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Gravitation
Gravitation is the study of force between the earth
( gravity ) and other bodies
Newton’s law of gravitation
Statement : Every body in the universe attract
other body with the force directly proportional to
product of their masses and inversely
proportional to square of the distance between
them. Force acts along the line joining the
centers of two masses.
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Explanation
Consider two bodies of masses m1 and m2
separated by the distance d.
Then from Newton’s law of gravitation,
F ∝ m1 m2 ---- ( 1 ) and
d F ∝ 1 /d2 ---- (2 )
From ( 1 ) and ( 2 )
m1 m2
F ∝
d2
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m1 m2
F= G
d2
Where G is the constant of proportionality called
gravitational constant.
11 N m2 /kg
It value
Its
l is
i 6.67
6 67 x 10 -11
/k 2
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Gravitational constant
Consider
m1 m2
F = G
d2
F d2
G =
m1 m2
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If m1 = m2 = 1 Kg and d = 1m
then G = F
G it ti
Gravitational
l constant
t t is
i numerically
i ll
equal to force of attraction between two
bodies of unit
nit masses separated b
by unit
nit
distance
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y Newton’s law of gravitation is called universal law,
because the law is not only applicable to the bodies
on earth but also to the heavenly bodies. The
gravitational force between two masses is
independent of any other external conditions like
temperature, pressure, magnetic field, material
medium etc.
y ‘G’ is not a mere number. It has dimension of 3 in
length -1
length,
1 in mass
mass, -2
2 in time
time.
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y Gravitational force is one of the basic forces of
nature and it is the weakest force.
f
y First accurate measurement of G was made by
Cavendish in 1798.
1798
y Newton’s law of gravitation is applicable for both
macroscopic bodies ( planets, stars etc) and
microscopic bodies ( electrons, protons etc ).
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y Newton’s law of gravitation does not violate
Newton’s third law of motion
y If F12 is the force exerted by body 1 on body 2 and
F21 is the force exerted by body 2 on body 1 then,
F21 = - F12
They form action – reaction pair
- ve sign
i shows
h
th
thatt F12 is
i opposite
it iin
direction to F21
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Relation between g and G
G M
g =
R2
g is independent of mass of the body
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y g ∝ 1/ R2 . Acceleration due to gravity is inversely
proportional to square off radius off planet.
y Equatorial
E
t i l radius
di off earth
th iis more th
than th
the polar
l
radius. Hence g is more at poles than the equator.
y Body is carried from equator to pole, then its mass
remains same but its weight
g increases.
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y Acceleration due to gravity is independent of mass,
size or shape off the falling
f
body.
y Above
Ab
th
the surface
f
off earth
th g varies
i (d
decrease )
inversely as the square of distance from centre of
earth. But inside the earth it decrease linearly with
distance from centre of earth.
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y Galileo showed that, gravitation accelerates all
objects at the same rate.
y This was a major departure from Aristotle’s belief
that heavier objects accelerate faster and lighter
that,
objects accelerate slower.
y Galileo correctly postulated that “air
air resistance
resistance” as
reason that lighter objects may fall more slowly in an
atmosphere.
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y Rate of decrease of acceleration due to gravity with
height is twice as compared to that with depth.
y Value
V l off acceleration
l ti d
due tto gravity
it iis minimum
i i
ffor
the planet mercury and maximum for the planet
Jupiter.
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Variation of g with altitude ( height )
g1 = g ( 1- 2h / R )
From above equation, it is found that acceleration
due to gravity decreases with increase of altitude.
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Variation of g with depth
g1 = g ( 1- h / R )
From above equation, it is found that acceleration
due to gravity decreases with increase of depth.
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Variation of g with latitude
g1 = g - ω2R Cos2θ
At equator θ = 0°
g1 = g - ω2R. Hence g is minimum
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At poles θ = 90˚
g1 = g - ω2R Cos2 90˚
g1 = g - ω2R . 0
g1 = g
Hence g is maximum
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y If the rate of rotation of earth increases, the value of
acceleration due to gravity decreases at all places
on the surface of earth except at poles.
y If earth stops rotating, at equator acceleration due to
gravity increases by 0.034m/s2 . But at poles there is
no change in acceleration due to gravity.
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Kepler’s laws of planetary motion
First law : All planets revolve around the sun in
elliptical orbits with sun at one of its focus.
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Second law :
Planets revolve around the sun in such a way that
i
imaginary
i
liline d
drawn ffrom sun tto planet
l
t sweeps outt
equal area in equal time intervals.
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Third law : Square of the period of rotation of the
planet revolving round the sun is directly proportional
to cube of their mean distances from the sun.
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Orbital velocity : It is the velocity with which a
satellite revolve around the earth in a particular orbit.
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Expression for orbital velocity is
GM
GM
Vo =
or
(R+h)
r
where r = R + h
h = height of satellite from the surface of earth
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It is also given by
g R2
Vo =
r
Orbital velocity is independent of the mass of body
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Expression for period of revolution of satellite is
2πr
T=
2π ( R + h )
=
Vo
Vo
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Escape velocity
It is the minimum velocity with which an object is
projected from the surface of planet so that it
escapes from the gravitational force of attraction of
the planet.
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Expression for escape velocity is
2GM
Ve =
R
or Ve = √ 2 g R
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y The value of escape velocity from the surface of
earth is 11.2
11 2 Km/s.
/
y Escape
E
velocity
l it iis iindependent
d
d t off th
the angle
l th
through
h
which body is projected.
y Escape velocity is independent of mass of the body.
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Geo – Stationary satellite
y Geo - stationary satellite : Geo stationary satellite
appears to be stationary over the same place to an
observer on earth. They are also called synchronous
satellites.
satellites
y Geo – stationary satellite is one whose orbital period
is same as the p
period of rotation of earth about its
own axis. Hence satellite appears to be stationary to
the observer on earth.
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y Geo – stationary satellite has height of about
36,000Km
36
000
above the equator.
y It orbits round the earth from west to east ( just like
the earth )).
y Its orbital velocity is nearly 3.1Km/s.
y Plane of orbit of geo stationary satellite is normal to
axis of rotation of earth.
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y Geo – stationary satellites are communication
satellites. They provide relay facilities
f
for
f
international communication, internal
communication TV
communication,
TV, radio communication etc
etc.
y APPLE was the first Indian experimental
communication satellite launched in the y
year 1981.
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y INSAT is the series of Indian communication
satellites.
y INSAT – 1A,
1A 1B
1B, 1C
1C, 1D
1D, 2A
2A, 2B
2B, 2C and
d 2D are
Indian other communication satellites.
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I R S Satellites
y Remote sensing is a process by which objects on
earth are sensed ( acquisition of information )
remotely by satellites or rockets
rockets.
y Remote sensing satellites are used in the filed of
Agriculture, Forestry, mineral location, Geology,
Meteorology,
gy Hydrology
y
gy etc.
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y Bhaskara – 1 was the first Indian remote sensing
satellite launched in the year 1979.
y ISRO had developed operational remote sensing
satellites The first in the series is IRS – 1A.
satellites.
1A
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y The mass of the body which measure gravitational
force of attraction with earth is called gravitational
mass.
mass
y The mass of the body which measure its inertia is
called inertial mass.
y Gravitational mass and inertial mass are same.
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Weightlessness
y For a body in an artificial satellite, the
gravitational force of attraction is balanced
by the centrifugal force due to rotation i.e
gravity is not felt. This situation is called
weightlessness
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