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You are given a total mass M to divide it into two parts so that the gravitational force between them at a distance is maximum, the masses should be

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$\frac{M}{2}, \frac{M}{2}$

$\frac{M}{5}, \frac{4 M}{5}$

$\frac{M}{4}, \frac{3 M}{4}$

$\frac{M}{3}, \frac{2 M}{3}$

answer is D.

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Detailed Solution

$F = \frac{Gm (M - m)}{R^{2}}$

$F_{max} = \frac{dF}{dm} = 0$

$= \frac{d}{dm} (Mm - m^{2}) = 0$

$\begin{aligned} M - 2 m = 0 \\ ∴ M = 2 m \\ ∴ m = \frac{M}{2} \end{aligned}$

$Among all the values, \frac{M^{2}}{4} is maximum.$

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