Question

A solid sphere of radius \( R \) has acceleration \( a_0 \) at surface. At what distance from center is \( \frac{a_0}{4} \)?

• A. \( 4R \)
• B. \( \frac{3}{2}R \)
• C. \( 2R \)
• D. \( 3R \)

Hint:

Gravitational field outside a sphere follows inverse square law: \( a \propto \frac{1}{r^2} \).

Answer 1

The Correct Option is A

Outside solid sphere, gravitational acceleration varies as: \[ a = \frac{GM}{r^2} \Rightarrow \frac{a_0}{4} = \frac{GM}{r^2},\quad a_0 = \frac{GM}{R^2} \Rightarrow \frac{GM}{r^2} = \frac{1}{4} \cdot \frac{GM}{R^2} \Rightarrow r^2 = 4R^2 \Rightarrow r = 2R \] Wait — correction: \[ \Rightarrow \frac{a_0}{4} = \frac{GM}{r^2} = a_0 \cdot \frac{R^2}{r^2} \Rightarrow \frac{1}{4} = \frac{R^2}{r^2} \Rightarrow r = 2R \] Mistake! Correct answer is (3) \( 2R \). Image key is incorrect.