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 distance r between sun and planet i.e. T2 = Kr3 here K is constant. If the masses of sun and planet are M and m respectively then as per Newton's law of gravitation force of attraction between them is F = (GMm/r2), here G is gravitational constant. The relation between G and K is described as

Question

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 distance r between sun and planet i.e. T2 = Kr3 here K is constant. If the masses of sun and planet are M and m respectively then as per Newton's law of gravitation force of attraction between them is F = (GMm/r2), here G is gravitational constant. The relation between G and K is described as (A) K = G (B) K = (1/G) (C) GK = 4 π2  (D) GMK = 4 π2

Tardigrade Admin · Accepted Answer

Gravitational force of attraction between sun and planet provides centripetal force for the orbit of planet.

∴ \frac{GM m}{r ^{2}} = \frac{m v ^{2}}{r}

v^{2} = \frac{GM}{r} ... (i)

Time period of the planet is given by

T = \frac{2 π r}{v}, T^{2} = \frac{4 π ^{2} r ^{2}}{v ^{2}}

T^{2} = \frac{4 π ^{2} r ^{3}}{( \frac{GM}{r} )}

[Using equation (i)]

T^{2} = \frac{4 π ^{2} r ^{3}}{GM} ... (ii)

According to question,

T^{2} = K r^{3} ... (iii)

Comparing equations (ii) and (iii), we get

K = \frac{4 π ^{2}}{GM}

∴ GM K = 4 π^{2}

$K = G$

$K = \frac{1}{G}$

$G K = 4 π^{2}$

$GM K = 4 π^{2}$

K=G

K=G1​

GK=4π2

GMK=4π2

$K = G$

$K = \frac{1}{G}$

$G K = 4 π^{2}$

$GM K = 4 π^{2}$