Problems

01

Quasi-frequency

An underdamped harmonic oscillator vibrates at some “quasi-frequency” $\mu$. This is not a true frequency because the motion is not truly periodic, due to its decay over time. Find a formula for the ratio $\mu /{\omega }_0$ where ${\omega }_0$ is the natural frequency of the undamped harmonic oscillator with the same mass and spring. Does the damped oscillator vibrate with a faster or slower frequency than the same oscillator without damping? What happens to $\mu$ as $c\to 2\sqrt{km}$ and thus $b\to {\omega }_0$?

02

Bobbing buoy

A cylindrical buoy $1\mathrm{ft}$ in diameter bobs in water with its axis vertical. The density of water is $62.4\mathrm{lb}/{\mathrm{ft}}^3$. The period of its bobbing oscillation motion is $1.7\sec$. How heavy is the buoy?