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

Question

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

Quizplus · Accepted Answer

The total mechanical energy $E$ of a harmonic oscillator is given by $E = \frac{1}{2}kA^2$, where $k$ is the spring constant and $A$ is the amplitude. The angular frequency $\omega$ is related to the spring constant and mass $m$ by $\omega = \sqrt{\frac{k}{m}}$, and the frequency $f$ is $\frac{\omega}{2\pi}$. Given $E = 4.1 \, \text{J}$ and $A = 0.2 \, \text{m}$, we can find $k$ from the energy equation and then solve for $f$. Rearranging the energy equation gives $k = \frac{2E}{A^2}$, and substituting $E = 4.1 \, \text{J}$ and $A = 0.2 \, \text{m}$ yields $k = \frac{2 \times 4.1}{0.2^2} = 205 \, \text{N/m}$. Then, $\omega = \sqrt{\frac{205}{0.25}} \approx 28.6 \, \text{rad/s}$, and $f = \frac{\omega}{2\pi} \approx \frac{28.6}{2\pi} \approx 4.6 \, \text{Hz}$.

A 025 Kg Harmonic Oscillator Has a Total Mechanical Energy