Question

A moon of Saturn has a nearly circular orbit of radius R and an orbit period of T. Which of the following expressions gives the mass of Saturn?

• A. $\frac{2\pi R}{T}$
• B. $\frac{4{\pi }^{2}R}{T}$
• C. $\frac{2\pi R^3}{\left(G{T}^2\right)}$
• D. $\frac{4{\pi }^2{R}^2}{\left(G{T}^2\right)}$
• E. $\frac{4{\pi }^2{R}^3}{\left(G{T}^2\right)}$

Answer 1

The correct option is E $\frac{4{\pi }^2{R}^3}{\left(G{T}^2\right)}$

Here the gravitational pull by Saturn provides the centripetal force on its motion i.e, $\frac{GMm}{R^2}=\frac{m{v}^2}{R}$ or $GM/R=v^2...\left(1\right)$

Also time period, $T=\frac{2\pi R}{v}...\left(2\right)$

From (1) and (2), $GM/R=(2\pi R/T{)}^2$

or $M=\frac{4{\pi }^2{R}^3}{G{T}^2}$