Four identical particles of mass \(m\) are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is \(\left(\frac{2 \sqrt{2}+1}{32}\right) \frac{\mathrm{Gm}^2}{\mathrm{~L}^2}\), the length of the sides of the square is _______.

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Answer & Solution

Correct Answer

(B) \(4 \mathrm{~L}\)

Step-by-step Solution

Detailed explanation

\(F_{\text {net }}=\sqrt{2} F+F^{\prime}\)

\(F=\frac{G m^2}{a^2} \text { and } F^{\prime}=\frac{G m^2}{(\sqrt{2} a)^2}\)

\(F_{\text {net }}=\sqrt{2} \frac{G m^2}{a^2}+\frac{G m^2}{2 a^2}\)

\(\left(\frac{2 \sqrt{2}+1}{32}\right) \frac{G m^2}{L^2}=\frac{G m^2}{a^2}\left(\frac{2 \sqrt{2}+1}{2}\right)\)

\(a=4 L\)

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