Textbook

Chapter No.

05 Gravitation
Subtopics Centripetal Acceleration

5.1 Introduction
Gravitational Orbit
Force Velocity
5.2 Kepler's Laws
S.3 Universal Law of Gravitation
RadiuS
5.4 of Earth
Measurement of the Gravitational Constant (G)
Alritude Radius of Orbit
5.5 Acceleration due to Gravity
5.6 Variation in the Acceleration due to gravity
with Altitude, Depth, Latitude and Shape
$.7 Gravitational Potential and Potential Energy
5.8 Earth Satellites

For asatellite in circular orbit around

the earth, the gravitational force
provides the necessary centripetal force.

Formulae 3. Gravitational constant: G= Fr?

m,m,
1. Kepler's formula for planetary motion: 4. Measurement of G using Cavendish balance:
i. Kepler's law of equal areas:
Kestoring torque. t= GM -L=K0
Areal velocity, AA L
= constant
At 2m Where, r=initial distance of separation between
the centres of the large (with mass M) and the
Kepler's law of periods:
neighbouring small (with mass m) sphere,
T°«r' or = Constant K=restoring torque per unit angle and
9 = the angle of twist.
b Ratio of time periods of two planets,
5.
\3/2 Acceleration due to gravity:
On the surface of the earth, g= GM
T,
Where, r = average distance of planet ii
from the Sun. In terms of density, g= pGR
2
Gravitational force:
Variation in acceleration due to gravity:
i. Scalar form: F = Gm,m, i At a height above the earth,
GM GM
Gravitational force between two equal masses, (R +h)
where r=R+h
Gm
F=
ii.
R+h.
iii Vector form: Fx=G"(-,)
iii.
where, i,, =the unit vector from m, to m: ...(For h<<R)
and
force F is directed from m, to m,. iv. Ala depth below the earth,
40
 Chapter 05: Gravitation
1 A latitde . g
At cquator,
Hence g- Ro VR+h
Alpoles, e 0° (if g, is not known)
Hence g'g In terms of density of earth (only close to the
Ifcarth stops rotating. then the value of g Surface of earth).
at equator increases by Ro
v.- 2R G
Gravitational potential encrgy of a body:
For body stationary on the carth: where p Density of earth
-GMm
-mgR iv. In terms of escape velocity, v. =É
R
For body revolving around the earth:
11. Weightlessness:
GMm
i. W = m(g+ a)
when lift moves upward with acceleration 'a"
8 Gravitational Potential: ii. W =m(g - a)
1. In terms of potential energy: V=
U when lift moves downward with acceleration 'a
Im iii. W =0

On surface of the earth: V= GM when lift moves as free fall.

R where, W= Weight of the body in the lift
iii. At certain height above the surface of the earth:
12. Time Period of a satellite:
GM
V
At height h from surface of earth,
4n'
9 Escape Velocity: i. T= (R+h)
VGM
For abody stationary on the earth's surface
|2GM (R+h'
a. V =
R
b. ii.
T= 2r GM

R+h
Ve =R 8npG iii. T=2r
V3
For a body revolving around the Earth's surface R
iv. Close to surface of earth, T = 2
at a height h Vg
a b Ve= 2gr
V. Ratio of time periods of two satellites at
different radius of orbits ,
3/2 3/2

R +h,
C
R+h,

10. Critical (orbital) velocity: 13. Binding energy:

J. a When satellite is orbiting close to the i For a body stationary on the Earth,
surface of earth, v, = GM B.E. =
GMm
VR R
b When salellite is orbiting at height h' ii. For a body revolving around the Earth at a
fromthe surface of earth, GMm GMm
|GM
height h, B.E. =
2(R +h) 2
VR h
GM
,rradius of orbit.
14 Potential energy of a body on the Earth's
surface:
In terms of acceleration due to gravity, For a body on the carth's surtace,
-GMm
v, = VgR (whensatellite is orbiting close PE. -mgR
to the surlace of carth)
b. V. = gh (R +h) where For a body revolving around the Earth,
-GMm
P.E. =
g, acceleration due to graviy at height h.
41
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