Gravitation


   
 
Kepler's Laws of Planetary Motion
 
First law
 
Kepler's first law
 
The path of each planet around the sun is an ellipse with the sun at one focus.
 
If F1 and F2 are the two foci, P is the planet and S is the sun:
 
F1P + F2P = constant.
 
Second law
 
Kepler's second law
 
A line joining the sun to the planet sweeps out equal areas in equal intervals of time.
 
If the time taken by a planet to travel from P1 to Q1 is equal to the time taken to travel from P2 to Q2, the areas covered are equal (shaded region).
 
Third law
A planet moves around the sun in such a way that the square of its time period is proportional to the cube of the semi-major axis of its elliptical orbit.
 
If T is the time period of revolution and 'a' the semi-major axis then,
 
 
for circular orbits a = r (radius)
 
 
Derivation of Newton's law of gravitation from Kepler's laws
 
Let m be the mass of a body in an orbit of radius r, velocity v and angular velocity 'w'. the centripetal force
 
 
If T is the time period then,
 
 
 
 
According to Kepler's third law,
 
 
 
 
 
If M is the mass of the sun and the body of mass m is revolving around the sun, then the constant of proportionality is assumed to be GM, where G is a constant.
 
 
which is Newton's law of gravitation.
 
 
     
   
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