Explain how Newton arrived at his law of gravitation from Kepler’s third law.

#### Solution

Newton considered the orbits of the planets as circular. For a circular orbit of radius r, the centripetal acceleration towards the center is

a = `-"v"^2/"r"` ........(1)

**Point mass orbiting in a circular orbit**

Here v is the velocity and r, the distance of the planet from the center of the orbit.

The velocity in terms of known quantities r and T is

v = `(2π"r")/"T"` .......(2)

Here T is the time period of the revolution of the planet. Substituting this value of ‘v’ in equation (1) we get,

a = `-((2π"r")/"T")^2/"r"`

= `-(4π^2"r")/"T"^2` ........(3)

Substituting the value of ‘a’ from (3) in Newton’s second law, F = ma where ‘m’ is the mass of the planet.

F = `-(4π^2"mr")/"T"^2` .......(4)

From Kepler’s third law,

`"r"^3/"T"^2` = k (constant) ......(5)

`"r"/"T"^2 = "k"/"r"^2` .....(6)

By using equation (6) in the force expression, we can arrive at the law of gravitation.

F = `-(4π^2"mk")/"r"^2` .......(7)

Here negative sign implies that the force is attractive and it acts towards the center. He equated the constant 4π^{2}k to GM which turned out to be the law of gravitation.

F = `-"GMm"/"r"^2`