Find the limiting values of amplitude of a forced damped oscillator, when $(i) ω_{d} < < ω_{0} a n d (i i) ω_{d} > > ω_{0}$

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

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

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

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

answer is A.

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Detailed Solution

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