Question

Two hard metal spheres of the same material, of the density $\mathrm{\rho }$, and of equal radius R are just touching each other. Calculate the force of gravitational attraction between them.

Answer 1

Given:

The density of the metal spheres = $\mathrm{\rho }$.

The radius of the spheres = R.

Required to find:

The force of gravitational attraction between the two metal spheres.

Solution:

Let the mass of each sphere be (m).

The mass (m) of the sphere = volume of the sphere $\left(\frac{4}{3}\pi R^3\right)$ x the density $\mathrm{\rho }$ of the sphere.

As the spheres are touching to each other, the distance between them is ‘2R’.

From the equation $F=\frac{G{m}_1{m}_2}{{\left(R\right)}^2},whereRisthedis\tan cebetweenthetwospheres,$ we get

$$\begin{gathered} F=\frac{Gx\left(\frac{4}{3}\pi R^3\rho \right)x\left(\frac{4}{3}\pi R^3\rho \right)}{{\left(2R\right)}^2}=\frac{Gx\left(\frac{16}{9}{\pi }^2{R}^6{\rho }^2\right)}{{4R}^2} \\ =\frac{Gx\left(\frac{4}{9}{\pi }^2{R}^6{\rho }^2\right)}{R^2}=\frac{4G{\pi }^2{R}^4{\rho }^2}{9}N \end{gathered}$$

Hence, the force of attraction between the two spheres is $\frac{4G{\pi }^2{R}^4{\rho }^2}{9}N$.