144 CBSE New Pattern ~ Physics 11th (Term-I)
08
Gravitation
Quick Revision
1. Universal Law of Gravitation It states that, As, F12 and F21 are directed towards the
every body in this universe attracts every other centres of the two particles, so gravitational
body with a force whose magnitude is directly force is a central force.
proportional to the product of their masses and 3. Principle of Superposition According to
inversely proportional to the square of the this principle, the resultant gravitational force
distance between their centres. F can be expressed in vector addition of all
mm forces, at a point (as shown below).
Gravitational force, F = G 1 2 2
r i.e. F = F12 + F13 + F14 + ... + F1n
where, G is a constant of proportionality and is
known as universal gravitational constant. m4
^
r41 ^ m3
In CGS system, the value of G is r31
6.67 ´ 10 -8 dyne cm 2g -2 and its SI value is F14 F13
^
r21
6.67 ´ 10 -11 N-m 2kg -2 . m1 m2
Dimensional formula for G is [M -1L3T - 2 ]. ^
F1n
F12
2. Vector Form of Newton’s Law of
Gravitation In vector notation, Newton’s law ^
rn 1
of gravitation is written as follows
mm mn
F12 = - G 12 2 r$21 …(i)
r21
Resultant force,
where, F12 = gravitational force exerted on A by æm m m ö
B and r$21 is a unit vector pointing towards A. F = - Gm 1 ç 22 r$21 + 23 r$31 +... + 2n r$n 1 ÷
è r21 r31 rn 1 ø
Negative sign shows that the gravitational force
is attractive in nature. 4. Acceleration due to gravity The
mm acceleration produced in the motion of a body
Similarly, F21 = - G 12 2 r$12 …(ii)
r12 under the effect of gravity is called
acceleration due to gravity ( g ).
where, r$12 is a unit vector pointing towards B. GM
Equating Eqs. (i) and (ii), we have At the surface of the earth, g = 2
R
F12 = - F21
,5. Weight of a body It is the gravitational force 7. Intensity of Gravitational Field at a
with which a body is attracted towards the Point The gravitational force acting per unit
centre of the earth w = mg mass at any point in gravitational field is called
It is a vector quantity and its SI unit is intensity of gravitational field at that point.
newton (N). Intensity of gravitational field at a distance r ,
6. Factors Affecting Acceleration Due to from a body of mass M is given as
Gravity F GM
E = = 2
● Shape of Earth Acceleration due to gravity, m r
1 It is a vector quantity and its direction is
g µ 2
R towards the centre of gravity.
Therefore, g is minimum at equator and 8. Gravitational Potential Gravitational
maximum at poles. potential at a point in the gravitational field is
defined as the amount of work done per unit
● Rotation of Earth about Its Own Axis If
mass in bringing a body of unit mass from
w is the angular velocity of rotation of earth infinity to that point without acceleration.
about its own axis, then acceleration due to W
gravity at a place having latitude l is given i.e. V =-
m
by
F ×d r -GM
g ¢ = g - Rw2 cos 2 l =-ò = - ò E ×d r =
m r
At poles, l = 90° and g ¢ = g . It is a scalar quantity. The unit of gravitational
Therefore, there is no effect of rotation of potential in SI system is Jkg –1 and in CGS
earth about its own axis at poles. system is erg-g –1 .
At equator, l = 0° and g ¢ = g - Rw2 Dimensional formula for gravitational potential
The value of g is minimum at equator. is [M 0L2T -2 ].
If earth stops its rotation about its own axis, Special Cases
then g will remain unchanged at poles but ● When r = ¥, then V = 0, hence gravitational
increases by Rw2 at equator. potential is maximum (zero) at infinity.
● At surface of the earth r = R , then
● Effect of Altitude The value of g at height
h from earth’s surface, -GM
V =
g R
g¢= 2
æ hö 9. Gravitational Potential Energy
ç1 + ÷
è Rø Gravitational potential energy of a body at a
point is defined as the amount of work done in
Therefore, g decreases with altitude.
bringing the given body from infinity to that
● Effect of Depth The value of g at depth h point against the gravitational force.
from earth’s surface, Gravitational potential energy,
æ hö æ GM ö
g ¢ = g ç1 - ÷ U = ç- ÷ ´m
è Rø è r ø
Therefore, g decreases with depth from 10. Escape Speed Escape speed on the earth
earth’s surface. (or any other planet) is defined as the
The value of g becomes zero at earth’s minimum speed with which a body should be
centre. projected vertically upwards from the surface
, of the earth, so that it just escapes out from GMm
PE of the satellite, U = - mv 2 = -
gravitational field of the earth and never R +h
returns on its own. Total mechanical energy of satellite,
\ Escape velocity, v e = 2 gR E = K +U
where, R is the radius of the earth. GMm
E =-
8 2(R + h )
Also, escape velocity, v e = R p Gr
3 Satellites are always at finite distance from the
where, r is the mean density of the earth. earth and hence their energies cannot be
11. Earth’s Satellites A satellite is a body which positive or zero.
is constantly revolving in an orbit around a 14. Geo-stationary Satellite These satellites
comparatively much larger body. e.g. The revolves in a circular orbits around the earth in
moon is a natural satellite while INSAT-1B is the equatorial plane with period of revolution
an artificial satellite of the earth. Condition for same as that of earth, i.e. T = 24 h and also
establishment of satellite is that the centre of known as geo-synchronous satellites.
orbit of satellite must coincide with centre of ● It should revolve in an orbit concentric and
the earth or satellite must move around in coplanar with the equatorial plane of earth.
greater circle of the earth. ● These satellites appears stationary due to its
12. Orbital Velocity of a Satellite Orbital law relative velocity w.r.t. that place on
velocity is the velocity required to put the earth.
satellite into its orbit around the earth or a ● It should be at a height around 36000 km.
planet.
● These satellites are used for communication
Satellite purpose like radio broadcast, TV broadcast,
m
vo etc.
R+h 15. Polar Satellite They are low-altitude satellites
(h » 500 to 800 km) which circle in a
R h North-South orbit passing over the North and
O South poles. It is also known as sun
Earth synchronous satellite.
● The time period is about 100 min.
● These satellites are used for military purpose.
GM 16. Weightlessness A body is said to be in a state
Mathematically, it is given by v o =
R +h of weightlessness when the relation of the
supporting surface is zero or its apparent
13. Energy of an Orbiting Satellite When a
weight is zero. At one particular position, the
satellite revolves around a planet in its orbit, it
two gravitational pulls may be equal &
possesses both potential energy (due to its
opposite and the net pull on the body becomes
position against gravitational pull of the earth)
zero. This is zero gravity region or the null
and kinetic energy (due to orbital motion). If m
point where the body is said to be weightless.
is the mass of the satellite and v is its orbital
velocity, then KE of the satellite, The state of weightlessness can be observed in
1 1 GM the following situations
K = mv 2 = m (Q v = GM /r ) ● When objects fall freely under gravity.
2 2 r
● When a satellite revolves in its orbit around
GMm
K = (Qr = R + h ) the earth.
2 (R + h )
● When bodies are at null points in outer space.
08
Gravitation
Quick Revision
1. Universal Law of Gravitation It states that, As, F12 and F21 are directed towards the
every body in this universe attracts every other centres of the two particles, so gravitational
body with a force whose magnitude is directly force is a central force.
proportional to the product of their masses and 3. Principle of Superposition According to
inversely proportional to the square of the this principle, the resultant gravitational force
distance between their centres. F can be expressed in vector addition of all
mm forces, at a point (as shown below).
Gravitational force, F = G 1 2 2
r i.e. F = F12 + F13 + F14 + ... + F1n
where, G is a constant of proportionality and is
known as universal gravitational constant. m4
^
r41 ^ m3
In CGS system, the value of G is r31
6.67 ´ 10 -8 dyne cm 2g -2 and its SI value is F14 F13
^
r21
6.67 ´ 10 -11 N-m 2kg -2 . m1 m2
Dimensional formula for G is [M -1L3T - 2 ]. ^
F1n
F12
2. Vector Form of Newton’s Law of
Gravitation In vector notation, Newton’s law ^
rn 1
of gravitation is written as follows
mm mn
F12 = - G 12 2 r$21 …(i)
r21
Resultant force,
where, F12 = gravitational force exerted on A by æm m m ö
B and r$21 is a unit vector pointing towards A. F = - Gm 1 ç 22 r$21 + 23 r$31 +... + 2n r$n 1 ÷
è r21 r31 rn 1 ø
Negative sign shows that the gravitational force
is attractive in nature. 4. Acceleration due to gravity The
mm acceleration produced in the motion of a body
Similarly, F21 = - G 12 2 r$12 …(ii)
r12 under the effect of gravity is called
acceleration due to gravity ( g ).
where, r$12 is a unit vector pointing towards B. GM
Equating Eqs. (i) and (ii), we have At the surface of the earth, g = 2
R
F12 = - F21
,5. Weight of a body It is the gravitational force 7. Intensity of Gravitational Field at a
with which a body is attracted towards the Point The gravitational force acting per unit
centre of the earth w = mg mass at any point in gravitational field is called
It is a vector quantity and its SI unit is intensity of gravitational field at that point.
newton (N). Intensity of gravitational field at a distance r ,
6. Factors Affecting Acceleration Due to from a body of mass M is given as
Gravity F GM
E = = 2
● Shape of Earth Acceleration due to gravity, m r
1 It is a vector quantity and its direction is
g µ 2
R towards the centre of gravity.
Therefore, g is minimum at equator and 8. Gravitational Potential Gravitational
maximum at poles. potential at a point in the gravitational field is
defined as the amount of work done per unit
● Rotation of Earth about Its Own Axis If
mass in bringing a body of unit mass from
w is the angular velocity of rotation of earth infinity to that point without acceleration.
about its own axis, then acceleration due to W
gravity at a place having latitude l is given i.e. V =-
m
by
F ×d r -GM
g ¢ = g - Rw2 cos 2 l =-ò = - ò E ×d r =
m r
At poles, l = 90° and g ¢ = g . It is a scalar quantity. The unit of gravitational
Therefore, there is no effect of rotation of potential in SI system is Jkg –1 and in CGS
earth about its own axis at poles. system is erg-g –1 .
At equator, l = 0° and g ¢ = g - Rw2 Dimensional formula for gravitational potential
The value of g is minimum at equator. is [M 0L2T -2 ].
If earth stops its rotation about its own axis, Special Cases
then g will remain unchanged at poles but ● When r = ¥, then V = 0, hence gravitational
increases by Rw2 at equator. potential is maximum (zero) at infinity.
● At surface of the earth r = R , then
● Effect of Altitude The value of g at height
h from earth’s surface, -GM
V =
g R
g¢= 2
æ hö 9. Gravitational Potential Energy
ç1 + ÷
è Rø Gravitational potential energy of a body at a
point is defined as the amount of work done in
Therefore, g decreases with altitude.
bringing the given body from infinity to that
● Effect of Depth The value of g at depth h point against the gravitational force.
from earth’s surface, Gravitational potential energy,
æ hö æ GM ö
g ¢ = g ç1 - ÷ U = ç- ÷ ´m
è Rø è r ø
Therefore, g decreases with depth from 10. Escape Speed Escape speed on the earth
earth’s surface. (or any other planet) is defined as the
The value of g becomes zero at earth’s minimum speed with which a body should be
centre. projected vertically upwards from the surface
, of the earth, so that it just escapes out from GMm
PE of the satellite, U = - mv 2 = -
gravitational field of the earth and never R +h
returns on its own. Total mechanical energy of satellite,
\ Escape velocity, v e = 2 gR E = K +U
where, R is the radius of the earth. GMm
E =-
8 2(R + h )
Also, escape velocity, v e = R p Gr
3 Satellites are always at finite distance from the
where, r is the mean density of the earth. earth and hence their energies cannot be
11. Earth’s Satellites A satellite is a body which positive or zero.
is constantly revolving in an orbit around a 14. Geo-stationary Satellite These satellites
comparatively much larger body. e.g. The revolves in a circular orbits around the earth in
moon is a natural satellite while INSAT-1B is the equatorial plane with period of revolution
an artificial satellite of the earth. Condition for same as that of earth, i.e. T = 24 h and also
establishment of satellite is that the centre of known as geo-synchronous satellites.
orbit of satellite must coincide with centre of ● It should revolve in an orbit concentric and
the earth or satellite must move around in coplanar with the equatorial plane of earth.
greater circle of the earth. ● These satellites appears stationary due to its
12. Orbital Velocity of a Satellite Orbital law relative velocity w.r.t. that place on
velocity is the velocity required to put the earth.
satellite into its orbit around the earth or a ● It should be at a height around 36000 km.
planet.
● These satellites are used for communication
Satellite purpose like radio broadcast, TV broadcast,
m
vo etc.
R+h 15. Polar Satellite They are low-altitude satellites
(h » 500 to 800 km) which circle in a
R h North-South orbit passing over the North and
O South poles. It is also known as sun
Earth synchronous satellite.
● The time period is about 100 min.
● These satellites are used for military purpose.
GM 16. Weightlessness A body is said to be in a state
Mathematically, it is given by v o =
R +h of weightlessness when the relation of the
supporting surface is zero or its apparent
13. Energy of an Orbiting Satellite When a
weight is zero. At one particular position, the
satellite revolves around a planet in its orbit, it
two gravitational pulls may be equal &
possesses both potential energy (due to its
opposite and the net pull on the body becomes
position against gravitational pull of the earth)
zero. This is zero gravity region or the null
and kinetic energy (due to orbital motion). If m
point where the body is said to be weightless.
is the mass of the satellite and v is its orbital
velocity, then KE of the satellite, The state of weightlessness can be observed in
1 1 GM the following situations
K = mv 2 = m (Q v = GM /r ) ● When objects fall freely under gravity.
2 2 r
● When a satellite revolves in its orbit around
GMm
K = (Qr = R + h ) the earth.
2 (R + h )
● When bodies are at null points in outer space.
