MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Circular Motion Page 1

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1.

Newton's Law of Universal Gravitation

Every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them.

Mathematically,

 F a m1m2 r2

Where F is the magnitude of the force acting between two bodies of masses m1 and m2, separated by a distance r.

 \ F = Gm1m2 r2

G is called the universal gravitational constant.

2.

Universal Gravitational Constant (G)

Universal Gravitational Constant

 G = Fr2 m1m2

Magnitude of G: 6.673 × 10- 11 Nm2 / kg2
SI Unit of G:
Newton - metre2 / kilogram2
Dimensions of G: [M
- 1 L- 3 T - 2]

3.

Relation between G and g

Acceleration due to earth's gravity at the surface of the earth = g
The gravitational force of attraction between the earth and a body of mass m on the surface of the earth is given by

 F = GMm r2

Where M is the mass of the earth and R is the radius of the earth.

But weight of the body = gravitational force

 \ mg = GMm R2

 \ g = GM R2

… (1)

4.

Relation between G and g'

At a height h above the surface of the earth, acceleration due to earth's gravity = g'.
\ The corresponding equations are,

 F' = GMm (R + h)2

and F' = mg'

 \ mg' = GMm (R + h)2

 \ g' = GM (R + h)2

… (2)

5.

Relation between g and g'

From (1) and (2)

 g' = R2 g (R + h)2

 \ g' = g R 2 (R + h)

… (3)

Conclusion: From (3), as height '
h' above the surface of the earth increases, acceleration due to earth's gravity decreases.

 6. Projection of a Satellite Definition of Satellite: A lighter body which revolves around a heavier body due to the gravitational influence of the heavier body is called its satellite. Why is it necessary to have at least a two stage rocket to launch a satellite? A satellite fired from the earth's surface with low velocity will fall back to the earth. If it is fired with a high velocity, it will escape from the earth's gravitational field. In neither case will it become a satellite of the earth. A rocket with at least two stages is required to launch a satellite, because the first stage is used to carry the satellite up to the desired height. In the second stage, rocket is turned horizontally and the satellite is fired with the proper horizontal velocity to perform UCM around the earth.

7.

Orbital Velocity or Critical Velocity (proper speed) of a Satellite (Vc)

The horizontal velocity with which a satellite should be projected from a point above the earth's surface, so that it orbits in a circular path around the earth is called the orbital velocity or critical velocity (Vc) of the satellite.

8.

Escape Velocity (Ve)

Definition: The minimum velocity with which a body should be projected from the surface of the earth so that it escapes the gravitational field of the earth is called the escape velocity of the body.

9.

Horizontal Projection of a Satellite at Height h above the Surface of the Earth

 Case : I V < Vc V ® Velocity of satellite Vc ® Critical Velocity The satellite will fall back to the earth. Case : II V = Vc The satellite moves in a circular path and performs UCM around the earth. Case : III Ve > V > Vc Ve ® Escape Velocity Here, the satellite will revolve in an elliptical orbit around the earth. Case : IV V ³ Ve The satellite will escape from the gravitational field of the earth.

 V - Velocity of projection in a horizontal direction Vc - Critical Velocity Ve - Escape Velocity

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