A driven oscillator is acted upon by a force F- F 0 sin - The amplitude of oscillation is given by A- F 0 - a- 2 -b- -c - the resonant angular frequency is

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Forced Oscillations

A driven osci...

Question

A driven oscillator is acted upon by a force $F = F_{0} s i n ω$ . The amplitude of oscillation is given by $A = \frac{F_{0}}{\sqrt{a ω^{2} - b ω + c}}$ , the resonant angular frequency is

d \frac{a}{b}

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\frac{2 a}{b}

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\frac{b}{a}

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\frac{b}{2 a}

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Solution

The correct option is D $\frac{b}{2 a}$
The driven force $F = F_{0} sin ω t$
The amplitude of oscillation
$A = \frac{F_{0}}{\sqrt{{a ω}^{2} - b ω + c}}$
At resonance frequency A become maximum.So,
$\frac{d A}{d ω} = 0 \Rightarrow F_{0} ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{2 a ω - b}{{(a ω^{2} - b ω + 1)}^{\frac{3}{2}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ = 0 \Rightarrow ω = \frac{b}{2 a} \to$
This is the resonence frequency.

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