A mass $M$ is split into two parts, $m$ and $ (M-m)$, which are then separated by a certain distance. What ratio of $\frac{m}{M}$ maximizes the gravitational force between the two parts
A mass $M$ is split into two parts, $m$ and $ (M-m)$, which are then separated by a certain distance. What ratio of $\frac{m}{M}$ maximizes the gravitational force between the two parts
$\frac{1}{3}$
$\frac{1}{2}$
$\frac{1}{4}$
$\frac{1}{5}$
A mass $M$ is split into two parts, $m$ and $ (M-m)$, which are then separated by a certain distance. What ratio of $\frac{m}{M}$ maximizes the gravitational force between the two parts
$F = \frac{{Gm(M - m)}}{{{r^2}}}$
For maximum force $\frac{{dF}}{{dm}} = 0$
$\frac{d}{{dm}}\left( {\frac{{GmM}}{{{r^2}}} - \frac{{G{m^2}}}{{{r^2}}}} \right) = 0$
$M - 2m = 0\, \Rightarrow \,\frac{m}{M} = \frac{1}{2}$
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