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Question

A weakly damped harmonic oscillator is executing resonant oscillations. The phase difference between the oscillator and the external periodic force is:


A
Zero
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B
π/4
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C
π/2
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D
π
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Solution

The correct option is C $$\pi/2$$
The equation for forced oscillation in a damped system is given as-
$$m\dfrac{d^2x}{dt^2}+b\dfrac{dx}{dt}+kx=F_{0}cos\omega t$$
Dividing by m throughout,
$$\implies \dfrac{d^2x}{dt^2}+2\beta\dfrac{dx}{dt}+\omega_{0}^2x=Acos\omega t$$
The expected solution is of form $$x=Dcos(\omega t-\delta)$$
Put this is in above equation gives,
$$tan\delta=\dfrac{2\beta\omega}{\omega_{0}^2-\omega^2}$$
For resonant oscillation, $$\omega_{0}=\omega$$
$$\implies \delta=\pi/2$$ which is the phase difference between $$x$$ and $$F$$.

Physics

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