# Chapter 22 - Vibrations - Section 22.1 - Undamped Free Vibration - Problems - Page 652: 7

$\omega_n=19.7rad/s$ $y=(0.0833 cos19.7t)ft$ $C=1in$

#### 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: $C=1in$

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