Free Undamped/Damped Motion
- Thread starter AmyA
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$x(t) = A\sin(\omega t + \phi)$
$v(t) = A\omega \cos(\omega t + \phi)$
$x(0) = -\sqrt{3}$, $v(0) = 3$, $\omega = \sqrt{\dfrac{k}{m}} = \sqrt{\dfrac{9}{1}} = 3$, $\phi = -\dfrac{\pi}{3}$
using the above information gives
$x(t) = 2\sin \left(3 t - \dfrac{\pi}{3} \right)$
$v(t) = 6\cos \left(3 t - \dfrac{\pi}{3}\right)$
passing through equilibrium, $x(t) = 0 \implies 3t - \dfrac{\pi}{3} = \{0, \pi, 2\pi, ... \} \implies t = \bigg\{ \dfrac{\pi}{9}, \dfrac{4\pi}{9}, \dfrac{7\pi}{9}, ... \bigg \}$
the {1st, 3rd, 5th, ...} passes through equilibrium are in the upward direction
(h) set $x(t) = -1$ and solve for time
(i) use solutions from (h) and evaluate if v(t) > 0
(j) use solutions from (h) and evaluate if v(t) < 0
$v(t) = A\omega \cos(\omega t + \phi)$
$x(0) = -\sqrt{3}$, $v(0) = 3$, $\omega = \sqrt{\dfrac{k}{m}} = \sqrt{\dfrac{9}{1}} = 3$, $\phi = -\dfrac{\pi}{3}$
using the above information gives
$x(t) = 2\sin \left(3 t - \dfrac{\pi}{3} \right)$
$v(t) = 6\cos \left(3 t - \dfrac{\pi}{3}\right)$
passing through equilibrium, $x(t) = 0 \implies 3t - \dfrac{\pi}{3} = \{0, \pi, 2\pi, ... \} \implies t = \bigg\{ \dfrac{\pi}{9}, \dfrac{4\pi}{9}, \dfrac{7\pi}{9}, ... \bigg \}$
the {1st, 3rd, 5th, ...} passes through equilibrium are in the upward direction
(h) set $x(t) = -1$ and solve for time
(i) use solutions from (h) and evaluate if v(t) > 0
(j) use solutions from (h) and evaluate if v(t) < 0
