Answer
(a) $y(t)=6e^{-2.8t}cos(4\pi t)$
(b) $0.12 s$
Work Step by Step
Identify the given quantities as $k=6, f=2Hz, c=2.8$
(a) At t=0, the displacement is at a maximum of 6in, we can use the cosine damping model
$y(t)=ke^{-ct}cos(\omega t)$ where $\omega=2\pi f=4\pi$, so that $y(t)=6e^{-2.8t}cos(4\pi t)$
(b) Let $y(t)=0.5$, we have $6e^{-2.8t}cos(4\pi t)=0.5$. Graph the function $f(t)=6e^{-2.8t}cos(4\pi t)-0.5$ as shown in the figure, and the first zero can be found at $t\approx0.12 s$ which gives the time it takes for the amplitude of the vibration to decrease to 0.5 in.
