Question

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

A
Zero
B
π/4
C
π/2
D
π

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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