Question

In a solar system, the time period of revolution of a planet tracing a circular orbit of radius \(R\) is proportional to:

• A. \(R^3\)
• B. \(R^{1/2}\)
• C. \(R^{3/2}\)
• D. \(R^2\)

Hint:

Remember Kepler's Third Law: \[ T^2\propto R^3 \] For quick questions directly write \[ T\propto R^{3/2} \]

Answer 1

The Correct Option is C

Concept:

• Kepler's Third Law states: \[ T^2 \propto R^3 \] for planets revolving around the same star.

Step 1: Write the gravitational force.
\[ \frac{GMm}{R^2} \] This provides the necessary centripetal force. \[ \frac{GMm}{R^2} = m\frac{v^2}{R} \]

Step 2: Express velocity in terms of time period.
\[ v=\frac{2\pi R}{T} \] Substituting, \[ \frac{GM}{R^2} = \frac{4\pi^2R}{T^2} \]

Step 3: Find the relation between \(T\) and \(R\).
\[ T^2 = \frac{4\pi^2R^3}{GM} \] Therefore, \[ T^2\propto R^3 \] \[ T\propto R^{3/2} \] \[ \boxed{\text{Option (C)}} \]