Question

A small mass and a thin uniform rod each of mass '$m$' are positioned along the same straight line as shown. The force of gravitational attraction exerted by the rod on the small mass is

• A. $\frac{G{m}^2}{2L^2}$
• B. $\frac{G{m}^2}{3L^2}$
• C. $\frac{G{m}^2}{4L^2}$
• D. $\frac{3G{m}^2}{4L^2}$

Answer 1

The correct option is B $\frac{G{m}^2}{3L^2}$

The segment of the rod with length dx has a mass dm. the mass per unit length is constant.

Force of gravitational attraction exerted by rod on the small mass dm is

$F=-Gm{\int }_L^{3L}\frac{dm}{x^2}$ ,

$F=-Gm{\int }_L^{3L}\frac{m}{2L}\frac{dx}{x^2}$ ,

$F=\frac{-G{m}^2}{2L}{\int }_L^{3L}\left(\frac{dx}{x^2}\right)$ ,

$F=\frac{-G{m}^2}{2L}\left(\frac{-1}{x}\right)$ ,

$F=\frac{G{m}^2}{3L^2}$ .