Write the three laws given by Kepler.
How did they help Newton to arrive at the
inverse square law of gravity?
Answers
Answer:
There are actually three, Kepler’s laws that is, of planetary motion: 1) every planet’s orbit is an ellipse with the Sun at a focus; 2) a line joining the Sun and a planet sweeps out equal areas in equal times; and 3) the square of a planet’s orbital period is proportional to the cube of the semi-major axis of its orbit. As it’s the third which is most often used, Kepler’s law usually means Kepler’s third law (of planetary motion).
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