Question:
A uniform solid sphere of radius \(R\) produces a gravitational acceleration of \(a_0\) on its surface. The distance of the point from the centre of the sphere where the gravitational acceleration becomes
\[
\frac{a_0}{4}
\]
is
A uniform solid sphere of radius \(R\) produces a gravitational acceleration of \(a_0\) on its surface. The distance of the point from the centre of the sphere where the gravitational acceleration becomes
\[
\frac{a_0}{4}
\]
is
Show Hint
Outside a spherical body, gravitational acceleration varies inversely as the square of the distance:
\[
g\propto \frac{1}{r^2}
\]
Updated On: Jun 22, 2026
- \(4R\)
- \(\dfrac{3R}{2}\)
- \(2R\)
- \(3R\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Write the gravitational acceleration on the surface.
For a sphere of mass \(M\) and radius \(R\), gravitational acceleration at the surface is \[ a_0=\frac{GM}{R^2} \] where \(G\) is the gravitational constant.
Step 2: Write the gravitational acceleration at a distance \(r\) from the centre.
For a point outside the sphere, \[ a=\frac{GM}{r^2} \] According to the question, \[ a=\frac{a_0}{4} \] Substitute the expressions: \[ \frac{GM}{r^2}=\frac{1}{4}\left(\frac{GM}{R^2}\right) \]
Step 3: Simplify the equation.
Cancel \(GM\) from both sides: \[ \frac{1}{r^2}=\frac{1}{4R^2} \] Taking reciprocal, \[ r^2=4R^2 \] \[ r=2R \]
Step 4: Final conclusion.
Hence, the required distance from the centre of the sphere is \[ \boxed{2R} \]
For a sphere of mass \(M\) and radius \(R\), gravitational acceleration at the surface is \[ a_0=\frac{GM}{R^2} \] where \(G\) is the gravitational constant.
Step 2: Write the gravitational acceleration at a distance \(r\) from the centre.
For a point outside the sphere, \[ a=\frac{GM}{r^2} \] According to the question, \[ a=\frac{a_0}{4} \] Substitute the expressions: \[ \frac{GM}{r^2}=\frac{1}{4}\left(\frac{GM}{R^2}\right) \]
Step 3: Simplify the equation.
Cancel \(GM\) from both sides: \[ \frac{1}{r^2}=\frac{1}{4R^2} \] Taking reciprocal, \[ r^2=4R^2 \] \[ r=2R \]
Step 4: Final conclusion.
Hence, the required distance from the centre of the sphere is \[ \boxed{2R} \]
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