Kepler's third law states that square of period of revolution (T) of a planet around the sun is proportional to the third power of average distance r between sun and planet, i.e., here K is a constant. If the masses of the sun and planet are M and m resp

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Solution

Q.
Kepler's third law states that square of period of revolution (T) of a planet around the sun is proportional to the third power of average distance r between sun and planet, i.e., $T^{2} = K r^{3}$ here K is a constant. If the masses of the sun and planet are M and m respectively, then as per Newton’s law of gravitation, the force of attraction between them is $F = \frac{G M m}{r^{2}},$ here G is the gravitational constant. The relation between G and K is described as [2015]

1 $K = G$

2 $K = \frac{1}{G}$

3 $G K = 4 π^{2}$

4 $G M K = 4 π^{2}$

Ans.
(4)

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

$∴$ $\frac{G M m}{r^{2}} = \frac{m v^{2}}{r}; v^{2} = \frac{G M}{r}$ ...(i)

Time period of the planet is given by

$T = \frac{2 π r}{v}, T^{2} = \frac{4 π^{2} r^{2}}{v^{2}} = \frac{4 π^{2} r^{2}}{(\frac{G M}{r})} (Using (i))$

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

According to question, $T^{2} = K r^{3}$ ...(iii)

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

$K = \frac{4 π^{2}}{G M}, ∴ G M K = 4 π^{2}$

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