GRAVITATION
1. Kepler's laws of planetary motion :
Gravitational force holds the entire solar system together and makes possible orbiting earth
satellites. Such motions are governed by Kepler's three laws, which are direct consequence of
Newton's law of motion and gravitation.
(i) The law of orbit :
Each planet revolves around sun in an elliptical orbit with the sun is at one of the
focus.
(ii) The law of areas :
Line joining any planet to the sun sweeps out equal areas in equal times.
Or angular momentum of planet about sun remains constant.
(iii) The law of periods :
The square of the period T of any planet about the sun is proportional to the cube of
the semi-major axis a of the orbit. For circular orbits with radius r, the semimajor axis
a is replaced by r and the law is written as
4 2 3
T2 = r
GM
2. Newton's law of gravitation :
Every particle in the universe attracts every other particle with a gravitational force whose
magnitude is given by
m1m 2
F=G
r2
Think about variation of g in different cases (write the formula ypurself)
(a) Height
(b) Depth
(c) Shape
(d) Rotation
3. Shell theorem
(i) If a point mass lies outside the spherical shell/sphere with a spherical symmetric
internal mass distribution, the shell/sphere attracts the point mass as if the whole mass
of the shell/sphere were concentrated at its centre.
1
(ii) If the point mass lies inside the uniform spherical shell, the gravitational force on the
point mass is zero.
4. Gravitational shielding is not possible.
5. Gravitational forces obey the principle of superposition; that the force F1 on a particle is the
sum of the forces exerted on it by all other particles.
n
F1 = F1i
i= 2
6. Intensity of gravitational field on the surface of the earth :
GM
ag = (gravitational acceleration)
R2
Assuming earth is a uniform sphere of radius R.
7. Free fall acceleration :
Earth is not uniform and also rotating, so the free fall acceleration g on the earth differs
slightly from the gravitational acceleration ag.
8. Effect of rotation:
If is the angular velocity of earth about its axis, then value of acceleration due to gravity at a
latitude angle λ is given by
g; = g – 2Rcos2λ
(i) At poles g' = g
(ii) At equator g' = g - 2R
The body on the equator will fly off if the time period of rotation of earth become 1.41h, or
rotation velocity become 17 times the present value of rotation.
9. Gravitational potential energy:
The gravitational potential energy U of two particles, with masses M and m and separated at
a distance r is given by
−GMm
U=
r
10. Potential energy of a system of particles :
If a system contains many particles, the total gravitational potential energy is the sum of
terms representing the potential energies of all the pairs; for three particles m1, m2, m3.
Gm1m 2 Gm1m3 Gm 2 m3
U = − + +
r12 r13 r23
−3 GM 2
Self energy of Earth, U =
5 R
2
11. Escape speed :
The speed with which the projected object will escapes from the gravitational pull of an
astronomical body. For an astronomical body of mass M and radius R.
2GM
ve = or ve = 2gR
R
12. Orbital speed :
When a satellite moves in a circular orbit of radius r around earth of mass M, its orbital speed
is given by
GM gR 2
v0 = or v0 =
r r
and time period of revolution
2r r3
T= = 2
v0 GM
For a satellite close to earth, r ≈ R
v0 = gr = 8km / s
R
and T = 2 = 84.6 min .
g
Relation between escape velocity and orbital velocity
𝑣𝑒 = √2
13. If satellite or planet revolves in an elliptical orbit with eccentricity e, then (optional)
1 + e GM 1 − e GM
v max = and v min =
1 − e a 1 + e a
v max 1 + e
= , where a is the semimajor axis.
v min 1 − e
14. Energy of a satellite :
GMm −GMm −GMm
K= ,U = and E = K + U =
2r r 2r
−U
Also K = =E
2
GMm
Binding energy, B.E. =
2r
3
15. Geostationary satellite :
(i) Orbital speed, v0 = 3km / s
(ii) Time period of revolution, T = 24h
(iii) Height of geostationary satellite, h = 36000km
16. Existence of atmosphere : A planet will have atmosphere if vrms < ve.
17. Black hole : A body to be a black hole
ve > c (c is the speed of light)
2GM 2GM
or c or R 2
R c
18. Weightlessness : A body is said to be in a state of weightlessness when the reaction of the
supporting surface is zero.
For a body in a satellite, by Newton's second law
GMm
− N = ma
r2
2
mv02 m GM
− =
r r r
N=0
Thus the surface of satellite exerts no force on the body and hence it experiences
weightlessness.
