A metal wire of uniform mass density having length L and mass M is… | JEE Main 2024 (4 Apr, Shift 1)

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Newton's Law of Gravitation

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: (d)

Semicircle of length $L$ : radius $r = \frac{L}{\pi}$ , linear density $\lambda = \frac{M}{L}$ .
Components along the diameter cancel; along the symmetry axis $F = \int_0^\pi \frac{Gm\lambda r\,d\theta}{r^2}\sin\theta = \frac{2Gm\lambda}{r}$ .
$F = \frac{2Gm(M/L)}{L/\pi} = \frac{2\pi GMm}{L^2}$
So the answer is (d).

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