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Two metal spheres of equal radius $r$ and equal density are touching each other. If the force of gravitationbetween them is $F$ , it can be said that

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F \propto r^{4}

F \propto r

F \propto r^{2}

F \propto \frac{1}{r^{2}}

answer is A.

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Detailed Solution

If the force of gravitation

between them is

F

, it can be said that

F \propto r^{4}

.
The force of gravitation

(F)

between two objects is given by

F = \frac{G m_{1} m_{2}}{r^{2}}

where

G

is the universal gravitation constant,

m_{1}

is the mass of the first object,

m_{2}

is the mass of second object and

r

is the distance between them.
The distance between them will be the distance between their centres which is

2 r

.
We know that mass of a sphere is

\frac{4}{3} π r^{3} \times density

, where

r

is the radius of the sphere.
Now, mass of spheres, considering its density as

d,

will be

\frac{4}{3} πr^{3} d

\Rightarrow F = \frac{G (\frac{4}{3} πr^{3} d) (\frac{4}{3} πr^{3} d)^{}}{{(2 r)}^{2}} ​

\Rightarrow F = \frac{1}{9} G​π^{2} d^{2} \times r^{4}

Therefore,

F \propto r^{4}

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