30.State Kepler's law of planetry motion. (OR) How did Newton arrive at the inverse
square rule ?
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Answers
Explanation:
Newton used Kepler's third law of planetary motion to arrive at the inverse-square rule. He assumed that the orbits of the planets around the Sun are circular, and not elliptical, and so derived the inverse-square rule for gravitational force using the formula for centripetal force. This is given as:
F=
r
mv
2
.... {i) where, m is the mass of the particle, r is the radius of the circular path of the particle and v is the velocity of the particle.
Newton used this formula to determine the force acting on a planet revolving around the Sun. Since the mass m of a planet is constant, equation (i) can be written as:
F α
r
v
2
... (ii)
Now, if the planet takes time T to complete one revolution around the Sun, then its velocity v is given as:
v=
T
2πr
... (iii) where, r is the radius of the circular orbit of the planet
or, v α
T
r
... (iv) [as the factor 2π is a constant]
On squaring both sides of this equation, we get:
v
2
α
T
2
r
2
... (v)
On multiplying and dividing the right hand side of this relation by r, we get:
v
2
α
r
1
r ... (vi)
According to Kepler's third law of planetary motion, the factor
T
2
r
3
is a constant.
Hence, equation (vi) becomes:
v
2
α
r
1
... (vii)
On using equation (vii) in equation (ii) we get:
F α
r
2
1
Hence. the gravitational force between the sun and a planet is inversely proportional to the square of the distance between them.