The amplitude of velocity of a particle is given by, Vm=V0 /(a ω2-b ω+c) where V0, a, b and c are positive: The condition for a single resonant frequency is

Question

The amplitude of velocity of a particle is given by, Vm=V0 /(a ω2-b ω+c) where V0, a, b and c are positive: The condition for a single resonant frequency is (A) b2<4 a c (B) b 2=4 ac (C) b 2=5 ac (D) b2=7 a c

Tardigrade Admin · Accepted Answer

V m = V 0 /( a ω2- b ω+ c ) If there is a single resonant frequency, then this equation should be satisfied for only one that particular resonant frequency, hence a ω2-b ω+c=0 will have equal roots therefore; D=0 ⇒(-b)2-4 a c=0 ⇒ b2=4 a c

Q. The amplitude of velocity of a particle is given by, $V_{m} = V_{0} / (a ω^{2} - bω + c)$ where $V_{0}, a, b$ and $c$ are positive : The condition for a single resonant frequency is

$b^{2} < 4 a c$

$b^{2} = 4 a c$

$b^{2} = 5 a c$

$b^{2} = 7 a c$

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

Q. The amplitude of velocity of a particle is given by, Vm​=V0​/(aω2−bω+c) where V0​,a,b and c are positive : The condition for a single resonant frequency is

b2<4ac

b2=4ac

b2=5ac

b2=7ac