Find the limiting values of amplitude of a force damped oscillator, when (i) $ω_{d} < < ω_{0}$ and (ii) $ω_{d} > > ω_{0}$ .

Difficult

A

$\frac{F}{m ω_{0}^{2}}, \frac{F}{m ω_{d} \sqrt{ω_{d}^{2} + {(\frac{b}{m})}^{2}}}$
B

$\frac{2 F}{m ω_{0}^{2}}, \frac{F}{2 m ω_{d} \sqrt{ω_{d}^{2} + {(\frac{b}{m})}^{2}}}$
C

$\frac{F}{\sqrt{2} m ω_{0}^{2}}, \frac{F}{m ω_{d} \sqrt{2 [ω_{d}^{2} + {(\frac{b}{m})}^{2}]}}$
D

$\frac{F}{2 m ω_{0}^{2}}, \frac{F}{2 m ω_{d} \sqrt{ω_{d}^{2} + {(\frac{b}{m})}^{2}}}$

Solution

$\begin{array}{l} A = \frac{F_{0} / m}{\sqrt{{(ω_{0}^{2} - ω_{d}^{2})}^{2} + {(\frac{b ω_{d}}{m})}^{2}}} \\ When ω_{d} << ω_{0} \\ A = \frac{F_{0} / m}{\sqrt{{(ω_{0}^{2})}^{2}}} = \frac{F_{0}}{m ω_{0}^{2}} \end{array}$
$W h e n ω_{d} > > ω_{0}$
$A = \frac{F_{0} / m}{\sqrt{ω_{d}^{4} + {(\frac{b}{m})}^{2} ω_{d}^{2}}} = \frac{F}{m ω_{d} \sqrt{ω_{d}^{2} + {(\frac{b}{m})}^{2}}}$

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