A uniform ring of mass 2m and radius a is placed directly above the uniform sphere of mass M and of same radius. The centre of ring is at distance `sqrt3a` from the centre of sphere. The gravitational force exerted by the sphere on the ring is `N(GMm)/a^2` units. Calculate is the value of N?

Answer

0.43

Explanation

`dF=G(Mdm)/(4a^2)`
`F = sumdFcos theta`
`= sum(GMdm)/(4a^2)cos theta`
`=(GM)/(4a^2).(sqrt3a)/(2a)sumdm`
`= sqrt(3GM)/(8a^2).2m`
`= sqrt3/4.(GMm)/a^2`
`:. N = 0.43`
Alternate method

Gravitational field `(barE)=(G2md)/((a^2+d^2)^(3//2))`
`= (sqrt3xx2)/(8a^2)`
`:. vecE=sqrt3/4(GM)/a^2`
Force `=MbarE=sqrt3/4(GMm)/a^2`
`:. F= 0433 (GMm)/(a^2)`
`N = 0.43`