A mass M is broken into two parts, m and ( M - m ). How is m rela… - Vidyadip

Question

A mass M is broken into two parts, m and ( M - m ). How is m related to M so that the gravitational force between two parts is maximum?

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Answer

Let $m_1 = m, m_2 = M - m \text{F}=\text{G}=\frac{\text{m}(\text{M}-\text{M})}{\text{r}^2}=\frac{\text{G}}{\text{r}}(\text{Mm}-\text{m}^2)$
Differentiating w.r.t. $\text{m},\frac{\text{dF}}{\text{dm}}=\frac{\text{G}}{\text{r}^2}(\text{M})-2\text{m})$
For F to be maximum, $\frac{\text{dF}}{\text{dm}}=0$
$\frac{\text{G}}{\text{r}^2}(\text{M}-2\text{m})=0$
$\text{M}=2\text{m},$ or $\text{m}=\frac{\text{M}}{2}$
$\therefore\text{m}_1=\text{m}_2=\frac{\text{M}}{2}$

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