# Simple Harmonic Oscillator

## What You’ll Do:

• Determine the spring constant of a spring
• Make a video of a mass oscillating on the end of the spring
• Analyze the video to find the oscillation period, amplitude, and maximum velocity of the mass

## Resources

• Vertically-mounted ruler with hook
• Spring with hook
• 50g Mass hanger and 50g/100g masses
• Phone or digital camera
• Tracker Software

## Background

Any system that has a restoring force acting upon it can potentially produce simple harmonic motion. A mass $m$ on a spring is one example, since the spring produces a restoring force of $F=-kx$ on the mass, where $k$ is the spring constant in units of Newtons/meter. If the spring is stretched or compressed away from its resting length, the mass will oscillate up and down with an oscillation period of $$T=2\pi\sqrt{\frac{m}{k}}$$

Thus, the oscillation frequency of a given mass and spring is constant. However, the total energy of the harmonic oscillator depends on the amount by which the spring is initially stretched or compressed before release, which is also equal to the maximum oscillation amplitude A. From this, it can be shown that the maximum speed of the oscillation mass is $$v_{max} = A\frac{2\pi}{T}=A\sqrt{\frac{k}{m}}$$

Harmonic Oscillator Worksheet