First slide

Gravitational force and laws

Question

A solid sphere of mass M and a ring of mass m have their centres lying on the x-axis separated by a distance $2 \sqrt{2} r$ where r is the radius of the ring as shown in Fig. The gravitational force exerted by the sphere on the ring is

Moderate

A

$\frac{2 \sqrt{2} GMm}{27 r^{2}}$
B

$\frac{GMm}{9 \sqrt{2} r^{2}}$
C

$\frac{GMm}{8 r^{2}}$
D

$\frac{\sqrt{2} GMm}{9 r^{2}}$

Solution

The entire mass of the sphere can be assumed to be concentrated at its centre of mass O. Consider a small element of the ring at A. Let dm be the mass of the element (see Fig)
The gravitational force exerted by M on dm is
$dF = \frac{GMdm}{{(3 r)}^{2}} = \frac{GMdm}{9 r^{2}}$
This force can be resolved into x and y components as shown. Now consider another element at B. We find that components (namely $dFcos θ$ ) add up but y-components (namely of $dFsin θ$ ) cancel each other. Hence the x components of forces on all elements add up and y-components cancel. Hence, the total force exerted by the sphere on the ring is
$\begin{array}{l} F = \oint dFcos θ = \oint \frac{GMdm}{9 r^{2}} \times \frac{2 \sqrt{2}}{3} \\ = \frac{2 \sqrt{2} GM}{27 r^{2}} \oint dm = \frac{2 \sqrt{2} GMm}{27 r^{2}} \end{array}$
So the correct choice is (1).

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App