Question

5. 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

(1) K=1/G
(2) GK = 4π²
(4) K = G
(3) GMK = 4π²​

Answer 1

Answer:

(3) GMK = 4π²

this is your answer

Answer 2

Hey there!

Answer $\implies \boxed{\bold{GMK}= \bold{4\pi^{2} }}$

Consider the given values. Here, the gravitational forces act as the centripetal forces for the planet around the sun. We equate the two quantities and obtain:

$\implies$                    $\bold {\frac{GMm}{r^{2} } = m \omega^{2} r }$

Equating the two we obtain:

         

$\implies$                   $\bold {\frac{GMm}{r^{2} } = m(\frac{2\pi }{T})^{2} r }$

$\implies$                   $\bold {T^{2}GM = 4\pi ^{2}r^{3} }$

$\implies$                   $\bold {T^{2} =\frac{4\pi ^{2} }{GM} \,r^{3} }$

$\implies$                   $\textsf{where, } \bold {K = \frac{4\pi ^{2} }{GM} }$

$\implies$                   $\bold {GMK = 4\pi ^{2} }$