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Question

A mass M is broken in two parts of mass m and (M-m) respectively. Relation between m and M so that the force of gravitational between the two parts is maximum is


A
mM=2
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B
m=M2
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C
M=m2
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D
none of these
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Solution

The correct option is B $$m=\frac { M }{ 2 } $$
$$F=G\frac { m(M-m) }{ { r }^{ 2 } } =km(M-m)\\ where,\quad k=\frac { G }{ { r }^{ 2 } } \\ \therefore \quad F=k\left[ \frac { { M }^{ 2 } }{ 2 } -\frac { { M }^{ 2 } }{ 2 } +mM-{ m }^{ 2 } \right] \\ =k\left[ \frac { { M }^{ 2 } }{ 4 } -\left( \frac { M }{ 4 } -m \right) ^{ 2 } \right] \\ F\quad is\quad maximum,\quad when\quad \left( \frac { M }{ 4 } -m \right) =0\\ \Rightarrow m=\frac { M }{ 2 } $$

Physics

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