Question

An artificial satellite of mass \(m\) revolves around a planet in a circular orbit of radius \(R\) under the influence of an attractive central force given by \[ F\propto R^{-5/2}. \] How do the orbital velocity \(V\) and time period \(T\) depend on the orbital radius \(R\)?

• A. \[ V\propto R^{-3/4}, \qquad T\propto R^{7/4} \]
• B. \[ V\propto R^{-5/4}, \qquad T\propto R^{3/2} \]
• C. \[ V\propto R^{-1/2}, \qquad T\propto R^{5/4} \]
• D. \[ V\propto R^{-3/4}, \qquad T\propto R^{-7/4} \]

Hint:

If the central force varies as \[ F\propto R^{-n}, \] then from \[ \frac{mV^2}{R}=F \] we get \[ V\propto R^{\frac{1-n}{2}}. \] After finding \(V\), use \[ T=\frac{2\pi R}{V} \] to obtain the dependence of the time period.

Answer 1

The Correct Option is A