Question

The gravitational force between earth of radius $R$ and mass $M$ and rod of length $R$ and mass $m$ placed as shown in the figure is $\frac{12GMm}{n{R}^2}$ . Find the value of $n$ .

Answer 1

Answer: 24

Net force on the rod is equal to centripetal force.
Here, mass of small element at distance $x$ is $dm=\frac{m}{R}dx$
So, gravitational force, $d{F}_g=\frac{GMdm}{{\left(R+x\right)}^2}=\frac{GMmdx}{R{\left(R+x\right)}^2}$
$d{F}_g=\frac{GMm}{{\left(R+x\right)}^{2}R}dx$

$=\frac{GMm}{R}\underset{0}{\overset{R}{\int }}\frac{dx}{{\left(R+x\right)}^2}$
$F_g=\frac{GMm}{2R^2}$
$F_g=\frac{12GMm}{24R^2}$
Hence, $n=24$ .