Question

A uniform ring of mass $2m$ and radius $a$ is placed directly above the uniform sphere of mass $M$ and of same radius. The centre of ring is at distance $\sqrt{3}a$ from the centre of sphere. The gravitational force exerted by the sphere on the ring is $N\frac{GMm}{a^2}$ units. Write the value of $\left[100N\right],$ where $\left[\right]$ is the greatest integer function.

Answer 1

Answer: 43

$dF=G\frac{Mdm}{4a^2}$
$F=\Sigma dFcos\theta$
$=\sum \frac{GMdm}{4a^2}cos\theta$
$=\frac{GM}{4a^2}\cdot \frac{\sqrt{3a}}{2a}\sum dm$
$=\frac{\sqrt{3}GM}{8a^2}\cdot 2m$
$=\frac{\sqrt{3}}{4}\cdot \frac{GMm}{a^2}$
$\therefore F=0.43\frac{GMm}{a^2}$
$\therefore N=0.43$
Alternate method

Gravitational field $(\bar{E})=\frac{G2md}{{\left(a^2+d^2\right)}^{3/2}}$
$=\frac{\sqrt{3}\times 2GM}{8a^2}$
$\therefore \bar{E}=\frac{\sqrt{3}}{4}\frac{Gm}{a^2}$
Force $=M\bar{E}=\frac{\sqrt{3}}{4}\frac{GMm}{a^2}$
$\therefore F=0.433\frac{GMm}{a^2}$
$N=0.43$
$\left[100N\right]=43$