- The document discusses Newton's law of gravitation and Kepler's laws of planetary motion. - Newton's law of gravitation states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. - Kepler's laws describe the motion of planets in the solar system, including that planets move in elliptical orbits with the sun at one focus, their radius vectors sweep out equal areas in equal times, and the squares of their orbital periods are proportional to the cubes of their semi-major axes.

1. Gravitation
1.1 : Introduction
Copernicus gave his hello-centric theory in 1543, according to
which the sun is at the centre of the planetary system and the
planets move round the sun in closed orbits. Earlier the
scientist A. D. Ptolemy, 2000 years ago proposed a geocentric
theory for planetary motion, in which earth was assumed to
be centre of the universe and that the sun, stars, planets and
the moon all moved around it. Tycho Brahe made the clear
observations for the motion of the planets round the sun.
From his data, Kepler gave three laws of planetary motion. In
1666, Newton derived his universal law of gravitation from the
Kepler's laws.
1.2 : Newton's Law of Gravitation
The law states that, "Every particle in this universe attracts
every other particle with the force which is directly
proportional to the product of their masses and inversely
proportional to the square of the distance between their
centres." Mathematically,
F = G
𝑚1𝑚2
𝑟2
Where m1 and m2 are masses of the two bodies, r is the
distance between their centres, F is the force with which the
bodies attract each other and G is the universal constant of
gravitation or gravitational constant.

2. If in Eqn. (1.1), m1 = m2 = 1 kg. and r = 1 m, then F = G.
Thus, the gravitational constant is the gravitational force of
attraction between two bodies each of unit mass separated by
a unit distance. Value of G is 6.667 x 10-11
Nm2
kg-2
.
The dimensions of G can be obtained as follows.
F = G
𝑚1𝑚2
𝑟2 ∴ 𝐺 =
𝐹×𝑟2
𝑚1𝑚2
Dimensions of G =
[M1L1T−2] [𝑀0𝐿2𝑇0]
[𝑀1𝐿0𝑇0] [𝑀1𝐿0𝑇0
= [𝑀−1
𝐿3
𝑇−2
The Newton's law of gravitation is universal in nature. It is
valid, whatever be the nature of the two attracting masses,
their chemical composition, temperature, chemical reactions
taking place inside the masses and medium between the two
masses. It holds good for huge interplanetary distances to
extremely small distances. The law does not holds good for
inter atomic distance (when r < 10-7 cm) and also for very high
velocities, when the masses are moving about in the space.
Einstein explained this deviation on the basis of his theory of
relativity. With the help of law of gravitation and the three
laws of motion, Newton explained the motion of the planets
and deduced Kepler's laws. The law was helpful to discover
new planets Uranus, Neptune and Pluto.

3. 1.3 : Motion of a Particle in a Central Force Field Central
force : A force is said to be central force if it is always directed
towards or away from a fixed point 0 and its magnitude only
depends on the distance from that fixed point. The particle is
said to move in central force field if the force acting on the
particle satisfies properties of central force. The fixed point is
referred as the centre of force.
Fig. 1.1 : Central force field
Mathematically, F is central force only if 𝐹
⃗ (r) =𝑛
⃗⃗ 𝐹
⃗ (r)
Where n ▪is a unit vector along position vector r- with respect
to the fixed point (i. e. origin)
If F (r) < 0, the force is attractive force and if F (r)> 0, the force
is repulsive force. The gravitational force between two point
masses and the force between two unlike electric charges are
central attractive forces, while force between two like electric
charges is a repulsive central force. If a particle moves in a
central force field then the following properties hold :

4. 1. The path of the particle must be a plane curve, i.e., it must
lie in a plane.
2. The angular momentum of the particle is conserved, i.e.,
angular momentum is constant in time.
3. The particle moves in such a way that the position vector
(from the point 0) sweeps out equal areas in equal times. In
other words, areal velocity is constant. This is referred to as
the law of areas.
We will restrict to conservative central forces, where potential
energy V (r), is function of r only, hence the force is always
directed along r .
The central force is given as 𝐹
⃗ (r) =𝑛
⃗⃗ 𝐹
⃗ (r)
Since the central force is conservative, we can write
F(r)= -∇
⃗
⃗⃗ V(r)
Where V (r) is a scalar function.
The curl of a gradient of a scalar function is always zero
i.e. ∇
⃗
⃗⃗x ∇
⃗
⃗⃗V(r)=0
∇
⃗
⃗⃗x F(r)=0
Thus, a central force whose magnitude is a function of distance
from the centre is conservative one, for which the principle of
conservation of total energy holds good.

5. Conservation of angular momentum : If a particle is moving
under the influence of central force 𝐹
⃗ (r) =𝑛
⃗⃗ 𝐹
⃗ (r),
so that torque acting on it is given by,
𝑟
⃗=
𝑑𝐿
𝑑𝑡
= 𝑟
⃗ 𝑥 𝐹
⃗ (r)= 𝑟
⃗ 𝑥 𝑛
⃗⃗ 𝐹
⃗ (r)=0
𝑛
⃗⃗ is unit vector along 𝑟
⃗ Hence 𝑟
⃗ x 𝑛
⃗⃗= 0
𝑟
⃗=
𝑑𝐿
𝑑𝑡
=0
Change in angular momentum is constant and derivative of
constant is zero.
Where 𝐿
⃗⃗ is the angular momentum about the origin.
Therefore L = constant (vector) Thus, when a particle moves
under the action of a central force, its angular momentum 𝐿
⃗⃗
is conserved i.e., 𝐿
⃗⃗ remains the constant in
magnitude and direction if the torque 𝑟
⃗ acting on the particle
is zero.
But 𝐿
⃗⃗ =𝑟
⃗⃗⃗ x p
⃗⃗ ------------1.4
where p
⃗⃗ is the linear momentum.
Taking dot product with 𝑟
⃗⃗⃗ of the both sides of equation (1.4),
we get
𝑟
⃗⃗⃗. 𝐿
⃗⃗ =𝑟
⃗⃗⃗. (𝑟
⃗⃗⃗𝑥 p
⃗⃗) = 𝑟
⃗⃗⃗𝑥 𝑟
⃗⃗⃗⃗. p
⃗⃗=0----1.5
As in a scalar triple product the positions of dot and cross can
be

6. interchanged and 𝑟
⃗⃗⃗𝑥 𝑟
⃗⃗⃗⃗ = 0 .
Hence 𝑟
⃗⃗⃗ is perpendicular to the constant vector 𝐿
⃗⃗ i.e., the
motion of a particle takes place in a plane of central force.
Conservation of areal velocity :
Let, 0 be the centre of force. When the vector r changes to r +
∇r, in time dt, the area swept by the radius vector in this time
is
d𝐴
⃗=
1
2
𝑟
⃗⃗⃗𝑥 𝑑𝑟
⃗⃗⃗⃗⃗⃗
Fig. 1.2 : Area swept by the radius vector.
This area is swept in time dt, therefore dividing both sides of
equation by dt and taking limit as dt → 0, we get ,
𝑑𝐴
⃗
𝑑𝑡
=
1
2
𝑟
⃗⃗⃗𝑥
𝑑𝑟
⃗
𝑑𝑡
as
𝑑𝑟
⃗
𝑑𝑡
= 𝑉
=
1
2
𝑟
⃗⃗⃗ 𝑥 𝑣
⃗⃗⃗⃗ =
1(𝑟
⃗⃗⃗ 𝑥 𝑚𝑣
⃗⃗⃗⃗)
2𝑚
=
1
2𝑚
(𝑟
⃗⃗⃗ 𝑥 𝑝
⃗⃗⃗⃗)

7. Multiply and devide by m
𝑑𝐴
⃗
𝑑𝑡
=
𝑑𝐿
⃗⃗
2𝑚
Here
𝑑𝐴
⃗
𝑑𝑡
gives the areal velocity of the particle. But angular
momentum 𝐿
⃗⃗ is constant for the motion under central force.
Therefore, the areal velocity remains constant.
1.4 : Kepler's Laws:
Johannes Kepler's three laws of planetary motion are stated
below.
1. Law of Elliptical Orbits : All planets move in elliptical orbits
with the sun situated at one of the foci of the ellipse. This is
known as the law of elliptical orbits.
2. Law of areas : The radius vector, from the sun to the planet
sweeps equal area in equal time. That is the areal velocity of
the planet (area swept by the radius vector of a planet per unit
time) is constant. This is known as law of equal areas.
Fig. 1.3 : Kepler's laws

8. 3. Law of periods : The square of the time period of the
revolution of the planet is proportional to the cube of the
semi-major axis of the ellipse traced out by the planet. This is
known as harmonic law.