Question

A wire of length $l$ and mass $m$ is bent in the form of a semicircle. The gravitational field intensity at the centre of semicircle is:

• A. $\frac{Gml}{\pi l}$ along x-axis
• B. $\frac{Gm}{\pi l}$ along y-axis
• C. $\frac{2\pi Gm}{l^2}$ along y-axis
• D. $\frac{2\pi Gm}{l^2}$ along x-axis

Answer 1

The correct option is C $\frac{2\pi Gm}{l^2}$ along y-axis

Let the mass per unit length of the wire be $\lambda =\frac{m}{l}=\frac{m}{\pi R}$, where $R$ is the radius of the circle.

Thus horizontal components of the gravitational field intensity at the center cancel each other.

The vertical component of the same =${\int }_{-\pi /2}^{\pi /2}\frac{G\left(\lambda Rd\theta \right)}{R^2}cos\theta$$=\frac{2Gm}{\pi R^2}=\frac{2\pi Gm}{l^2}$