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A uniform ring of mas m and radius a is ...

A uniform ring of mas m and radius a is placed directly above a uniform sphere of mass M and of equal radius. The center of the ring is at a distance `sqrt3 a` from the center of the sphere. Find the gravitational force exerted by the sphere on the ring.

Text Solution

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The gravitatioN/Al field at any on the ring due to the sphere is equal to the field due to a isngle particle of mass M laced at the centre of the sphre.Thus, theforce on the ring due to the sphere is also equla to the force on it by a particle of mas M placed at this point.By Newton's third laaw it is equal to the force on the particle by the rig. Now the gravitatioN/Al field due to the ring at a distance `d=sqrt3` a on its axis is
`E=(Gmd)/((a^2+d^2)^(3/2)`=(sqrt3Gm)/(8a^2)`

The force on a article of masss M placed here is
`F=ME`
`=(sqrt3GMm)/(8a^2)`
This is also the force due to the sphere on the ring
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