First slide

Gravitational field intensity

Question

A mass m is placed at point P which lies on the axis of a ring of mass M and radius R at a distance R from its centre. The gravitational force on mass m is

Moderate

A

$\frac{GMm}{\sqrt{2} R^{2}}$
B

$\frac{GMm}{2 R^{2}}$
C

$\frac{GMm}{2 \sqrt{2} R^{2}}$
D

$\frac{GMm}{4 R^{2}}$

Solution

The situation is as shown in the figure.
Gravitational force on an object of mass m at point P at distance d from centre O lying on the axis of the circular ring of radius R and mass M is given by
$F = \frac{GMmd}{{(R^{2} + d^{2})}^{3 / 2}}$
When d = R, then
$F = \frac{GMmR}{{(R^{2} + R^{2})}^{3 / 2}} = \frac{GMmR}{{(2 R^{2})}^{3 / 2}} = \frac{GMmR}{2^{3 / 2} R^{3}} = \frac{GMm}{2 \sqrt{2} R^{2}}$

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