Physics - Gravitation Question with Solution | TestHub

Mass of each metal sphere is given by $M=\text{Volume}\times \text{Density}=\frac{4}{3}\pi R^3\rho$

Since the two identical spheres are kept in contact, the distance between their centers is $d=R+R=2R$.

The gravitational force of attraction between them is:

$F=\frac{G{M}_1{M}_2}{d^2}=\frac{G{M}^2}{(2R{)}^2}$

Substituting the value of $M$:

$F=\frac{G{\left(\frac{4}{3}\pi R^3\rho \right)}^2}{4R^2}$

$F=\frac{G\times \frac{16}{9}{\pi }^2{R}^6{\rho }^2}{4R^2}$

$F=\frac{4}{9}G{\pi }^2{R}^4{\rho }^2$

Therefore, $F\propto R^4{\rho }^2$.

Answer: $R^4{p}^2$