__Centripetal Force:__

__Definition:__

*“The force that causes an object to move along a curve (or a curved path) is called centripetal force.”*

__Mathematical Expression:__
We know that the magnitude of centripetal acceleration of a body in a uniform circular motions is directly proportional to the square of velocity and inversely proportional to the radius of the path Therefore,

a(c) α v2 (Here α represents the sign of proportionality do not write this in your examination and 2 represents square of v)

a(c) α 1/r

Combining both the equations:

a(c) α v2/r From Newton’s Second Law of Motion: F = ma => F(c) = mv

^{2}/r
Where,

Fc = Centripetal Force

m = Mass of object

v = Velocity of object

r = Radius of the curved path

__Factors On Which Fc Depends:__
Increase in the mass increases Fc.

It increases with the square of velocity.

It decreases with the increase in radius of the curved path.

__Examples:__
The centripetal force required by natural planets to move constantly round a circle is provided by the gravitational force of the sun.

If a stone tied to a string is whirled in a circle, the required centripetal force is supplied to it by our hand. As a reaction the stone exerts an equal force which is felt by our hand.

The pilot while turning his aeroplane tilts one wing in the upward direction so that the air pressure may provide the required suitable Fc.

__Centrifugal Force:__

__Definition:__

*“A force supposed to act radially outward on a body moving in a curve is known as centrifugal force.”*

__Explanation:__
Centrifugal force is actually a reaction to the centripetal force. It is a well-known fact that Fc is directed towards the centre of the circle, so the centrifugal force, which is a force of reaction, is directed away from the centre of the circle or the curved path.

According to Newton’s third law of motion action and reaction do not act on the same body, so the centrifugal force does not act on the body moving round a circle, but it acts on the body that provides Fc.

*Examples*

If a stone is tied to one end of a string and it is moved round a circle, then the force exerted on the string on outward direction is called centrifugal force.

The aeroplane moving in a circle exerts force in a direction opposite to the pressure of air.

When a train rounds a curve, the centrifugal force is also exerted on the track.

__Law of Gravitation__

__Introduction:__
Newton proposed the theory that all objects in the universe attract each other with a force known as gravitation. the gravitational attraction exists between all bodies. Hence, two stones are not only attracted towards the earth, but also towards each other.

__Statement:__
Every body in the universe attracts every other body with a force, which is directly proportional to the product of masses and inversely proporti:onal to the square of the distance between their centres.

__Mathematical Expression:__
Two objects having mass m1 and m2 are placed at a distance r.

According to Newton’s Law of Universal Gravitation.

F α m

_{1}m_{2}
Also F α 1/r

^{2}
Combining both the equations :

F α m

_{1}m_{2}/r^{2}
Removing the sign of proportionality and introducing a constant:

F = G (m

_{1}m_{2}/r^{2})

__Mass of Earth:__
Consider a body of mass ‘m’ placed on the surface of the earth. Let the mass of the earth is ‘M

_{e}’ and radius of the earth is ‘R_{e}’ .Neglecting the radius of body if compared with that of the earth.
Gravitational force of attraction between earth and body is

**F = G m M**

_{e}/ R_{e}^{2 }^{........(1)}

**F = W**........................(2)

Therefore Comparing (1) and (2), we get,

**W = G m M**

_{e}/ R_{e}^{2}
But W = mg

**mg = G m M**

_{e}/ R_{e}^{2}**or**

**g = G M**_{e}/R_{e}^{2}

**or**

**M**

_{e}= g x R_{e}^{2}/G
From astronomical data:

g= 9.8 m/s

^{2}
R

_{e }= 6.38 x 10^{6}m
G = 6.67 x 10

^{-11}N-m^{2}/kg^{2}
Putting these values in the above equation.

**M**

_{e}= 9.8 (6.38 x 10^{6})^{2}/6.67 x 10^{-11}
or

M

_{e}= 6x10^{24}Kg
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