Question

The figure shows a uniform rod of length $I$ whose mass per unit length is $\lambda$. What is the gravitational force of the rod on a particle of mass $m$ located a distance $d$ from one end of the rod?

Answer 1

consider a segment dx of the rod at a distance x from the mass. The force due to this segment is

$dF=\frac{G\lambda dxm}{x^2}$

Thus, resultant force is

$F={\int }_d^{d+l}\frac{G\lambda dxm}{x^2}=G\lambda m(\frac{1}{d}-\frac{1}{d+l})=\frac{G\lambda lm}{d(d+l)}$

Answer is $\frac{G\lambda lm}{d(d+l)}$.