GRAVITATION
Topic Flow
❖The universal law of gravitation
❖Acceleration due to gravity and its variation with altitude and
depth
❖Kepler’s law of planetary motion
❖Gravitational potential energy
❖Gravitational potential
❖Escape velocity
❖Satellite
❖Orbital speed
❖Time Period of revolution
❖Energy of a satellite
Newton’s Law Of Universal Gravitation
“Every particle of matter in the universe attracts every other particle
with a force equal to the product of masses of particles and inversely
proportional to the square of the distance between them”
𝐺𝑚1𝑚2
F ∝ m1m2/r² 𝐹 = 2
𝑟
Where G is a constant and is called the Universal gravitational
constant.
Magnitude (and unit) of G :- 6.67 x 10^-11 Newton m²/kg²
−1 3 −2
Dimension of G :- 𝑀 𝐿𝑇
Illustration:- Spheres of the same material and same radius r are
touching each other. Show that gravitational force between them is
4
directly proportional to 𝑟 .
Variation in the value of Acceleration due to Gravity(g)
a)Shape of the Earth :- The earth is not a perfect sphere. It is
somewhat flat at the two poles. The equatorial radius is
approximately 21 km more than the polar radius. And since
𝐺𝑚 1
𝑔= 2 or 𝑔∝ 2
𝑅 𝑅
The value of g is minimum at the equator and maximum at the poles.
b)Height above the surface of the earth :- The gravitational
force on mass m due to Earth of mass M at height
h above the surface of earth is
Thus, g′ < g i.e., the value of acceleration due to gravity g goes
on decreasing as we go above the surface of earth.
ℎ −2 2ℎ
𝑔' = 𝑔(1 + 𝑅
) or 𝑔'≈𝑔(1 − 𝑅
) if h << R
c)Depth below the surface of the earth:
ℎ
𝑔' = 𝑔(1 − 𝑅
) , g’<g
Kepler’s Law Of Planetary Motion
a) Law of Orbits :- All the planets move in elliptical orbits with the
sun as one of its focii.
b)Law of Areas :- The radius vector from the sun at the focus of
elliptical orbit to the planet sweeps out equal areas in equal
intervals of time .
❖Areal velocity of planet is always constant .
2 2
𝐴𝑟𝑒𝑎 𝑑𝐴 ω𝑅 ω𝑅 𝑚 𝐿
Areal velocity = 𝑡𝑖𝑚𝑒
= 𝑑𝑡
= 2
= 2𝑚
= 2𝑚
Where, L = angular momentum
ω = angular velocity
c)Laws of Periods :- The square of the time period of revolution of
a planet is proportional to the cube of the mean distance of the
planet from the sun.
If a is the mean distance of sun from the planet,
2 3 2 3
𝑇 ∝𝑎 or 𝑇 = 𝑘𝑎
Where , k is a constant.
If a1 and a2 are semi-major axis of the orbits of two planets
around the sun with respective time periods T1 and T2 ,
Then,
3 3
𝑇1 𝑎1
3 = 3
𝑇2 𝑎2
Illustration:- The period of revolution of planet A around the sun
is eight times that of B. How many times is the distance of A from
the sun, greater than that of B?
Solution
Gravitational Potential Energy
When a body of mass (m) is moved from infinity to a point inside the
gravitational influence of a source mass (M) without accelerating it, the
amount of work done in displacing it into the source field is stored in
the form of potential energy . This is known as gravitational potential
energy. It is represented by the symbol U .
PE = U = - GMm/d = - Wconservative∞→r = Wext,slowly∞→r
Gravitational Potential
The potential at a point may also be defined as the work done per unit
mass by an external agent in bringing a particle slowly from the
reference point to the given point.
“change in potential” between two points ( VB - VA) = UB - UA/m
a)Potential due to point mass M at a point P which is at a distance
r
𝐺𝑀
𝑉 =− 𝑟
b)Potential due to Uniform ring of radius “a” and mass M at a point
P on its axis.
𝐺𝑀
𝑉 = − 2 2
𝑎 +𝑟
Escape Velocity
The minimum velocity needed to take a particle infinitely away from
the earth is called the escape velocity. On the surface of earth its
value 11.2 km/s.
The binding energy of a particle on the surface of earth kept at rest is
GMm/R . If this much energy in the form of kinetic energy is supplied
to the particle, it leaves the gravitational field of the earth. So, if ve is
the escape velocity of the particle, then
2
𝑚𝑣𝑒 𝐺𝑀𝑚 2 2𝐺𝑀 𝐺𝑀
2
= 𝑅
or 𝑣𝑒 = 𝑅
or 𝑣𝑒 = 2𝑔𝑅 as 𝑔 = 2
𝑟
❖Escape velocity is independent of angle of projection.
Illustration :- The escape velocity from the earth is about 11 km s−1.
The escape velocity from a planet having twice the radius and the
same mean density as the earth is
Solution
Satellites
Satellites are generally of two types:
Natural Satellites: Moon is a natural satellite of the earth.
Artificial Satellite: These are launched in to space by humans and
they help us in weather forecasting,telecommunications etc. The path
of these satellites is elliptical with the center of earth at a focus.
a)Orbital speed :- The necessary centripetal force to the satellite
is being provided by the gravitational force exerted by the earth
on the satellite.
𝐺𝑀 1
𝑣0 = 𝑟
or v0 ∝
𝑟
b)Period of Revolution :-
3
2π𝑟 2π𝑟 𝑟
𝑇 = 𝑣0
or 𝑇 = 𝐺𝑀
or 𝑇 = 2π 𝐺𝑀
𝑟
3
𝑟
Or 𝑇 = 2π 2 (as GM = gR²)
𝑔𝑅
c)Energy of satellite :- The potential energy of the system is
U = -GMm/r
2
𝑚𝑣0 1 𝐺𝑀
The Kinetic Energy of the satellite is , 𝐾 = 2
= 2 𝑚( 𝑟
)
𝐺𝑀𝑚
The total energy is , E = K + U = - 2𝑟
❖This energy is constant and negative, i.e., the system is closed.
The farther the satellite from the earth the greater its total
energy.
