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 \(R\)?
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 \(R\)?
Show 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.
Updated On: Jul 29, 2026
- \[ V\propto R^{-3/4}, \qquad T\propto R^{7/4} \]
- \[ V\propto R^{-5/4}, \qquad T\propto R^{3/2} \]
- \[ V\propto R^{-1/2}, \qquad T\propto R^{5/4} \]
- \[ V\propto R^{-3/4}, \qquad T\propto R^{-7/4} \]
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
For circular motion, the central force provides the centripetal force.
\[
F=\frac{mV^2}{R}.
\]
Given,
\[
F\propto R^{-5/2}.
\]
Step 1: Find the dependence of orbital velocity on \(R\). Using \[ \frac{mV^2}{R} \propto R^{-5/2}, \] we get \[ V^2 \propto R^{-5/2}\cdot R. \] \[ V^2 \propto R^{-3/2}. \] Therefore, \[ V \propto R^{-3/4}. \]
Step 2: Find the dependence of time period on \(R\). For circular motion, \[ T=\frac{2\pi R}{V}. \] Substituting \[ V\propto R^{-3/4}, \] \[ T \propto R^{1+\frac34}. \] \[ T \propto R^{7/4}. \]
Step 3: Choose the correct option. Thus, \[ V\propto R^{-3/4} \] and \[ T\propto R^{7/4}. \] Therefore, \[ \boxed{ V\propto R^{-3/4}, \qquad T\propto R^{7/4} } \] \[ \boxed{\text{Answer = (A)}} \]
Step 1: Find the dependence of orbital velocity on \(R\). Using \[ \frac{mV^2}{R} \propto R^{-5/2}, \] we get \[ V^2 \propto R^{-5/2}\cdot R. \] \[ V^2 \propto R^{-3/2}. \] Therefore, \[ V \propto R^{-3/4}. \]
Step 2: Find the dependence of time period on \(R\). For circular motion, \[ T=\frac{2\pi R}{V}. \] Substituting \[ V\propto R^{-3/4}, \] \[ T \propto R^{1+\frac34}. \] \[ T \propto R^{7/4}. \]
Step 3: Choose the correct option. Thus, \[ V\propto R^{-3/4} \] and \[ T\propto R^{7/4}. \] Therefore, \[ \boxed{ V\propto R^{-3/4}, \qquad T\propto R^{7/4} } \] \[ \boxed{\text{Answer = (A)}} \]
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