Question

Write the three laws given by Kepler. How did they help Newton to arrive at the inverse square law of gravity?

Answer 1

Three laws given by Kepler is as follows:
First Law: The orbits of the planets are in the shape of ellipse, having the Sun at one focus.
Second Law: The area swept over per hour by the radius joining the Sun and the planet is the same in all parts of the planet's orbit.
Third Law: The squares of the periodic times of the planets are proportional to the cubes of their mean distances from the Sun.
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:
$F=m{v}^2/r$ ...(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:
$F\propto v^2/r$ ...(ii)
Now, if the planet takes time $T$ to complete one revolution around the Sun, then its velocity $v$ is given as:
$v=2\pi r/T$ ...(iii) where, $r$ is the radius of the circular orbit of the planet
or, $v\propto r/T$ ...(iv) [as the factor $2\pi$ is a constant]
On squaring both sides of this equation, we get
$v^2\propto r^2/T^2$...(v)
On multiplying and dividing the right-hand side of this relation by $r$, we get:
$v^2\alpha$ $r^3/r{T}^2$ ....(vi)
According to Kepler's third law of planetary motion, the factor $r^3/T^2$ is a constant. Hence, equation (vi) becomes:
$v^2\propto 1/r$...(vii)
On using equation (vii) in equation (ii), we get:
$F\alpha \frac{1}{r^2}$
Hence, the gravitational force between the sun and a planet is inversely proportional to the square of the distance between them.