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Two particles of masses in the ratio $1 : 2$ are moving in circles of radii in the ratio $2 : 3$ with time periods in the ratio $3 : 4$ . The ratio of their centripetal forces is

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9 : 4

1 : 4

9 : 16

16 : 27

answer is D.

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Detailed Solution

F = mr ω^{2}

$F \propto \frac{mr}{T^{2}}$

$m_{1} : m_{2} = 1 : 2$

$r_{1} : r_{2} = 2 : 3$

$t_{1} : t_{2} = 3 : 4$

$F_{1} : F_{2} = ?$

$F_{1} = \frac{m_{1} r_{1}}{t_{1}^{2}} = \frac{1 \times 2}{3^{2}} = \frac{2}{9}$

$F_{2} = \frac{m_{2} r_{2}}{t_{2}^{2}} = \frac{2 \times 3}{4^{2}} = \frac{6}{16}$

$\frac{F_{1}}{F_{2}} = \frac{2 / 9}{6 / 16} = \frac{2}{9} \times \frac{16}{6} = \frac{16}{27}$

$\frac{F_{1}}{F_{2}} = \frac{16}{27} \Rightarrow F_{1} : F_{2} = 16 : 27$

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