Page 1
133
6.1 Newton’s Law of Gravitation
Newton’s law of gravitation states that every body in this 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 their centres. The direction of the force is along the line joining the
particles.
Thus the magnitude of the gravitational force F that two particles of masses
m
1
and m
2
separated by a distance r exert on each other is given by
or F =
Also clear that Which is Newton’s third law of motion.
Here G is constant of proportionality which is called ‘Universal gravitational
constant’.
(i) The value of G is 6.67 × 10
–11
N-m
2
kg
–2
in S.I and 6.67 × 10
–8
dyne
–
cm
2
g
–2
in C.G.S. system.
(ii) Dimensional formula [M
–1
L
3
T
–2
].
(iii) The value of G does not depend upon the nature and size of the bodies.
(iv) It does not depend upon the nature of the medium between the two bodies.
6.2 Acceleration Due to Gravity
The force of attraction exerted by the earth on a body is called gravitational
pull or gravity.
The acceleration produced in the motion of a body under the effect of gravity
is called acceleration due to gravity, it is denoted by g.
If M = mass of the earth and R = radius of the earth and g is the acceleration
Page 2
133
6.1 Newton’s Law of Gravitation
Newton’s law of gravitation states that every body in this 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 their centres. The direction of the force is along the line joining the
particles.
Thus the magnitude of the gravitational force F that two particles of masses
m
1
and m
2
separated by a distance r exert on each other is given by
or F =
Also clear that Which is Newton’s third law of motion.
Here G is constant of proportionality which is called ‘Universal gravitational
constant’.
(i) The value of G is 6.67 × 10
–11
N-m
2
kg
–2
in S.I and 6.67 × 10
–8
dyne
–
cm
2
g
–2
in C.G.S. system.
(ii) Dimensional formula [M
–1
L
3
T
–2
].
(iii) The value of G does not depend upon the nature and size of the bodies.
(iv) It does not depend upon the nature of the medium between the two bodies.
6.2 Acceleration Due to Gravity
The force of attraction exerted by the earth on a body is called gravitational
pull or gravity.
The acceleration produced in the motion of a body under the effect of gravity
is called acceleration due to gravity, it is denoted by g.
If M = mass of the earth and R = radius of the earth and g is the acceleration
due to gravity, then
? g =
(i) Its value depends upon the mass radius and density of planet and it is
independent of mass, shape and density of the body placed on the surface
of the planet.
(ii) Acceleration due to gravity is a vector quantity and its direction is always
towards the centre of the planet.
(iii) Dimension [g] = [LT
–2
]
(iv) It’s average value is taken to be 9.8 m/s
2
or 981 cm/sec
2
, on the surface
of the earth at mean sea level.
6.3 V ariation in g with Height
Acceleration due to gravity at height h from the surface of the earth
g =
Also g' =
= [As r = R + h]
(i) If h << R g' =
(ii) If h << R. Percentage decrease .
6.4 V ariation in g with Depth
Acceleration due to gravity at depth d from the surface of the earth
g' =
also g' =
(i) The value of g decreases on going below the surface of the earth.
(ii) The acceleration due to gravity at the centre of earth becomes zero.
Page 3
133
6.1 Newton’s Law of Gravitation
Newton’s law of gravitation states that every body in this 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 their centres. The direction of the force is along the line joining the
particles.
Thus the magnitude of the gravitational force F that two particles of masses
m
1
and m
2
separated by a distance r exert on each other is given by
or F =
Also clear that Which is Newton’s third law of motion.
Here G is constant of proportionality which is called ‘Universal gravitational
constant’.
(i) The value of G is 6.67 × 10
–11
N-m
2
kg
–2
in S.I and 6.67 × 10
–8
dyne
–
cm
2
g
–2
in C.G.S. system.
(ii) Dimensional formula [M
–1
L
3
T
–2
].
(iii) The value of G does not depend upon the nature and size of the bodies.
(iv) It does not depend upon the nature of the medium between the two bodies.
6.2 Acceleration Due to Gravity
The force of attraction exerted by the earth on a body is called gravitational
pull or gravity.
The acceleration produced in the motion of a body under the effect of gravity
is called acceleration due to gravity, it is denoted by g.
If M = mass of the earth and R = radius of the earth and g is the acceleration
due to gravity, then
? g =
(i) Its value depends upon the mass radius and density of planet and it is
independent of mass, shape and density of the body placed on the surface
of the planet.
(ii) Acceleration due to gravity is a vector quantity and its direction is always
towards the centre of the planet.
(iii) Dimension [g] = [LT
–2
]
(iv) It’s average value is taken to be 9.8 m/s
2
or 981 cm/sec
2
, on the surface
of the earth at mean sea level.
6.3 V ariation in g with Height
Acceleration due to gravity at height h from the surface of the earth
g =
Also g' =
= [As r = R + h]
(i) If h << R g' =
(ii) If h << R. Percentage decrease .
6.4 V ariation in g with Depth
Acceleration due to gravity at depth d from the surface of the earth
g' =
also g' =
(i) The value of g decreases on going below the surface of the earth.
(ii) The acceleration due to gravity at the centre of earth becomes zero.
(iii) Percentage decrease .
(iv) The rate of decrease of gravity outside the earth (if h << R) is double to
that of inside the earth.
6.5 Gravitational Field
The space surrounding a material body in which gravitational force of
attraction can be experienced is called its gravitational field.
Gravitational Field intensity : The intensity of the gravitational field of a
material body at any point in its field is defined as the force experienced by
a unit mass (test mass) placed at that point. If a test mass m at a point in a
gravitational field experiences a force
6.6 Gravitational Potential
At a point in a gravitational field potential V is defined as negative of work
done per unit mass in shifting a test mass from some reference point (usually
at infinity) to the given point.
Negative sign indicates that the direction of intensity is in the direction where
the potential decreases.
Gravitational potential V =
6.7 Gravitational Potential Energy
The gravitational potential energy of a body at a point is defined as the
amount of work done in bringing the body from infinity to that point against
the gravitational force.
W =
This work done is stored inside the body as its gravitational potential
energy
? U =
If r = 8 then it becomes zero (maximum).
6.8 Escape V elocity
The minimum velocity with which a body must be projected up so as to
enable it to just overcome the gravitational pull, is known as escape velocity.
I
Page 4
133
6.1 Newton’s Law of Gravitation
Newton’s law of gravitation states that every body in this 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 their centres. The direction of the force is along the line joining the
particles.
Thus the magnitude of the gravitational force F that two particles of masses
m
1
and m
2
separated by a distance r exert on each other is given by
or F =
Also clear that Which is Newton’s third law of motion.
Here G is constant of proportionality which is called ‘Universal gravitational
constant’.
(i) The value of G is 6.67 × 10
–11
N-m
2
kg
–2
in S.I and 6.67 × 10
–8
dyne
–
cm
2
g
–2
in C.G.S. system.
(ii) Dimensional formula [M
–1
L
3
T
–2
].
(iii) The value of G does not depend upon the nature and size of the bodies.
(iv) It does not depend upon the nature of the medium between the two bodies.
6.2 Acceleration Due to Gravity
The force of attraction exerted by the earth on a body is called gravitational
pull or gravity.
The acceleration produced in the motion of a body under the effect of gravity
is called acceleration due to gravity, it is denoted by g.
If M = mass of the earth and R = radius of the earth and g is the acceleration
due to gravity, then
? g =
(i) Its value depends upon the mass radius and density of planet and it is
independent of mass, shape and density of the body placed on the surface
of the planet.
(ii) Acceleration due to gravity is a vector quantity and its direction is always
towards the centre of the planet.
(iii) Dimension [g] = [LT
–2
]
(iv) It’s average value is taken to be 9.8 m/s
2
or 981 cm/sec
2
, on the surface
of the earth at mean sea level.
6.3 V ariation in g with Height
Acceleration due to gravity at height h from the surface of the earth
g =
Also g' =
= [As r = R + h]
(i) If h << R g' =
(ii) If h << R. Percentage decrease .
6.4 V ariation in g with Depth
Acceleration due to gravity at depth d from the surface of the earth
g' =
also g' =
(i) The value of g decreases on going below the surface of the earth.
(ii) The acceleration due to gravity at the centre of earth becomes zero.
(iii) Percentage decrease .
(iv) The rate of decrease of gravity outside the earth (if h << R) is double to
that of inside the earth.
6.5 Gravitational Field
The space surrounding a material body in which gravitational force of
attraction can be experienced is called its gravitational field.
Gravitational Field intensity : The intensity of the gravitational field of a
material body at any point in its field is defined as the force experienced by
a unit mass (test mass) placed at that point. If a test mass m at a point in a
gravitational field experiences a force
6.6 Gravitational Potential
At a point in a gravitational field potential V is defined as negative of work
done per unit mass in shifting a test mass from some reference point (usually
at infinity) to the given point.
Negative sign indicates that the direction of intensity is in the direction where
the potential decreases.
Gravitational potential V =
6.7 Gravitational Potential Energy
The gravitational potential energy of a body at a point is defined as the
amount of work done in bringing the body from infinity to that point against
the gravitational force.
W =
This work done is stored inside the body as its gravitational potential
energy
? U =
If r = 8 then it becomes zero (maximum).
6.8 Escape V elocity
The minimum velocity with which a body must be projected up so as to
enable it to just overcome the gravitational pull, is known as escape velocity.
I
136
If v
e
is the required escape velocity, then
(i) Escape velocity is independent of the mass and direction of projection
of the body.
(ii) For the earth, v
e
= 11.2 km/sec
(iii) A planet will have atmosphere if the velocity of molecule in its atmosphere
is lesser than escape velocity. This is why earth has atmosphere while
moon has no atmosphere.
6.9 Kepler’s laws of Planetary Motion
(1) The law of Orbits : Every planet moves around the sun in an elliptical
orbit with sun at one of the foci.
(2) The law of Area : The line joining the sun to the planet sweeps out equal
areas in equal interval of time. i.e., areal velocity is constant. According
to this law planet will move slowly when it is farthest from sun and more
rapidly when it is nearest to sun. It is similar to law of conservation of
angular momentum.
Areal velocity =
(3) The law of periods : The square of period of revolution (T) of any planet
around sun is directly proportional to the cube of the semi-major axis of
the orbit.
T
2
? a
3
or T
2
?
where a = semi-major axis
r
1
= Shortest distance of planet from sun (perigee).
r
2
= Largest distance of planet from sun (apogee).
Page 5
133
6.1 Newton’s Law of Gravitation
Newton’s law of gravitation states that every body in this 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 their centres. The direction of the force is along the line joining the
particles.
Thus the magnitude of the gravitational force F that two particles of masses
m
1
and m
2
separated by a distance r exert on each other is given by
or F =
Also clear that Which is Newton’s third law of motion.
Here G is constant of proportionality which is called ‘Universal gravitational
constant’.
(i) The value of G is 6.67 × 10
–11
N-m
2
kg
–2
in S.I and 6.67 × 10
–8
dyne
–
cm
2
g
–2
in C.G.S. system.
(ii) Dimensional formula [M
–1
L
3
T
–2
].
(iii) The value of G does not depend upon the nature and size of the bodies.
(iv) It does not depend upon the nature of the medium between the two bodies.
6.2 Acceleration Due to Gravity
The force of attraction exerted by the earth on a body is called gravitational
pull or gravity.
The acceleration produced in the motion of a body under the effect of gravity
is called acceleration due to gravity, it is denoted by g.
If M = mass of the earth and R = radius of the earth and g is the acceleration
due to gravity, then
? g =
(i) Its value depends upon the mass radius and density of planet and it is
independent of mass, shape and density of the body placed on the surface
of the planet.
(ii) Acceleration due to gravity is a vector quantity and its direction is always
towards the centre of the planet.
(iii) Dimension [g] = [LT
–2
]
(iv) It’s average value is taken to be 9.8 m/s
2
or 981 cm/sec
2
, on the surface
of the earth at mean sea level.
6.3 V ariation in g with Height
Acceleration due to gravity at height h from the surface of the earth
g =
Also g' =
= [As r = R + h]
(i) If h << R g' =
(ii) If h << R. Percentage decrease .
6.4 V ariation in g with Depth
Acceleration due to gravity at depth d from the surface of the earth
g' =
also g' =
(i) The value of g decreases on going below the surface of the earth.
(ii) The acceleration due to gravity at the centre of earth becomes zero.
(iii) Percentage decrease .
(iv) The rate of decrease of gravity outside the earth (if h << R) is double to
that of inside the earth.
6.5 Gravitational Field
The space surrounding a material body in which gravitational force of
attraction can be experienced is called its gravitational field.
Gravitational Field intensity : The intensity of the gravitational field of a
material body at any point in its field is defined as the force experienced by
a unit mass (test mass) placed at that point. If a test mass m at a point in a
gravitational field experiences a force
6.6 Gravitational Potential
At a point in a gravitational field potential V is defined as negative of work
done per unit mass in shifting a test mass from some reference point (usually
at infinity) to the given point.
Negative sign indicates that the direction of intensity is in the direction where
the potential decreases.
Gravitational potential V =
6.7 Gravitational Potential Energy
The gravitational potential energy of a body at a point is defined as the
amount of work done in bringing the body from infinity to that point against
the gravitational force.
W =
This work done is stored inside the body as its gravitational potential
energy
? U =
If r = 8 then it becomes zero (maximum).
6.8 Escape V elocity
The minimum velocity with which a body must be projected up so as to
enable it to just overcome the gravitational pull, is known as escape velocity.
I
136
If v
e
is the required escape velocity, then
(i) Escape velocity is independent of the mass and direction of projection
of the body.
(ii) For the earth, v
e
= 11.2 km/sec
(iii) A planet will have atmosphere if the velocity of molecule in its atmosphere
is lesser than escape velocity. This is why earth has atmosphere while
moon has no atmosphere.
6.9 Kepler’s laws of Planetary Motion
(1) The law of Orbits : Every planet moves around the sun in an elliptical
orbit with sun at one of the foci.
(2) The law of Area : The line joining the sun to the planet sweeps out equal
areas in equal interval of time. i.e., areal velocity is constant. According
to this law planet will move slowly when it is farthest from sun and more
rapidly when it is nearest to sun. It is similar to law of conservation of
angular momentum.
Areal velocity =
(3) The law of periods : The square of period of revolution (T) of any planet
around sun is directly proportional to the cube of the semi-major axis of
the orbit.
T
2
? a
3
or T
2
?
where a = semi-major axis
r
1
= Shortest distance of planet from sun (perigee).
r
2
= Largest distance of planet from sun (apogee).
137
• Kepler’s laws are valid for satellites also.
6.10 Orbital Velocity of Satellite
? v = [r = R + h]
(i) Orbital velocity is independent of the mass of the orbiting body.
(ii) Orbital velocity depends on the mass of planet and radius of orbit.
(iii) Orbital velocity of the satellite when it revolves very close to the surface
of the planet.
v =
6.11 Time Period of Satellite
T = = [As r = R + h]
(i) Time period is independent of the mass of orbiting body
(ii) T
2
? r
3
(Kepler’s third law)
(iii) Time period of nearby satellite, T =
For earth T = 84.6 minute ˜ 1.4 hr.
6.12 Height of Satellite
h =
6.13 Geostationary Satellite
The satellite which appears stationary relative to earth is called geostationary
or geosynchronous satellite, communication satellite.
A geostationary satellite always stays over the same place above the earth.
The orbit of a geostationary satellite is known as the parking orbit.
(i) It should revolve in an orbit concentric and coplanar with the equatorial
plane.
(ii) It sense of rotation should be same as that of earth.
(iii) Its period of revolution around the earth should be same as that of earth.
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