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A particle of mass m is placed at a distance of 4R from the centre of a huge uniform sphere of mass M and radius R. A spherical cavity of diameter R is made in the sphere as shown in the figure. If the gravitational interaction potential energy of the system of mass m and the remaining sphere after making the cavity is λGMm28R Find the value of λ
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Solution
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Since the radius of the cavity is half the radius of the sphere it was made in, the mass removed to make it is one-eighth of the mass of the sphere.

Gravitational Potential Energy =PEsys

PEsys=PEdue to spherePEdue to removed mass

PEdue to sphere=GMm4RPEdue to removed mass=Gm(M8)4RR2=GMm28R.

So, PEsys=GMm4R+GMm28R=6GMm28R

Therefore, λ=6

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