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. $\pi /4$
• C. $\pi /2$
• D. $\pi$

Answer 1

The correct option is C $\pi /2$

The equation for forced oscillation in a damped system is given as-
$m\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+kx=F_{0}cos\omega t$

Dividing by m throughout,

$⟹\frac{d^{2}x}{d{t}^2}+2\beta \frac{dx}{dt}+{\omega }_0^{2}x=Acos\omega t$

The expected solution is of form $x=Dcos(\omega t-\delta )$

Put this is in above equation gives,

$tan\delta =\frac{2\beta \omega }{{\omega }_0^2-{\omega }^2}$

For resonant oscillation, ${\omega }_0=\omega$

$⟹\delta =\pi /2$ which is the phase difference between $x$ and $F$.