Question

Spheres of the same metarial and same radius r are touching each other. Show that grevitational force between them is directly proportional to r^(4).

Answer 1

Gravitational force between them is directly proportional to $r^4$

Given:

Spheres of the same material and same radius r are touching each other.

To find:

Show that gravitational force between them is directly proportional to $r^4$

Formula used:

Gravitational force = $\frac{G m_{1} m_{2}}{r^{2}}$

Density = $\frac{Mass}{Volume}$

Where G = gravitational constant

           $m_1$ and $m_2$ = Mass of the object

                     r = center distance between two mass

Explanation:

Density = $\frac{Mass}{Volume}$

    Mass = Density × Volume

              $=\left(\frac{4}{3} \pi r^{3}\right) \rho$

Gravitational force = $\frac{G m_{1} m_{2}}{r^{2}}$

Gravitational force = $\frac{G\left(\frac{4}{3} \pi r^{3}\right)^{r^{2}}\left(\frac{4}{3} \pi r^{3}\right) \rho^{2}}{r^{2}}$

Center distance between two Sphere = 2r because both sphere are touching each other.

Gravitational force = $\frac{G\left(\frac{4}{3} \pi r^{3}\right)^{r^{2}}\left(\frac{4}{3} \pi r^{3}\right) \rho^{2}}{2r^{2}}$

So from above equation

Gravitational force between them is directly proportional to $r^4$

To learn more...

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