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A ring has a mass $M$ and radius $R$. The distance of the point on its geometric axis from its centre at which the gravitational field is strongest is
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Verified Answer
The correct answer is:
$\frac {R}{\sqrt 2}$
Gravitational field from the centre is given by
$\mathrm{E}=\frac{\mathrm{GMx}}{\left(\mathrm{R}^2+\mathrm{x}^2\right)^{3 / 2}}$
The gravitational field is strongest. So
$\begin{aligned}
& \frac{d E}{d x}=0 \\
& 0=\frac{\left(R^2+x^2\right)^{3 / 2} G M-G M x \frac{3}{2}\left(R^2+x^2\right)^{1 / 2} 2 x}{\left(R^2+x^2\right)^3} \\
& \Rightarrow 0=G M\left(R^2+x^2\right)^{1 / 2}\left[R^2+x^2-3 x^2\right] \\
& \Rightarrow x=\frac{R}{\sqrt{2}}
\end{aligned}$
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