Question

If Mass (M) is split into two parts(m) and (M-m) , which are then separated by certain distance. What ratio the m/M willaximise the gravitational force between them?

Answer 1

Given:

• Mass of the whole body= M
• Mass of two parts = m and M-m

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Need to find:

• What ratio of m÷M will maximise the gravitational force between them.

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Solution:

Let r be the distance between m and M-m,

Then,

Gravitational force between them will be:

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

$\implies F = \dfrac{G(m)(M - m)}{ {r}^{2} }$

$\implies F = \dfrac{G(mM - {m}^{2}) }{ {r}^{2} }$

For F to be maximum,

$\dfrac{dF}{dm} = 0$

and

$\dfrac{ {d}^{2}F}{d {m}^{2} } < 0$

As M, m and r are constants so,

$\dfrac{d}{dm}\dfrac{G \: Mm - {m}^{2} }{ {r}^{2} } = 0$

$\implies \dfrac{G}{ {r}^{2} } M - 2m = 0$

But

$\implies \dfrac{G}{ {r}^{2} } \neq \: 0$

Thus,

$\implies M - 2m = 0$

$\implies M = 2m$

$\red{\bold{\boxed{\large{ \implies \dfrac{m}{M} = \dfrac{1}{2}}}}}$

Thus, the force will be maximum when the two particles will be identical (each having half of the mass of main body).