Question

A uniform ring of mass $m$ and radius a is placed directly above a uniform sphere of mass $M$ and of equal radius. The centre of the ring is at a distance $\sqrt{3}a$ from the centre of the sphere. Find the gravitational force exerted by the sphere on the ring.

• A. $\frac{\sqrt{2}GMm}{8a^2}$
• B. $\frac{\sqrt{3}GMm}{4a^2}$
• C. $\frac{\sqrt{3}GMm}{8a^2}$
• D. $\frac{\sqrt{3}GMm}{7a^2}$

Answer 1

The correct option is C $\frac{\sqrt{3}GMm}{8a^2}$

$dF=$ force on a small mass ${}^{\prime }d{m}^{\prime }$ of the ring by the sphere.
Net force on ring$=\sum \left(dF\sin \theta \right)$ or $\int dF\sin \theta$

$=\sum \frac{GM\left(dm\right)}{{\left(2a\right)}^2}\times \frac{\sqrt{3}}{2}$

$=\frac{\sqrt{3}GM}{8a^2}\sum \left(dm\right)$

But $\sum \left(dm\right)=m$, the mass of whole ring.

$\therefore$ Net force$=\frac{\sqrt{3}GMm}{8a^2}$