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Question

A uniform sphere of mass M and radius R exerts a force F on a small mass m situated at a distance of 2R from the centre O of the sphere. A spherical portion of diameter R is cut from the sphere as shown in the figure. The force of attraction between the remaining part of the sphere and the mass m will be:


A

7F9

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B

2F3

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C

4F9

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D

F3

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Solution

The correct option is A

7F9


The force of attraction between the complete sphere and mass m is:
F=GmM(2R)2=GmM4R2
Mass of complete sphere is: M=4π3R3×ρ
where ρ is the density of the sphere. [Density =MassVolume]
Mass of the cut-out portion is m=4π3(R2)3×ρ=M8
Now, the distance between the centre of the cut out portion and mass m: 2RR2=3R2
Hence, the force of attraction between the cut out portion and mass m is:
f=Gmm(3R2)2=G(M8)m9R24

=GmM4R2×29=2F9
Therefore, the force of attraction between the remaining part of the sphere and mass m: Ff=F2F9=7F9


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