Kepler's third law states that square of period revolution `(T)` of a planet around the sun is proportional to third power of average distance `i` between sun and planet i.e. `T^(2)=Kr^(3)`
here `K` is constant
if the mass of sun and planet are `M` and `m` respectively then as per Newton's law of gravitational the force of alteaction between them is `F=(GMm)/(r^(2))`, here `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=(1)/(G)`

Answer

C

Explanation

The gravitatinal forceo of attraction between the planet and sun provide the centripetal force
`(GMm)/(r^(2))=(mv^(2))/(r ) rarr V=sqrt(GM)/(r )`
The time period of planet will be
`T=(2pir)/(v) rarr T^(2)=(4pi^(2)r^(2))/(GM)=(4pi^(2)r^(3))/(GM)`
Also form kepler 's third law
`T^(2)=Kr^(3)`
From Eqs i and ii we get
`(4pi^(2)r^(3))/(GM)=Kr^(3) rarr GMK =4pi^(2)`