MICA PU COLLEGE, Mysuru Kiran Rao G

Gravitation
Gravitation:

It is the force of attraction between any two bodies in the universe by virtue of their masses.

Universal law of gravitation (or) Newton’s law of gravitation

Statement

Every body 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
m1 m2
F

Let m1 and m2 be the masses of two bodies separated by a distance d.

Let F be the force of attraction between them.

From Newton’s law of gravitation

Where G is the universal gravitational constant

Newton’s law of Gravitation in vector form

⃗ ̂

Where ̂ is the unit vector in the direction of ⃗

d is the distance between the bodies

m1 and m2 are the masses of two bodies

G is the universal gravitational constant

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Newton’s law of gravitation is called as universal law of gravitation. Why?

Newton’s law of gravitation is applicable for all bodies in the universe and hence it is called as universal law
of gravitation.

Gravitational constant (G):

The gravitational constant is defined as the force of attraction between the two bodies each of mass 1 Kg
and separated by a distance of 1 meter.

If m1 = m2 = 1 Kg, d = 1m then G = F

 SI unit of gravitational constant is N m2 Kg – 2

 Dimensional formula is G = [M – 1 L3 T – 2 ]
 G = 6.67 X 10 – 11 N m2 Kg – 2
 Among the basic forces in nature, the gravitational force is the weakest force.

Why gravitational constant is called as universal gravitational constant?

The value of G does not depend on

 The nature and size of the bodies

 The nature of the medium between the two bodies
 The physical conditions such as temperature, pressure etc. of surrounding medium.

There fore gravitational constant is called as universal gravitational constant.

Gravity:

It is the force of attraction between the earth and any object near the surface of earth.

What is meant by weight of the body?

Weight of the body means the gravitational force with which a body is attracted towards the center of the
earth

W=mg

Where W is the weight of the body

m is the mass of the body

g is the acceleration due to gravity

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Acceleration due to gravity (g):

It is the acceleration produced in the motion of body due to gravity.

Derive the relation between acceleration due to gravity (g) and gravitational constant (G)

R
M

In the above figure

M is the mass of the earth, m is the mass of the body, R is the radius of the earth.

From Newton’s law of gravitation

But F = mg

The value of g is 9.8 m/s2

Derive the expression for acceleration due to gravity at a point above the surface of the earth

A gh
h

B g
R
C M

In the above figure

A is a point at height h from the surface of the earth, B is the point on the surface of the earth, C is the
center of earth, M is the mass of the earth, R is the radius of the earth.

Acceleration due to gravity at the point A is given by

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( )
( )

Where G = universal gravitational constant

Acceleration due to gravity on surface of the earth is given by

( )

( )
( ) ( )

( )

( )

( )

On expanding above equation using Binomial theorem and neglecting higher order terms, we get

( )

Note: The acceleration due to gravity decreases as height increases.

When ( )

Derive the expression for acceleration due to gravity at a point below the surface of the earth

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In the above figure

A is a point on the surface of the earth, B is the point at a depth d from the surface of the earth, C is the
center of earth, M is the mass of the earth of radius R with respect to the point A. M I is the mass of shaded
point of the earth of radius (R – d).

Acceleration due to gravity at the point A

( )

Where G = universal gravitational constant

Acceleration due to gravity at the point B

( )
( )

( )
( ) ( )

( )
( )

We have

Mass of the earth of radius R = Density X volume of the sphere of radius R

( )

Mass of the earth of radius (R – d) = Density X volume of the sphere of radius (R – d)

( ) ( )

Eqn (4) and eqn (5) in eqn (3)

( )
( )

( )

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( )

Note: The acceleration due to gravity decreases as the depth increases.

What is the acceleration due to gravity at the center of the earth?

Acceleration due to gravity at the center of the earth is zero.

Gravitational Potential energy:

The gravitational potential energy at a point in the gravitational field is the work done in displacing the body
from infinity to that point.

Expression for gravitational potential energy

In the above figure

M is the mass of the earth. m is the mass of the particle placed at the point A. O is the center of the earth, R
is the radius of the earth, P is the point at a distance d from the center of the earth. A and B are the two
points separated by small distance dx, X is the distance between the earth and the particle.

From Newton’s law of gravitation, the force between the earth and unit mass body is given by

( )

Work done in displacing the particle from A to B is

( )

eqn (1) in eqn (2)

( )

Work done in displacing the particle from X = to X = d

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n
n
KT

[ ]

[ ]

Work done in displacing the particle from X = to X = R (Surface of the earth)

This work done is equal to the gravitational potential energy

Where negative sign indicates that the work is done by the gravitational force.

Escape speed (or) Escape Velocity:

Escape speed is the minimum speed with which a body is projected vertically upwards in order to escape
from the gravitational pull of earth.

Escape energy:

It is the minimum KE with which a body is projected vertically upwards in order to escape from the
gravitational pull of earth.

Where V0 is the escape speed of the projected body and

m is the mass.

Expression for escape speed:

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But

Where g = acceleration due to gravity

R = radius of the earth

Satellites:

A satellite is an object which revolves around the planet in circular or elliptical orbit.

Earth satellites:

Earth satellites are objects which revolve around the earth.

Polar satellites:

Polar satellites are the satellites that go around the earth poles of the earth in a north south direction.

Example: IRS satellites (Indian Remote Sensing satellites)

Geostationary satellites:

Geostationary satellites are the satellites that go around the circular orbit of the earth.

 The period of geostationary satellite is 24 hrs.

 Geostationary satellites are widely used for telecommunications.

Example:

INSAT series (Indian National Satellite)

APPLE (Ariane Passenger Payload Experiment)

Orbital speed of earth’s satellites:

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It is the velocity of revolution of a satellite around earth in a given orbit.

Note :

 PSLV (Polar satellite launch vehicle)

 GSLV (Geosynchronous satellite launch vehicle)

Expression for orbital speed of earth’s satellite:

Satellit In the above figure

e
m VO  V0 is the orbital speed of the satellite

 m is the mass of satellite

h
r
 h is the height of the satellite from the surface of
the earth.

R  R is the radius of the earth.

M  r is the distance between the center of the earth

and the satellite.

 M is the mass of the earth.

But r = R + h

If the satellite is revolving very close to the surface of earth, then h can be neglected

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But

GM = g R2

The orbital velocity is independent of mass of satellite.

Expression for energy of orbiting satellite:

Satellit In the above figure

e
m VO  V0 is the orbital speed of the satellite

 m is the mass of satellite

h
r
 h is the height of the satellite from the surface of
the earth.

 R is the radius of the earth.

R
 r is the distance between the center of the earth
M
and the satellite.

 M is the mass of the earth.

Satellites possess KE due to its orbital motion and possess PE due to its position with respect to earth
surface.

Energy of a satellite = KE + gravitational PE

E= ( ) ----- (1)

Where VO is the orbital velocity

and r = R + h

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( )

( )
( )

( )

Where negative sign indicates that the satellite is bound to the earth.

Weightlessness:

Weightlessness is the condition of zero weight. (Or)

A body experiencing a free fall towards the center of the earth has no ground reaction to support it. Under
this condition, a body experiences weightlessness.

Kepler’s laws

Kepler’s law of orbits:

All planets move in elliptical orbits with the sun as one of the foci of the

ellipse.

Kepler’s law of areas:

The line that joins any planet to the sun sweeps equal areas in equal intervals

of time. Planets move slower when they are far from the sun than when they

are nearer.

Kepler’s 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 by the planet.

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