Question

A uniform thin rod of mass $m$ and length $R$ is placed normally on surface of earth as shown. The mass of earth is $M$ and its radius is $R$. Then the magnitude of gravitational force exerted by earth on the rod is :

• A. $\frac{GMm}{4R^2}$
• B. $\frac{GMm}{2R^2}$
• C. $\frac{4GMm}{9R^2}$
• D. $\frac{GMm}{8R^2}$

Answer 1

The correct option is B $\frac{GMm}{2R^2}$

Consider an element of length $dx$ and mass $dm$
Here, $dm=\left(\frac{m}{R}dx\right)$
Force on element $dx$ of the rod is
$dF=\left(\frac{m}{R}dx\right)\frac{GM}{x^2}$

Total force on rod is
$F=\int dF=\frac{GMm}{R}\underset{R}{\overset{2R}{\int }}\frac{1}{x^2}dx$
$=\frac{GMm}{R}{[-\frac{1}{x}]}_R^{2R}$
$=\frac{GMm}{2R^2}$