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. \( \frac{p_2}{p_1} \)
• B. \( \frac{p_2^2}{p_1^2} \)
• C. \( \frac{p_2^{3/2}}{p_1^{3/2}} \)
• D. \( \frac{p_2^4}{p_1^4} \)

Hint:

The relationship between density and radius can be derived using the formula for gravitational force and the definition of density.

Answer 1

The Correct Option is A

- The weight of a body on the surface of a planet is given by \( W = \frac{GMm}{R^2} \), where \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the body, and \( R \) is the radius of the planet.
- The mass of the planet is related to its density and volume: \( M = \frac{4}{3} \pi R^3 p \), where \( p \) is the density.
- Since the body weighs the same on both planets, we equate the expressions for weight on both planets, which leads to the ratio of the radii \( R_2/R_1 = \sqrt[3]{p_1/p_2} \).
- Hence, the ratio of the radii of the planets is \( \frac{p_2}{p_1} \).