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A block of mass $m$ is connected to a light spring of force constant $k$. The system is placed inside a damping medium of 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. The incorrect
expression related to the damped oscillations is
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expression related to the damped oscillations is
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Verified Answer
The correct answer is:
$x=A e^{-\frac{b}{m}} \cos \left(\omega^{\prime} t+\phi\right)$
Equation of motion for damped oscillation is
$\frac{m d^2 x}{d t^2}+\frac{b d x}{d t}+k x=0$
Angular frequency is $\omega^{\prime}=\sqrt{\frac{k}{m}-\frac{b^2}{4 m^2}}$ Displacement is given by, $x=A_0 e^{-\frac{b}{2 m} t} \cos (\omega t+\phi)$ and energy is given by $E=\frac{1}{2} k A^2 e^{-\frac{b}{m} t}$
$\frac{m d^2 x}{d t^2}+\frac{b d x}{d t}+k x=0$
Angular frequency is $\omega^{\prime}=\sqrt{\frac{k}{m}-\frac{b^2}{4 m^2}}$ Displacement is given by, $x=A_0 e^{-\frac{b}{2 m} t} \cos (\omega t+\phi)$ and energy is given by $E=\frac{1}{2} k A^2 e^{-\frac{b}{m} t}$
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