Question

Find the gravitational force of interaction between the mass $m$ and an infinite rod of varying mass density $\lambda$ such that $\lambda \left(x\right)=\lambda /x$, where $x$ is the distance from mass $m$. Given that mass $m$ is placed at a distance $d$ from the end of the rod on its axis as shown in figure.

• A. $\frac{Gm\lambda }{d^2}$
• B. $\frac{Gm\lambda }{2d^2}$
• C. $\frac{Gm\lambda }{3d^2}$
• D. $\frac{3Gm\lambda }{2d^2}$

Answer 1

The correct option is B $\frac{Gm\lambda }{2d^2}$

Let's consider a small mass $dm$ at a distance of $x$.


So, total force is given by,
$dF=\frac{Gmdm}{x^2}$
where, $dm=\frac{\lambda }{x}dx$​​​​​
so, $dF=\frac{Gm\frac{\lambda }{x}dx}{x^2}$

By using integration for a whole body,
$\int dF={\int }_d^{\infty }\frac{Gm\lambda }{x^3}dxF=\frac{-Gm\lambda }{2}[\frac{1}{x^2}{]}_d^{\infty }$

$\Rightarrow F=\frac{Gm\lambda }{2d^2}$