Modeling Damped Harmonic Motion

00:00 5 marks

A mechanical system exhibits damped oscillations described by the function \( y(t) = e^{-kt}\sin(\omega t + \phi) \), where \(k, \omega, \phi\) are positive constants. Suppose a system is observed such that the maximum amplitude decays to half its initial value after 5 seconds. If \(\omega = 2\) and \(\phi = 0\), find the value of \(k\) and determine the time \(t\) when the function first reaches its maximum after \(t = 0\). Assume the maximum corresponds to the peak of the sine function and consider the effect of the exponential decay.

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Hint

Use the relationship between the amplitude decay and the exponential function, and recall that the sine function reaches its maximum at \(\frac{\pi}{2}\) for the first time.

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Complete Answer

k = \(\frac{\ln(2)}{5}\); the first maximum occurs at \(t = \frac{\pi}{4}\) seconds.

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