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(A) The equation of motion for a damped oscillator is given by $m \frac{d^2 x}{d t^2} + b \frac{d x}{d t} + k x = 0$. This matches option $D$.The angu

Vedclass · Accepted Answer

$x=A e^{-\frac{b}{m}} \cos \left(\omega^{\prime} t+\phi\right)$

Vedclass · Answer

(A) The equation of motion for a damped oscillator is given by $m \frac{d^2 x}{d t^2} + b \frac{d x}{d t} + k x = 0$. This matches option $D$.The angular frequency of damped oscillations is $\omega^{\prime} = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$. This matches option $B$.The displacement of a damped oscillator is given by $x(t) = A e^{-\frac{b}{2m}t} \cos(\omega^{\prime}t + \phi)$. Comparing this with option $A$,we see that the exponent in option $A$ is $-\frac{b}{m}$ instead of $-\frac{b}{2m}$. Therefore,option $A$ is incorrect.The energy of a damped oscillator decays as $E(t) = E_0 e^{-\frac{b}{m}t} = \frac{1}{2} k A^2 e^{-\frac{b}{m}t}$. This matches option $C$.

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