Question

Three uniform spheres each having a mass M and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two.

Answer 1

Step 1: Given information

Three uniform spheres,

M is the mass.

a is the radius.

Step 2: To find

We have to find the magnitude of the gravitational force on any of the spheres due to the other two.

Step 3: Calculation

We know the formula,

${\mathrm{F}}_1={\mathrm{F}}_2=\frac{{\mathrm{GM}}^2}{{\left(2\mathrm{a}\right)}^2}$

Here,

F₁ and F₂ are two forces acting at an angle than the resultant acting on the sphere.

Mow,

$$\begin{gathered} {\mathrm{F}}_{\mathrm{net}}={\mathrm{F}}_1\cos 30°+{\mathrm{F}}_2\cos 30° \\ =\left(\frac{\sqrt{3}}{2}\right)\times 2\times \frac{{\mathrm{GM}}^2}{{\left(2\mathrm{a}\right)}^2} \\ =\frac{\sqrt{3}{\mathrm{GM}}^2}{4{\mathrm{a}}^2} \end{gathered}$$

Therefore, the magnitude of the gravitational force on any of the spheres due to the other two is $\frac{\sqrt{3}{\mathrm{GM}}^2}{4{\mathrm{a}}^2}$.