Question:
A body weighs the same on the surfaces of two planets of densities \( p_1 \) and \( p_2 \). The ratio of the radii of the planets is:
A body weighs the same on the surfaces of two planets of densities \( p_1 \) and \( p_2 \). The ratio of the radii of the planets is:
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The relationship between density and radius can be derived using the formula for gravitational force and the definition of density.
Updated On: May 5, 2026
- \( \frac{p_2}{p_1} \)
- \( \frac{p_2^2}{p_1^2} \)
- \( \frac{p_2^{3/2}}{p_1^{3/2}} \)
- \( \frac{p_2^4}{p_1^4} \)
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The Correct Option is A
Solution and Explanation
- Weight on a planet: \[ W = mg = \frac{GMm}{R^2} \]
- Substitute mass of planet \( M = \frac{4}{3}\pi R^3 \rho \): \[ W = \frac{G \left(\frac{4}{3}\pi R^3 \rho\right) m}{R^2} = \frac{4}{3}\pi G m \rho R \]
- So, weight is proportional to: \[ W \propto \rho R \]
- Since weight is same on both planets: \[ \rho_1 R_1 = \rho_2 R_2 \]
- Hence: \[ \frac{R_2}{R_1} = \frac{\rho_1}{\rho_2} \]
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