.
iv.
Work W
Gravitational potential
mass m
It is a measured in Joule/kg. It is a scalar quantity.
v. If Vescape > Vrms of the molecules of a gas from a planet, there is an
atmosphere on the planet. If Ve < Vrms, there is no atmosphere on the planet.