Forced Oscillations - Differential equation for forced oscillations,Resonance,Velocity, Average energy, Quality factor: Link1

Forced Oscillations MCQ & Answers

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Q.1. Amplitude A of forced oscillations
$m\:\frac{d^2x}{dt^2} +$ $R\:\frac{dx}{dt} + kx = fo \:\:\:sin\: qt\: is\: given\: as ----$

$A. \:\:\:\:\:\: A=\frac{fo}{\sqrt(k-mq^2)^2 - R^2q^2}$

$B. \:\:\:\:\:\: A=\frac{fo}{\sqrt(k-mq)^2 - R^2q^2}$

$C. \:\:\:\:\:\: A=\frac{fo}{\sqrt(k-mq^2)^2 - Rq^2}$

$D. \:\:\:\:\:\: A=\frac{fo}{\sqrt(k-mq^2)^2 - R^2q}$

Q.2. The phase of oscillations
$m\:\frac{d^2x}{dt^2} +$ $R\:\frac{dx}{dt} + kx = fo \:\:\:sin\: qt\: is\: given\: as ----$

$A. \:\: tan\: \phi = \frac{Rq}{k-mq^2}$

$B. \:\: sin\: \phi = \frac{Rq}{k-mq^2}$

$C. \:\: cos\: \phi = \frac{Rq}{k-mq^2}$

$D. \:\: tan\: \phi = \frac{R^2q^2}{k-mq^2}$

Q.3. Which of the following differential equation represents forced oscillatory motion?

$A.\:\:\: m\:\frac{d^2x}{dt^2} +\:\: kx = 0$

$B.\:\:\: m\:\frac{d^2x}{dt^2} + R\frac{dx}{dt}+\:\: kx = fo\:\: sin\:\: qt\:\:$

$C.\:\:\: m\:\frac{d^2x}{dt^2} + R\frac{dx}{dt}-\:\: kx = 0\:$

$D.\:\:\: m\:\frac{d^2x}{dt^2} - R\frac{dx}{dt}-\:\: kx = 0\:$

Q.4. Natural frequency of free oscillations
$\:\:\: m\:\frac{d^2x}{dt^2} + R\frac{dx}{dt}+\:\: kx = fo\:\:sin\:\:qt\:\:is\:----$

$A.\:\:\: \omega = \sqrt {mk}$

$B.\:\:\: \omega = \sqrt {\frac{1}{km}$

$C.\:\:\: \omega = \sqrt {\frac{m}{k}$

$D.\:\:\: \omega = \sqrt {\frac{k}{m}$

Q.5. Maximum amplitude of resonance at amplitude resonance in forced oscillation is -------.

$A.\:\:\: \frac{fo\:\omega}{R}$

$B.\:\:\: \frac{fo\:}{\omega R}$

$C.\:\:\: \frac{foR\:}{\omega }$

$D.\:\:\: \frac{fo\: m\:}{\omega R }$

Q.6. Resonance takes place when naural angular frequency of oscillations ω is ------- angular frequency of driving force q in forced oscillation
$\:\:\: m\:\frac{d^2x}{dt^2} + R\frac{dx}{dt}+\:\: kx = fo\:\: sin\:\: qt\:.$

A. less than
B. greater than
C. equal to
D. less or greater than

Q.7. Maximum amplitude of velocity of resonance at velocity resonance in forced oscillation is ------.

$A.\: \:\frac{fo}{R}$

$B.\: \:\frac{fo}{2R}$

$C.\: \:\frac{foR}{2}$

$D.\: \:\frac{fo}{\omega R}$

Q.8. Half width Δω of the resonance for
$\:\:\: m\:\frac{d^2x}{dt^2} + R\frac{dx}{dt}+\:\: kx = fo\:\: sin\:\: qt\:\:is ----$

$A.\: \:\frac{R}{m}$

$B.\: \:\frac{R}{2m}$

$C.\: \:\frac{R}{4m}$

$D.\: \:\frac{R}{8m}$

Q.9. At velocity resonance, phase difference between displacement and applied force is -------.

A. π

$B.\: \:\frac{\pi}{4}$

$C.\: \:\frac{\pi}{2}$

D. 2π

Q.10. Bandwidth of power resonance q is the difference in angular frequencies of applied force for which power falls from
$\bar Pmax \: \: to\:\: \frac{1}{2}\bar Pmax \:\: is \:\:given \:\:as\:-----$