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

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  

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

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

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

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

$F=\frac{G{m}^2}{3L^2}$  .