CIRCULAR
MOTION AND GRAVITATION


GRAVITATION


Every object in our universe attracts the other object with certain fore towards its center. This force of attraction is known as GRAVITATIONAL FORCE and the phenomenon is called GRAVITATION. This is gravitational force which is responsible for the uniformity or regularity in our daily astronomical life. The whole system of the universe is in order only due to this force. Due to gravitation, the system of our universe is working uniformly and smoothly. The planets around the earth or around the sun moves in an orderly motion due to gravitation. 

NEWTON’S
LAW OF GRAVITATION


In
order to explain the gravitational force between two bodies, Newton formulated
a fundamental law known after his name i.e. "NEWTON'S LAW OF GRAVITATION" Newton’s law of gravitation states that every object in the universe attracts the other object with a force and : 

(1) The gravitational force of attraction between two bodies is directly proportional to the product of their masses.  
F a
m_{1} x m_{2}
 (1)


(2) The gravitational force of attraction between two bodies is inversely proportional to the square of the distance between their centers.  
F a
1/d^{2} 
(2)


MATHEMATICAL
REPRESENTATION



Combining (1) and (2)  
F a
m_{1}m_{2} /d^{2}


F = G m_{1}m_{2}/d^{2}


Where G = universal gravitational constant  
Value of G:  
G = 6.67 x 10^{11} Nm^{2}/kg^{2}  
MASS
OF THE 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 earth is ‘R_{e}’ .  
Gravitational force of attraction between earth and body is  
F
= G m M_{e}/ R_{e}^{2}


We know that the force of attraction of the earth on a body is equal to weight the weight of body.  
i.e  
F
= W


therefore  
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 }= 6.4 x 10^{6} m G = 6.67 x 10^{11} Nm^{2}/kg^{2} Putting these values in the above equation. 

M_{e}
= 9.8 (6.4 x 10^{6})^{2}/6.67 x 10^{11}
or 
