Question

Two solid spheres of same size of a certain metal are placed km contact with each other. Prove that the gravitational force acting between them KS directly proportional to the fourth poet LG the radius.

Answer 1

Correct Question:

Two solid spheres of same size of a certain metal are placed in contact with each other. Prove that the gravitational force acting between them is directly proportional to the fourth power of the radius.

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Given:

• Spheres have same size.
• Are made up of same materials

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Need to prove:

• Gravitational force$\propto \: {r}^{4}$

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Solution:

We know,

Density = Mass ÷ Volume

→ Mass = Density × Volume

→ Mass = $\rho \: ( \dfrac{4}{3}\pi {r}^{3} )$

As the two spheres are made up of same materials and have same size so their masses will be equal.

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$F= \dfrac{GMm}{ {r}^{2} }$

$\implies F = \dfrac{G( \dfrac{4}{3}\pi {r}^{3} \rho) ( \dfrac{4}{3}\pi {r}^{3} \rho)}{ {(2r)}^{2} }$

$\implies F = \dfrac{G \times \dfrac{16}{3} {\pi}^{2} {r}^{6} { \rho}^{2} }{4 {r}^{2} }$

$\implies F = G \times \dfrac{6}{9} \times {\pi}^{2} \times { \rho}^{2} \times \dfrac{ {r}^{6} }{ {r}^{2} }$

Considering everything as constant except the radius, we get...

$\implies F \propto \: {r}^{6 - 2}$

$\red{\large{\bold{\boxed{\implies F \propto \: {r}^{4}}}}}$