First slide

Damped and forced oscillations

Question

The amplitude of vibration of a particle is given by $a_{m} = \frac{a_{0}}{(a ω^{2} - b ω + c)}$ where $a_{0}, a, b$ and $c$ are positive. The condition for a single resonant frequency is

Moderate

A

$b^{2} = 4 a c$
B

$b^{2} > 4 a c$
C

$b^{2} = 5 a c$
D

$b^{2} = 7 a c$

Solution

For resonance,
Amplitude is maximum (infinity) as denominator function is quadratic equation
$a x^{2} + b x + c = 0 a ω^{2} - b ω + c = 0$
Roots of $ω$ ,
$ω = \frac{+ b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
$if ω = 0 \Rightarrow b^{2} - 4 a c = 0$
$b^{2} = 4 a c$

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