Question

A small planet is revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force between the planet and the star were proportional to $\mathrm { R } ^ { - 5 / 2 }$ then T would be proportional to

• A.
• B. $\mathrm { R } ^ { 3 / 5 }$
• C. $\mathrm { R } ^ { 7 / 2 }$
• D. $\mathrm { R } ^ { 7 / 4 }$

Answer 1

Answer: (D)

$\mathrm { R } ^ { 7 / 4 }$

Answer 2

According to the questions, the gravitational force between the planet and the star is

$\mathrm { F } = \frac { 1 } { \mathrm { R } ^ { 5 / 2 } }$ $\therefore \mathrm { F } = \frac { \mathrm { GMm } } { \mathrm { R } ^ { 5 / 2 } }$

Where M and m be masses of star and planet respectively.

For motions of a planet in a circular orbit.

$\mathrm { mR } \omega ^ { 2 } = \frac { \mathrm { GMm } } { \mathrm { R } ^ { 5 / 2 } }$

$\mathrm { mR } \left( \frac { 2 \pi } { \mathrm { T } } \right) ^ { 2 } = \frac { \mathrm { GMm } } { \mathrm { R } ^ { 5 / 2 } }$