If mass M is split into two parts, m and $(M - m)$ which are then separated by a certain distance. What ratio of m/M maximizes the gravitational force between the two parts.

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Solution

The correct option is B $1 / 2$
$F = \frac{G m (M - m)}{r^{2}} = \frac{G}{r^{2}} (m M - m^{2})$
Force will be maximum if
$\frac{d F}{m} = 0 \Rightarrow \frac{d}{d m} = [\frac{G}{r^{2}} (m M - m^{2})] = 0 \Rightarrow \frac{G}{r^{2}} (M - 2 m) = 0 ∴ M - 2 m = 0 \Rightarrow M = 2 m \Rightarrow \frac{M}{m} = 2 o r, \frac{m}{M} = \frac{1}{2}$
$F$ will be maximum when $\frac{m}{M} = \frac{1}{2}$

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If mass M is split into two parts, m and (M−m) which are then separated by a certain distance. What ratio of m/M maximizes the gravitational force between the two parts.

The correct option is B 1/2F=Gm(M−m)r2=Gr2(mM−m2)Force will be maximum ifdFm=0⇒ddm=[Gr2(mM−m2)]=0⇒Gr2(M−2m)=0∴M−2m=0⇒M=2m⇒Mm=2or,mM=12F will be maximum whenmM=12

If mass M is split into two parts, m and $(M - m)$ which are then separated by a certain distance. What ratio of m/M maximizes the gravitational force between the two parts.