Question

Two identical copper spheres with uniform density each of radius $R$ are in contact with each other. If the gravitational attraction between them is $F$, then

• A. $F\propto R^2$
• B. $F\propto R^4$
• C. $F\propto \frac{1}{R^2}$
• D. $F\propto \frac{1}{R^4}$

Answer 1

The correct option is B $F\propto R^4$

According to the Universal law of Gravitation, the gravitational force between two bodies is directly proportional to the mass of the two bodies and inversely proportional to the square of the distance between them.
The two identical spheres with uniform density behave as point masses with all the mass concentrated at the centre.

Mass=density $\times$ volume

Let $M$ be the mass and $\rho$ be the density of each sphere.
Volume of each sphere $=\frac{4}{3}\pi R^3$,
where $R$ is the radius of the sphere.

So, the mass of each sphere is $\frac{4}{3}\pi R^3\rho$

The distance between their centres is $2R$.

So, the force of attraction between them is $F=\frac{G{M}^2}{(2R{)}^2}$,
where $G$ is the universal gravitational constant.

$\therefore F=G\left(\frac{4}{3}\pi R^3\rho \right)\left(\frac{4}{3}\pi R^3\rho \right)\frac{1}{{\left(2R\right)}^2}$

$⟹F\propto \frac{R^6}{R^2}$

$⟹F\propto R^4$