Library / Foundations: Physics / FPH-53
Foundations: Physics
Total energy of a simple harmonic oscillator
Constant total energy of an undamped oscillator.
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- FPH-53
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- Foundations: Physics
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- E [J]
The total energy of an undamped oscillator is constant and equals ½mω²A²: all kinetic energy at the centre, all potential energy at the ends, and a mix in between. Because k = mω², it is the same as ½kA², the energy stored in a spring stretched to the amplitude. It rises with the square of the amplitude.
Constant total energy of an undamped oscillator. It equals the kinetic energy at the centre of the motion and the potential energy at the ends.
Calculator
- Stiffness implied, k = mω²50 N/m
- Maximum speed, ωA0.5 m/s
- Kinetic energy at half the amplitude0.04688 J
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 0.5 kg mass oscillating at ω = 10 rad/s with a 5 cm amplitude carries 0.0625 J, equivalent to a spring of stiffness 50 N/m stretched 5 cm.
- Doubling the amplitude quadruples the energy (to 0.25 J), which is why the damage from a vibration grows so quickly with its size.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Amplitude | 0.001 to 1 m | 2.5e-5 | 25 | 25 | +10% | +21 % |
| ω Angular frequency | 1 to 100 rad/s | 6.25e-4 | 6.25 | 6.249 | +10% | +21 % |
| m Oscillating mass | 0.01 to 10 kg | 0.00125 | 1.25 | 1.249 | +10% | +10 % |
Sweep of Amplitude
| Amplitude (m) | Total energy (J) |
|---|---|
| 0.001 | 2.5e-5 |
| 0.002371 | 1.406e-4 |
| 0.005623 | 7.906e-4 |
| 0.01334 | 0.004446 |
| 0.03162 | 0.025 |
| 0.07499 | 0.1406 |
| 0.1778 | 0.7906 |
| 0.4217 | 4.446 |
| 1 | 25 |
Sweep one variable
Variables and how each one contributes
Total energy
What the equation gives you. Everything else feeds it.
Oscillating mass
The oscillating mass, in kg. Raises the energy in direct proportion for a given ω and A. Raising it by 0.1 kg (to 0.6 kg) moves the result by +20 %, from 0.0625 to 0.075 J.
Angular frequency
The angular frequency, in rad/s. Raises the energy with its square: a stiffer spring gives a larger energy for the same amplitude. Raising it by 5 rad/s (to 15 rad/s) moves the result by +125 %, from 0.0625 to 0.1406 J.
Amplitude
The amplitude, in m. Raises the energy with its square: doubling the amplitude gives four times the energy. Raising it by 0.01 m (to 0.06 m) moves the result by +44 %, from 0.0625 to 0.09 J.
Worked example with the values above
- Square the amplitude and frequency: A² = 0.0025, ω² = 100.
- Multiply: m ω² A² = 0.5 × 100 × 0.0025 = 0.125 J.
- E = ½ × 0.125 = 0.0625 J.
Assumptions and limits
- An undamped simple harmonic oscillator: a restoring force proportional to the displacement.
- Constant amplitude and angular frequency; displacement is measured from the equilibrium position.
Common mistakes
- Using the frequency in hertz where ω in rad/s is needed.
- Forgetting that the energy depends on A², not A.
Statistics and uncertainty
Because the energy depends on A², a 1 % error in the amplitude gives a 2 % error in the energy, and the same holds for ω. The mass is usually known best.
Where it is used
- Energy stored in a vibrating spring-mass system.
- Energy that damping must remove in a cycle.
Maths you need
- Angular frequency: ω = 2π times the frequency, in rad/s
- Amplitude: the largest displacement from equilibrium