A block of mass $m$ is connected to a light spring with a force constant $k$. The system is placed inside a damping medium with a damping constant $b$. The instantaneous values of displacement, acceleration, and energy of the block are $x$, $a$, and $E$ respectively. The initial amplitude of oscillation is $A$, and $\omega^{\prime}$ is the angular frequency of oscillations. Which of the following expressions related to damped oscillations is incorrect?

Question

$E = \frac{1}{2} k A^2 e^{-\frac{b t}{m}}$

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

$x = A e^{-\frac{b}{2m} t} \cos \left(\omega^{\prime} t + \phi \right)$

$\omega^{\prime} = \sqrt{\frac{k}{m} - \frac{b^2}{4 m^2}}$

Solution

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