A harmonic oscillator is a physical system that oscillates sinusoidally around an equilibrium position under the effect of a restoring force proportional to the displacement from equilibrium. The simple pendulum (for small oscillations) and the mass-spring system are the classic examples, part of the senior high school physics curriculum.
Discover FizziQ
How to measure it in class
With the FizziQ app, the accelerometer allows you to study the oscillations of a harmonic oscillator.
Steps:
- Mass-spring system: attach the smartphone to a spring and let it oscillate
- Record the acceleration with FizziQ
- Measure the period T and verify that it is independent of amplitude
- Pendulum: attach the smartphone to a string and make it oscillate
- Verify the formula T = 2π√(L/g)
- Compare damped oscillations (with friction) and undamped oscillations
Scientific activities on this topic
- Pendulum study
- Period of a pendulum: isochronism
- Mass-spring oscillator: period
- Damping: three regimes
Learn more
Equation of motion:
The differential equation of a harmonic oscillator is: d²x/dt² + ω₀² × x = 0
where ω₀ = 2π/T₀ is the natural angular frequency.
General solution:
x(t) = A × cos(ω₀t + φ)
The motion is sinusoidal, with period T₀ = 2π/ω₀, amplitude A and initial phase φ.
Isochronism of oscillations:
The period of a harmonic oscillator is independent of the amplitude (within the validity limits of the model). This is the property of isochronism, fundamental for clocks.
Examples of oscillators:
| System | Period T₀ |
|---|---|
| Mass-spring | 2π√(m/k) |
| Simple pendulum | 2π√(L/g) |
| Torsion pendulum | 2π√(I/C) |
| LC circuit | 2π√(LC) |
Formula
Equation of motion as a function of time: x(t) = A × cos(ω₀t + φ)
Natural angular frequency: ω₀ = 2π/T₀ = 2πf₀
Period of the simple pendulum: T = 2π × √(L/g)
Period of the mass-spring system: T = 2π × √(m/k)
Mechanical energy: E = ½kA² = ½mω₀²A² (constant)
Application examples
- A 1 m pendulum has a period of 2 s
- Clock pendulums use isochronism
- Watch quartz crystals are harmonic oscillators at very high frequency
- The smartphone’s accelerometer contains micro-oscillators
FAQ
Q: Why is the period independent of amplitude? A: It is a mathematical property of the harmonic oscillator. The restoring force increases proportionally to the displacement: if you move farther away, you return faster.
Q: How to measure the period with FizziQ? A: By analyzing the accelerometer signal. The period is the time between two consecutive maxima (or two minima).
Q: Is a real pendulum truly harmonic? A: Only for small amplitudes (< 15°). For large oscillations, the equation is no longer harmonic and the period increases with amplitude.
Q: How to determine g with a pendulum? A: Measure T and L, then g = 4π²L/T². With FizziQ, you can automate the measurement of many periods for good precision.
Related concepts
Period - Frequency - Pendulum - Hooke’s Law - Mechanical Energy - Damping - Resonance Frequency