Question

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

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

Answer 1

Answer: (A)

Displacement of damped oscillator is given by
$x=x_m{e}^{-bt/2m}\sin \left({\omega }^{\prime }t+\varphi \right)$
where ${\omega }^{\prime }=$ angular frequency of damped oscillator
$=\sqrt{{\omega }_0^2-(b/2m{)}^2}$
$=\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}$