A metal wire of uniform mass density having length \(L\) and mass \(M\) is bent to form a semicircular arc and a particle of mass \(m\) is placed at the centre of the arc. The gravitational force on the particle by the wire is _______.

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Answer & Solution

Correct Answer

(D) \(\frac{2 \mathrm{GmM} \pi}{\mathrm{L}^2}\)

Step-by-step Solution

Detailed explanation

\(R = \frac{L}{\pi}\)

\(F = \int_{-\pi/2}^{\pi/2} \frac{G m (\frac{M}{L} R d\theta)}{R^2} \cos\theta\)

\(F = \frac{GmM}{LR} [\sin\theta]_{-\pi/2}^{\pi/2}\)

\(F = \frac{GmM}{LR} (1 - (-1)) = \frac{2GmM}{LR}\)

\(F = \frac{2GmM}{L(L/\pi)} = \frac{2GmM\pi}{L^2}\)

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