A point mass m and a uniforms thin rod (mass M and length L) are shown in figure. Find the gravitational force of attraction between them

Consider a small element in the Rod whose mass $\mathrm{dm}=\left(\frac{\mathrm{M}}{\mathrm{L}}\right)\mathrm{dx}$

The force between the point mass and element is

$\mathrm{df}=\frac{\mathrm{Gmdm}}{{\mathrm{x}}^2}=\frac{\mathrm{Gm}\left(\frac{\mathrm{M}}{\mathrm{L}}\right)\mathrm{dx}}{{\mathrm{x}}^2}$

$\therefore \mathrm{F}={\int }_{\mathrm{a}}^{\mathrm{a}+\mathrm{L}} \mathrm{dF}={\int }_{\mathrm{a}}^{\mathrm{a}+\mathrm{L}} \frac{\mathrm{Gmm}}{\mathrm{L}}\frac{\mathrm{dx}}{{\mathrm{x}}^2}\Rightarrow \mathrm{F}=\frac{\mathrm{GMm}}{\mathrm{a}(\mathrm{a}+\mathrm{L})}$