Questions: Question 1 (1 points) You want to predict the frequency with which a ball-on-spring system will oscillate. You measure the spring constant to be 106 N / m and use a ball of mass 1.21 kg. What is the linear frequency (in Hz)? Please enter a numerical answer below. Accepted formats are numbers or "e" based scientific notation e.g. 0.23,-2, 1e6, 5.23e-8

Solution Steps

To find the linear frequency of a ball-on-spring system, we use the formula for the angular frequency $\omega$ of a simple harmonic oscillator:

$$\omega = \sqrt{\frac{k}{m}}$$

where $k$ is the spring constant and $m$ is the mass of the ball.

Given:

• Spring constant, $k = 106 \, \mathrm{N/m}$
• Mass of the ball, $m = 1.21 \, \mathrm{kg}$

Substitute these values into the formula:

$$\omega = \sqrt{\frac{106}{1.21}}$$

Calculate $\omega$:

$$\omega = \sqrt{87.6033} \approx 9.3565 \, \mathrm{rad/s}$$

The linear frequency $f$ is related to the angular frequency $\omega$ by the formula:

$$f = \frac{\omega}{2\pi}$$

Substitute $\omega = 9.3565 \, \mathrm{rad/s}$ into the formula:

$$f = \frac{9.3565}{2\pi} \approx 1.489 \, \mathrm{Hz}$$

Final Answer