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 $\mathrm{m}$ is placed at the centre of the arc. The gravitational force on the particle by the wire is :

• A

  $\frac{\mathrm{GmM} \pi^2}{\mathrm{~L}^2}$
• B

  $\frac{\mathrm{GMm} \pi}{2 \mathrm{~L}^2}$
• C

  $0$
• Correct answer
  D

  $\frac{2 \mathrm{GmM} \pi}{\mathrm{L}^2}$

Solution

We have $R=\frac{L}{\pi}$ $\begin{aligned} & \mathrm{g}_0=\frac{2 \mathrm{G} \frac{\mathrm{M}}{\mathrm{L}}}{\mathrm{R}}=\frac{2 \mathrm{GM} \pi}{\mathrm{L}^2} \\ & \therefore \mathrm{F}_{\mathrm{m}}=\mathrm{mg}_0=\frac{2 \mathrm{GM} \pi \mathrm{m}}{\mathrm{L}^2} \end{aligned}$
Hence option (4)

Hence, the correct answer is option: 'D'.