Question

The amplitude of velocity of a particle is given by, $V_m=V_0 /\left(a \omega^2-b \omega+c\right)$ where $V_0, a, b$ and $c$ are positive : The condition for a single resonant frequency is

• A. $b^2<4 a c$
• B. $b ^2=4 ac$
• C. $b ^2=5 ac$
• D. $b^2=7 a c$

Answer 1

Answer: (B)

$V _{ m }= V _0 /\left( a \omega^2- b \omega+ c \right)$
If there is a single resonant frequency, then this equation should be satisfied for only one that particular resonant frequency, hence $a \omega^2-b \omega+c=0$ will have equal roots therefore;
$D=0 \Rightarrow(-b)^2-4 a c=0 \Rightarrow b^2=4 a c$