Consider a thin spherical shell of uniform density of mass M and radius R:

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The gravitational field inside the shell will be zero

The gravitational self energy of shell is GM22R

Attractive force experience by unit area of the shell pull the other half is GM22R2

Net gravitational force with which one hemisphere of the shell attracts other, is GM28R2

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Detailed Solution

Say the shell has acquired a mass m and further a mass dM is to be added dW=VdM=−GMdMR or,W=∫0MGMdMR =GM22R= Self energy =USay F is now the attractive force per unit area. if the shell expands from R to R+dR then work done by attractive force is−F×4πR2dR since this is the work done by gravitational field, this may be equal to reduction in gravitational potential energy.or, −πR2FdR=dU orF=−14πR2dUdR=GM28πR4 Now total force experienced by one hemisphere=F×πR2 (Vector addition adds only components perpendicular to the equatorial plane)⇒ Net gravitational force FG=GM28R2

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