Physics - SHM Question with Solution | TestHub

The differential equation for a damped harmonic oscillator is given by

$m\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+kx=0$

where $m$ is the mass, $b$ is the damping constant, and $k$ is the restoring force constant.

Dividing the entire equation by $m$, we get

$\frac{d^{2}x}{d{t}^2}+\frac{b}{m}\frac{dx}{dt}+\frac{k}{m}x=0$

The angular frequency of the damped oscillator is given by

$\omega =\sqrt{{\omega }_0^2-{\gamma }^2}$

where ${\omega }_0=\sqrt{\frac{k}{m}}$ is the natural angular frequency of the undamped oscillator and $\gamma =\frac{b}{2m}$ is the damping coefficient.

Substituting these values, we get the angular frequency of the damped oscillations as

$\omega =\sqrt{\frac{k}{m}-{\left(\frac{b}{2m}\right)}^2}$

Answer: $\sqrt{\frac{k}{m}-{\left(\frac{b}{2m}\right)}^2}$