Question

Two solid spheres each of radius \( R \) made of same material are placed in contact with each other. If the gravitational force acting between them is \( F \), then

• A. \( F \propto R^4 \)
• B. \( F \propto R^3 \)
• C. \( F \propto R^2 \)
• D. \( F \propto R \)

Hint:

Remember that mass of a sphere is proportional to \( R^3 \), and use Newton’s law of gravitation to analyze the force between touching spheres.

Answer 1

The Correct Option is A

Step 1: Mass of a solid sphere
Each sphere is of radius \( R \) and made of the same material, so their mass \( m \) is proportional to the volume: \[ m \propto R^3 \] Step 2: Gravitational force between two masses
\[ F = G \frac{m^2}{d^2} \] Since they are in contact, the distance between centers \( d = 2R \) \[ F \propto \frac{(R^3)^2}{(2R)^2} = \frac{R^6}{4R^2} = \frac{1}{4} R^4 \Rightarrow F \propto R^4 \]