Kepler's third law states that square of period of revolution (T) of a planet around the sun, is proportional to third power of average distace r between the sun and planet i.e. `T^(2) = Kr^(3)`, here K is constant. If the masses of the sun and planet are M and m respectively, then as per Newton's law of gravitationa force of attraction between them is `F = (GMm)/(r^(2))`, hence G is gravitational constant. The relation between G and K is described as

Options

A. `GK = 4pi^(2)`
B. `GMK = 4pi^(2)`
C. `K=G`
D. `K=I//G`

Answer

B

Explanation

The gravitational force of attraction between the planet and sun provide the centripetal force
i.e. `(GMm)/(r^(2))=(mv^(2))/(r)rArrv=sqrt((GM)/(r))`
The time period of planet will be
`T=(2pir)/(v)rArrT^(2)=(4pi^(2)r^(2))/((GM)/(r))=(4pi^(2)r^(3))/(GM)`...(i)
Also from Kepler's third law
`T^(2)=Kr^(3)`...(ii)
From Eqs. (i) and (ii), we get
`(4pi^(2)r^(3))/(GM)=Kr^(3)rArrGMK=4pi^(2)`.