Question

Derive an expression for orbital velocity of a planet in terms of gravitational constant (G) radius (R) of the orbit and mass (M) of the sun ​

Answer 1

To derive:

An expression for orbital velocity of a planet in terms of gravitational constant (G) radius (R) of the orbit and mass (M) of the sun.

Derivation:

• When a planet is revolving around a sun , we will consider that the trajectory of the planet is circular.

• Again, while revolving , the centripetal force on the planet can be attributed to the gravitational force between the sun and the planet.

• Let mass of planet be m.

$\rm \therefore \: F_{g} = F_{c}$

$\rm \implies \: \dfrac{GMm}{ {R}^{2} } = \dfrac{m {(v_{o})}^{2} }{R}$

$\rm \implies \: \dfrac{GMm}{ R } = m {(v_{o})}^{2}$

$\rm \implies \: { (v_{o})}^{2} = \dfrac{GM}{ R }$

$\rm \implies \:v_{o}= \sqrt{\dfrac{GM}{ R }}$

So, expression for orbital velocity is :

$\boxed{ \bold{ \:v_{o}= \sqrt{\dfrac{GM}{ R }}}}$

• This expression tells that orbital velocity of a planet is not dependent upon the mass of the planet.