A 0.25 kg harmonic oscillator has a total mechanical energy of $4.1 \mathrm {~J}$ If the oscillation amplitude is $20.0 \mathrm {~cm} ,$ what is the oscillation frequency?
A)4.6 Hz
B)1.4 Hz
C)2.3 Hz
D)3.2 Hz

Question

Quizplus · Accepted Answer

The oscillation frequency is given by:$$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$$where k is the spring constant and m is the mass. The spring constant is given by:$$k = \frac{m\omega^2}{A^2}$$where $\omega$ is the angular frequency and A is the oscillation amplitude. Substituting this into the first equation, we get:$$f = \frac{1}{2\pi} \sqrt{\frac{m\omega^2}{mA^2}}$$$$f = \frac{\omega}{2\pi A}$$The angular frequency is given by:$$\omega = \sqrt{\frac{k}{m}}$$Substituting this into the previous equation, we get:$$f = \frac{1}{2\pi A} \sqrt{\frac{k}{m}}$$$$f = \frac{1}{2\pi (0.200 \mathrm{~m})} \sqrt{\frac{4.1 \mathrm{~J}}{0.25 \mathrm{~kg}}}$$$$f = 4.6 \mathrm{~Hz}$$

A 025 Kg Harmonic Oscillator Has a Total Mechanical Energy Of Of 4.1