Question

The equation of a damped simple harmonic motion is $m \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x=0$. Then the angular frequency of oscillation is

• A. $\omega=\left(\frac{k}{m}-\frac{b^{2}}{4 m^{2}}\right)^{1 / 2}$
• B. $\omega=\left(\frac{k}{m}-\frac{b}{4 m}\right)^{1 / 2}$
• C. $\omega=\left(\frac{k}{m}-\frac{b^{2}}{4 m}\right)^{1 / 2}$
• D. $\omega=\left(\frac{k}{m}-\frac{b}{4 m^2}\right)^{1 / 2}$

Answer 1

Answer: (A)

Displacement of damped oscillator is given by
$x=x_{m} e^{-b t / 2 m} \sin \left(\omega^{\prime} t+\phi\right)$
where $\omega'=$ angular frequency of damped oscillator
$=\sqrt{\omega_{0}^{2}-(b / 2 m)^{2}}$
$=\sqrt{\frac{k}{m}-\frac{b^{2}}{4 m^{2}}}$