Answer
$\omega_n=19.7rad/s$
$ y=(0.0833 cos19.7t)ft$
$C=1in$
Work Step by Step
We can determine the required equation, natural frequency and amplitude as follows:
$y=\frac{v_{\circ}}{\omega_n}sin(\omega_n t)+y_{\circ}cos(\omega_n t)$..eq(1)
Now the natural frequency is given as
$\omega_n=\sqrt{\frac{K}{m}}$
$\implies \omega_n=\sqrt{\frac{2lb/in\times 12in/ft}{\frac{2}{32.2}}}$
$\implies \omega_n=19.7rad/s$
We plug in the known values in equation (1) to obtain:
$y=cos(19.7t)in$
$\implies y=(0.0833 cos19.7t)ft$
Now the amplitude can be determined as
$C=\sqrt{(\frac{v_{\circ}}{\omega_n})^2+(y_{\circ})^2}$
We plug in the known values to otbain:
$C=1in$
