MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Gravitation Know More

• Newton's Law of Gravitation: Newton's law of gravitation is valid only to particles or to point masses. Bodies, whose sizes are extremely small compared to the distance of separation between them, can be regarded as point masses.
In general,
Newton's law of gravitation cannot be used to determine the forces of attraction between real extended bodies, which are separated by a short distance. However, the law can be used for real bodies as per conditions listed below:

a.      It is valid for two bodies of any size provided they are spherically symmetric. For example, The earth and the moon.

b.      It is valid for two bodies when one is spherically symmetric and the other is small in size when compared to the distance of separation between their centers. For example, The earth and an earth satellite.

c.       It is valid for two bodies when neither of them is spherically symmetric, provided their size is considerably smaller when compared to the distance of separation between them. For example, Two pieces of rock separated by a few metres.

• Universal Law of Gravitation (G)

a.      The value of 'G' was first determined experimentally by an English Physicist, Henry Cavendish in 1708

b.      In SI, the value of G is 6.67 × 10-11Nm2kg-2.

c.       The value of G is independent of the nature and the size of the two bodies.

d.      The value of G is independent of the presence of other bodies in the vicinity.

e.      The value of G is independent of the properties of the intervening medium.

• Inertial Mass (mi)

a.      Inertial mass of a body is defined as the ratio of the applied force on a body placed on a frictionless horizontal surface to the acceleration produced in it. Gravity has no effect on the inertial mass of a body.

According to definition, Inertial mass =

 Applied force Acceleration produced

\ mi =

 F a

c.       Inertial mass of a body is independent of the size or shape of the body.

d.      Inertial mass of a body is independent of the temperature of the body.

e.      Inertial mass of a body is independent of the presence of other bodies near it.

f.        Inertial mass of a body remains unaffected during its change of state.

g.      Inertial mass remains unaffected when two or more bodies combine physically or chemically

• Gravitational Mass (mg)

a.      The gravitational mass of a body is defined as the ratio of the earth's gravitational force on the body (i.e. weight of a body) to the acceleration due to gravity at the place. According to definition,

mg =

 Gravitational force of the earth on the body Acceleration due to gravity

\ mg =

 F g

c.       The inertia of a body has no role to play in the determination of the gravitational mass of a body.
After performing several experiments,
Newton concluded that the inertial mass of a body is equivalent to the gravitational mass of a body.

• Acceleration due to Gravity (G)

i.            Variation of g with Altitude:
Consider the earth to be a sphere of radius R and mass M.
The acceleration due to gravity at the surface of the earth is

gs =

 GM R2

...(1)

ii.             The acceleration due to gravity at a height h above the surface of earth is

gh =

 GM (R + h)2

...(2)

iii.            Dividing eq. (1) by eq. (2), we get,

 gs gh

=

 (R + h)2 (R)2

\

 gs gh

=

(

 R + h R

)2

\

 gs gh

=

(1 +

 h R

)2        ...(3)

\ gh =

gs

(1 +

 h R

)2

...(4)

vii.            From eq. (4), we conclude that gh < gs i.e., acceleration due to gravity decreases with increasing height.
Reciprocating both sides of eq. (3) we get,

 gh gs

=

(1 +

 h R

)-2

 \ gh gs

=

(1 +

 h R

)-2

...(5)

ix.            Expanding the R.H.S. by Binomial theorem, we get,

 \ gh gs

=

(1 -2

 h R

+ terms containing higher powers of

 h R

)

 If h << R, h R

is very small as compared to 1.

Neglecting higher powers of

 h R

, we get,

 gh gs

= (1 - 2

 h R

)

 gh = gs

(1 - 2

 h R

)      ...(6)

xiii.            Note: When h << R, then variation of g with height is given by eq. (6) but if h is not very small as compared to R, then eq. (4) holds good.

xiv.            Variation of g with Depth:
Consider the earth to be a sphere of radius R and mass M.
The acceleration due to gravity at the surface of the earth is

gs =

 GM R2

...(1)

xv.            If r is the density of the earth, then,

M =

 4 3

pR3r     ...(2)

xvi.            Substituting eq. (2) in eq. (1), we get,
gs =

\ gs =

 4 3

pRGr         ...(3)

xvii.            Consider a point P at a depth d below the surface of the earth. The distance of the point P from the centre of the earth is then (R-d).
Let a sphere be drawn with O as centre and (R-d) as radius. The force of gravity acting on the body is only due to the inner solid sphere of radius (R-d).
The mass M' of the inner solid sphere of radius (R-d) is given by

M =

 4 3

p(R - d)3         ...(4)

\ The acceleration due to gravity gd at a depth d below the surface of the earth is gd

 GM' (R - d)2

xix.            Substituting eq. (4) in eq. (5), we get,
\ gd =

\ gd =

 4 3

p(R - d)rG         ...(6)

xx.            Dividing eq. (6) by eq. (3), we get

 gd gs

=

 R - d R

 \ gd gs

= 1 -

 d R

 \ gd = gs (1 - d R

)            ...(7)

xxiii.            From eq. (7) we conclude that gd < gs i.e., acceleration due to gravity decreases with increasing depth.

xxiv.            Note: The acceleration due to gravity is maximum at the surface of the earth.

xxv.            Variation of g with Latitude: Suppose earth is sphere of radius R and w is its angular speed about its axis of rotation. Let l be the latitude of a place and g be the acceleration due to gravity at the place in the absence of the rotational motion of the earth.
Then, the acceleration due to gravity at the place with latitude
l due to the rotational motion of the earth is given by
g'= g - R
w2 cos2l
(Derivation omitted) …(1)

From eq. (1) we conclude that g'< g
i.e., acceleration due to gravity at a place decreases due to earth's rotation about its axis.

Case I:
At the equator,
l= 0o.
\g'= g - Rw2
(... cos
l = cos0o =1)
Case II:
At the poles,
l = 90o.
\ g'= g
(... cos
l = cos90o = 0)

• Earth Satellite: An artificial satellite put in its orbit around the earth is called an earth satellite. India launched its first satellite Aryabhatta on April 19, 1975.
• Geostationary Satellites (Synchronous or Communication satellites)

.            Time period of revolution around the earth = 24 hrs.

i.            The orbit of a geostationary satellite is also called as parking orbit.

ii.            Height of a parking orbit above the surface of the earth is 36,000 km.

iii.            Speed in the parking orbit » 3.1 km/s.

iv.            The orbit of a geostationary satellite is also called as parking orbit.

Note: India launched its first experimental communication satellite APPLE (Ariane Passenger Pay Load Experiment) on June 19, 1981.

• Kepler's Laws of Planetary Motion:

.            Kepler's First Law(Law of Orbits): It states that each planet moves in an elliptical path with the sun at one of its foci.

i.            Kepler's Second Law(Law of Areas): It states that each planet moves around the sun in such a way that the radius vector drawn from the sun to the planet sweeps out equal areas in equal times.

i.e.,

 dA dT

= constant

ii.            Kepler's Third Law(Law of periods): It states that the square of the period of a planet about the sun is directly proportional to the cube of the planet's average distance from the sun.
T2
µ r3

i.e.,

 T2 r3

= constant

iii.

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