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a solid sphere of uniform density and radius of 4m is located having centre at origin of co ordinate system. two small spheres, each of radius 1m and centres at A(2m,0,0) and B(-2m,0,0) are taken out from solid sphere. mass of remaining body is (186/23) kg. find potential at centre of sphere.?
Asked by NSSORATHIYA sorathiya | 23 Jun, 2019, 10:17: AM
answered-by-expert Expert Answer
Potential V of full sphere at O = -(3/2)G×M/4  = -(3/8)G×M ....................(1)
 
where G is constsant of gravitation, M is mass of full sphere.
 
Potential VA of sphere of radius 1 m placed at A(2,0,0) at O = - G×m/2  ................(2)
 
Potential VB of sphere of radius 1 m placed at B(-2,0,0) at O = - G×m/2 ....................(3)
 
where m is mass of small sphere of radius 1 m. Since Mass of sphere is proportioanl to cube of radius, we have m = M/64
 
hence VA = VB = - (G/2) (M/64)  = - G×M/128 ...............(4)
 
If V ' is the potential of full sphere after cutting out two spheres of radius 1 m at A and B, then we have
 
V = V ' + VA + VB ..........................(5)
 
Then V ' = V - VA - VB = -(3/8) G×M + (1/128)G×M + (1/128) G×M = -(23/64)G×M ....................(6)
 
Mass of Sphere is calculated as,   M - 2×(M/64) = (31/32) M = (186/23) kg   or   M  = (186/23)×(32/31)= (192/23) kg
 
Hence we get V ' from eqn.(6) after substituting the values as,   V ' = -(23/64)×6.674×10-11 × (192/23) ≈ 20×10-11 Joules
Answered by Thiyagarajan K | 23 Jun, 2019, 13:19: PM

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