GRAVITATION

CHAPTER – 8 GRAVITATION

Observation of stars, planets and their motion has been the subject of attention in many
countries since the earliest of times.
The earliest recorded model for planetary motions proposed by Ptolemy about 2000 years
ago was a geocentric model. According to this theory, the sun, the moon and all planets, were in a
uniform motion in circles called epicycles with the motionless earth at the centre. However a more
elegant model in which the sun was the centre around which the planet revolved was mentioned
by Aryabhatta in 5 century AD in his treatise.
In 15 century Nicolas Copernicus proposed a definitive model, the helio-centric theory,
according to which the earth and all other planets move in a circular orbit around the sun. In 16
century Johannes Kepler analyzed the data collected by Tycho Brahe and put forth his discoveries
in the form of three laws known as Kepler’s laws.

Kepler’s laws of planetary motion.

1. Law of orbits: All planets move in elliptical orbits with the sun situated
at one of the foci of the ellipse.

Explanation: and are the foci, is semi major axis and is the semi
minor axis.

2. Law of areas: The line that joins any planet to the sun sweeps equal areas in equal intervals of
time.

Explanation: Let the sun be at one of the foci of the ellipse.

Let the position and momentum of the planet be ⃗ and ⃗ respectively.
Then the area sweep out by the planet of mass m in time t is,

(⃗ ⃗ )

(⃗ ⃗)
⃗⃗
(⃗ )

(⃗ ⃗)

⃗⃗ ( )
⃗⃗
⃗

But ⃗ ⃗ ⃗ where ⃗ is Gravitation Force

But the force ⃗ is central force, i e. ⃗ is along ⃗
⃗ ⃗
⃗⃗

⃗⃗ = constant
Using the above statement in equation (1)

Note: The law of areas is the consequence of conservation of angular momentum.

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3. Law of periods: The Square of the time period of revolution of a planet is proportional to the
cube of the semi major axis of the ellipse traced out by the planet.

Explanation: If is the time period and is semei major axis, then .

Gravitational force: Gravitational force is the force of attraction between the two bodies due to
their masses. It is one of the basic forces of nature and is always attractive.

Gravitation: The tendency of bodies to move toward each other is called gravitation.

Gravity: The attractive force between earth and any other body is called gravity.

Newton’s Universal law of Gravitation: Everybody in the universe attracts every other body with
a force which is directly proportional to the product of their masses and inversely proportional to
the square of the distance between them.

Explanation: If and are the masses of two bodies respectively and are separated by a
distance then,
| ⃗|

|⃗⃗|
where is universal gravitational constant.

Vector form: The force ⃗ is acting on a point mass m2 due to another point
mass m1, and the force is directed towards point mass m1. This is given by,
⃗ ( ̂)
| ⃗|
⃗⃗ ( ̂)
|⃗⃗|
where ̂ is the unit vector from to and ⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗

Note: The gravitational force on point mass m1 due to point mass m2 has the same magnitude as
the force on point mass m2 but the opposite direction.
i.e ⃗ ⃗

Gravitational force due to multiple point masses:

If we have a collection of point masses the force on any one of
them is the vector sum of the gravitational force exerted by the
other point masses.

̂ ̂ ̂

In case of gravitational force on a particle from a real (extended) object, we will divide the
extended object in to deferential parts each of mass dm and each producing deferential force ⃗ on
⃗ ∫ ⃗

Note: If the extended object is a uniform sphere or a spherical shell we can avoid the integration by
assuming that the objects mass in concentrated at the objects centre.
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Determination of Gravitational constant:

In 1798 Henry Cavendish determined the value of G. The experimental arrangement is as shown.

The Bar AB has two small lead sphere attached at it ends. The bar is suspended from a rigid
support by a fine wire. Two large lead spheres and are brought close to the small ones but on
opposite sides. The big sphere attracts the nearby small ones by equal and opposite forces. There is
no net force on the bar but only torque which is equal to the F times the length of the bar and F is
the force of attraction between a big and its neighboring small sphere. Due to this torque the
suspended wire gets twisted such that the restoring torque of the wire equals to the gravitational
torque.

If is the angle of twist, then restoring torque is equal to .

where mass of big sphere, mass of small spheres, length of the bar AB.

The measurement of G has been refined and the currently accepted value is,

Acceleration due to gravity: The acceleration experienced by a body due to gravitational force of
the earth is known as acceleration due to gravity.

Expression for Acceleration due to Gravity:

Let us assume that the earth is uniform sphere of mass ME.
Consider a body of mass m lying on the surface of the earth.
The magnitude of gravitational force acting on the body is given by,

The acceleration experienced by the body of mass m due to gravity is given by

Dependence of Acceleration due to gravity: The above equation suggests that g depends on
(i) mass of the earth and (ii) Radius of the earth

Note: The value of g is independent of mass of the body

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Acceleration due to gravity below and above the surface of earth
(i) Acceleration due to gravity above the surface of earth:

Now consider a body at a height h above the surface of earth.

The acceleration due to gravity at height h is given by,

( )

( )

( )

( )

( )

( )
This shows that the acceleration due to gravity decreases as we go away from the surface of earth.

( )

( ) ( )

( )

(ii) Acceleration due to gravity below the surface of the earth:

Now volume density

( )

When the body of mass is taken to a depth d, the mass of the earth
of radius ( ) will only be effective for the gravitational pull.
The outward shell will have no resultant effect on the mass of the body.
The acceleration due to gravity on the surface of the earth of radius ( ) is given by,

( )

( )

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( )
( )

( )

As we go down below earth’s surface the value of g decreases by a factor ( )

Note: When , ( ) , At the centre of the earth the value of g is zero. The value
of g is more on the surface of the earth.

Gravitational potential energy: Gravitational potential energy of a body at a point is defined as

the work done in displacing the body from infinity to that point in the gravitational field.
Potential energy of the body arising due to gravitational force is called gravitational potential
energy.

Expression for Gravitational potential energy:

Consider a body of mass m placed at a distance x
from the earth of mass ME.
The gravitational force of attraction between the
body and earth is given by ,

Now let the body of mass m be displaced from point C to B through a distance dx towards the
earth, then Work done,

The total work done in displacing the body of mass m from infinity to a distance r towards the
earth can be calculated by integrating the above equation between the limits to .

∫ ∫

∫ * +

( )

The work done is equal to the gravitation potential energy of the body and it is represented by V.

Gravitation potential: The gravitational potential due to the gravitational force of the earth is
defined as the potential energy of a particle of unit mass at that point.

( )

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The unit of gravitational potential is J kg-1 and dimensional formula is [M0L2T-2]

Escape speed: The minimum initial speed required for an object to escape from the earth’s
gravitational field (to reach infinity) is called escape speed.

Expression for Escape speed: Consider an object of mass m is thrown upward so that it can reach
infinity, then the speed there was vf.
The energy of an object is the sum of Potential Energy and Kinetic energy.

( ) ( )
Initially if the object was thrown with a speed vi from point at a distance ( ) from the centre
of the earth, the energy is given by,

( )
( )
By the principle of conservation of energy, equation (1) and (2) are equal.

( )
The RHS of the above equation is positive quantity with a minimum value zero, hence so must be
the LHS.

( )

( )
( )

( )
( )

( )

( )

( ) √

If the object is thrown from the surface of the earth then

( ) √

( ) √

Note: (1) Using the value of g and ( ) for the earth.

(2) ( ) for moon is 2.3 km/s, This is why moon has no atmosphere.

Satellite: Satellites are the celestial objects revolving around the planet.

Earth’s satellites: Earth satellite is an object which revolves around earth.

Earth has only one natural satellite - moon with a time period 27.3 days and its rotational period
about it axis is also same as that of the time period. But earth has so many artificial satellites which
have practical use in fields like telecommunication, geophysics, meteorology, cartography etc.
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Orbital velocity: The velocity required to put a satellite into its orbit around the earth is called
orbital velocity.

Expression for Orbital speed: Consider a satellite of mass m and speed in a circular orbit at a
distance ( ) from the centre of the earth.

( )

( )

( ) ( )

( ) √

√ ( )

Time period of a satellite: In every orbit the satellite travels a distance ( ) with speed ,
then its time period is,
( ) ( )

(√ )
( )

( )( )
√
( )
√

( )
√
This is the expression for time period of a satellite.

Note: Further, squaring on both sides

( )

( )( )

( )

Total energy of an orbiting satellite: The energy of a satellite in its orbit is the sum of the potential
energy due to the gravitational force of attraction and kinetic energy due to the orbital motion.

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Expression for total energy of the satellite:
We have,

( )

( ) ( )

( )
( )

( )

Note: If total energy of an orbiting satellite is equal or greater than zero then the satellite does not
remain in the orbit, it escapes from the earths pull. Negative energy implies that the satellite is
bound to the earth.

Geostationary satellites: Satellites in a circular orbits around the earth in the equatorial plane with
time period T = 24 hours are called Geostationary satellites and the orbit is called Geo-synchronous
orbit.
For geostationary orbit,
1. The time period of the satellite is equal to the rotational period of the earth.
2. The height form the equatorial plane must be about 35800km (nearly equal to 36000 km)
3. Direction of rotation of the satellite must be same as that of the earth.

Use of Geostationary Satellites: They are used for telecommunication purpose.

Note: The INSAT groups of satellites are Geo-stationary satellites.

Polar satellites: The low altitude satellites which go around the poles of the earth in a north south
direction are called polar satellites and the orbit is called polar orbit. The time period of a polar
satellite is about 100 minutes and hence it crosses any altitude many times a day.

Uses of polar satellites:

1. They are used for remote sensing. The IRS group of satellites are Remote sensing Satellites.
2. They are used for environmental studies, and also in the field of meterology.
3. They are used for natural resource survey
4. They are used for forest, waste land, drought assessment etc.

Weightlessness: When there is no normal reaction or upward force on the object from any surface,
then the weight of the object will become zero, this particular situation of the object is termed as
weight less ness.
When an object is in free fall, it is weightless.
While a man or an object accelerating downwards, if the lift is cutoff, feels weightless.
In a satellite revolving round the earth, gravitational force of the earth provides necessary
centripetal force to the satellite and this force is opposite to the force exerted by satellite on the
man, thus the person inside a satellite feels weightlessness.

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