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  3. Physics
  4. A uniform ring of mass m and radius r is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance r√3 as shown in the figure. The gravitational force exerted by the sphere on the ring will be

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Q. A uniform ring of mass m and radius $r$ is placed directly above a uniform sphere of mass $M$ and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance $r\sqrt{3}$ as shown in the figure. The gravitational force exerted by the sphere on the ring will be

Gravitation

Solution:

image
$dF = G \frac{Mdm}{4r^{2}}$
$F = \Sigma dF\,cos\,\theta$
$= \Sigma \frac{GMdm}{4r^{2}} cos\,\theta$
$= \frac{GM}{4r^{2}} \times \frac{\sqrt{3}r}{2r} \Sigma dm$
$= \frac{\sqrt{3}GMm}{8r^{2}}$