Question

Imagine a planetary system in which the gravitational force varies as 1/R instead of 1/R2. What relation would correspond to kepler, s third law.
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Answer 1

Given:

Imagine a planetary system in which the gravitational force varies as 1/R instead of 1/R2                                          

To find :Relation Gravitational force and kepler' s third law

                                                           

solution:              

The gravitational force provides the orbital circular motion of the planets with centripetal force

$\frac{GMm}{r^{2} }$ = $\frac{mv^{2} }{r}$

⇒$v = \sqrt{\frac{GM}{r} }$

In one revolution, the time span for the earth will be T=   $\frac{2\pi r}{v}$ = $\frac{2\pi r}{\sqrt{\frac{GM}{r} } }$

squaring each side

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

comparing with given equation $T^{2}$ = k $r^{3}$

K = $\frac{4\pi ^{2} }{GM}$

K G M = $4\pi ^{2}$

This shows the G is gravitational constant. The relation between G and K .