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Question

Consider two solid uniform spherical objects of the same density ρ. One has radius R and the other has radius 2R. They are in outer space where the gravitational fields from other objects are negligible. If they are arranged with their surface touching, what is the contact force between the objects due to their traditional attraction ?

A

Gπ2R4
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B

12881Gπ2R4ρ2
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C

12881Gπ2
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D

12887π2R2G
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Solution

The correct option is B
12881Gπ2R4ρ2
The arrangement of the given spherical objects is shown below.


Let m1 and m2 be the masses of the smaller and larger spheres, respectively.

Using mass- density relation,

m1=43πR3ρ, and m2=43π(2R)3ρ

Free body diagram of mass m1 is given below.


In the figure, Fg is gravitational force of attraction on m1 due to m2 and N is normal reaction on m1.

Since, m1 is in equilibrium, so from FBD, we get

N=Fg=Gm1m2r2

Substituting the values, from figure

Fg=G×(43πR3ρ)(43×8πR3ρ)(3R)2

Fg=12881Gπ2R4ρ2

Hence, option (b) is the correct choice.

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