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  4. If the gravitational force between two objects were proportional to 1/R (and not as 1/R2 ), where R is the distance between them, then a particle in a circular path (under such a force) would have its orbital speed v, proportional to

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Q. If the gravitational force between two objects were proportional to $ 1/R $ (and not as $ 1/R^2 $ ), where $R$ is the distance between them, then a particle in a circular path (under such a force) would have its orbital speed v, proportional to

AIPMTAIPMT 1994Gravitation

Solution:

Centripetal force $(F)=\frac{m v^{2}}{R}$
and the gravitational force
$(F)=\frac{G M m}{R^{2}}=\frac{G M m}{R}$
(where $R^{2} \rightarrow R$ ).
Since $\frac{m v^{2}}{R}=\frac{G M m}{R}$,
therefore $v=\sqrt{G M}$.
Thus velocity $v$ is independent of $R$.