First slide

Gravitational field intensity

Question

A uniform ring of mass m is lying at a distance $\sqrt{3}$ a form the centre of a sphere of mass M just over the sphere (where a is the radius of the ring as well as that of the sphere). Then magnitude of gravitational force between them is

Moderate

A

$\frac{GMm}{8 a^{2}}$
B

$\frac{GMm}{\sqrt{3} a^{2}}$
C

$\sqrt{3} \frac{GMm}{a^{2}}$
D

$\sqrt{3} \frac{GMm}{8 a^{2}}$

Solution

For a small mass dm, distance from the center of sphere is 2a. force along ‘dm’ $\frac{G, M, dm}{4 a^{2}} = F$
Component of force F sin get cancelled net force 2Fcos $θ$ .
For the entire ring- $\frac{GM}{4 a^{2}} \frac{m}{2} \times 2 cosθ$
$\frac{GMm}{4 a^{2}} \frac{\sqrt{3} a}{2 a} = \frac{\sqrt{3} GMm}{8 a^{2}}$
Alternative solution:-
field intensity due the ring at the centre of the sphere is
$g = \frac{GMx}{{(R^{2} + x^{2})}^{\frac{3}{2}}} = \frac{GMa}{{(a^{2} + 3 a^{2})}^{\frac{3}{2}}} = \frac{\sqrt{3} GM}{8 a^{2}} F = mg = \frac{\sqrt{3} GMm}{8 a^{2}}$

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