Library / Foundations: Physics / FPH-63
Foundations: Physics
Amplitude from displacement and speed
Amplitude of a simple harmonic oscillator found from its position x and speed v at any one instant, together with the angular frequency ω.
- Ref
- FPH-63
- Rev
- v1
- Field
- Foundations: Physics
- Result
- A [m]
x and v are the displacement and speed at the same instant; ω is the angular frequency. At the ends of the swing v = 0 and A = |x|; at the centre x = 0 and A = |v|/ω. The sign of v does not matter because only v² appears.
Amplitude of a simple harmonic oscillator found from its position x and speed v at any one instant, together with the angular frequency ω. It follows from conservation of energy.
Calculator
- Peak speed, ωA0.5 m/s
- Peak acceleration, ω²A5 m/s²
- Period, 2π/ω0.6283 s
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.
- One snapshot of position and speed is enough: the oscillator will swing out to 0.05 m, which is 1.67 times its current distance from the centre.
- The result follows from energy: ½mv² + ½kx² = ½kA², with k = mω².
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| v Speed now (either sign) | −50 to 50 m/s | 0.03 | 5 | 4.97 | +2 m/s | +384 % |
| ω Angular frequency | 0.1 to 1,000 rad/s | 0.03 | 4 | 3.97 | +10% | −5.72 % |
| x Displacement now | −1 to 1 m | 0.04 | 1.001 | 0.9608 | +0.04 m | +61.2 % |
Sweep of Speed now (either sign)
| Speed now (either sign) (m/s) | Amplitude (m) |
|---|---|
| −50 | 5 |
| −37.5 | 3.75 |
| −25 | 2.5 |
| −12.5 | 1.25 |
| 0 | 0.03 |
| 12.5 | 1.25 |
| 25 | 2.5 |
| 37.5 | 3.75 |
| 50 | 5 |
Sweep one variable
Variables and how each one contributes
Amplitude
What the equation gives you. Everything else feeds it.
Displacement now
The displacement at one instant, in m. Raises the amplitude, most strongly when the body is near the end of its swing. Raising it by 10 % (to 0.033) moves the result by +3.71 %, from 0.05 to 0.05186 m.
Speed now (either sign)
The speed at the same instant, in m/s. Raises the amplitude, most strongly when the body is near the centre of its swing. Raising it by 10 % (to 0.44) moves the result by +6.51 %, from 0.05 to 0.05325 m.
Angular frequency
The angular frequency, in rad/s. Lowers the amplitude: at the same speed a faster oscillator has a smaller swing. Raising it by 10 % (to 11) moves the result by −5.72 %, from 0.05 to 0.04714 m.
Worked example with the values above
- Speed term: v²/ω² = 0.4² / 10² = 0.0016 m².
- Position term: x² = 9×10−4 m².
- A = √(9×10−4 + 0.0016) = 0.05 m.
Assumptions and limits
- Undamped simple harmonic motion with constant ω.
- x and v measured at the same instant.
Common mistakes
- Using ω = 2πf with f read in the wrong units.
- Forgetting to divide v by ω before squaring.
- Mixing a displacement in centimetres with a speed in m/s.
Statistics and uncertainty
The amplitude combines two terms in quadrature, so it is more sensitive to whichever of x and v/ω is larger. A 1 % error in ω changes the speed term by 2 %, so it mainly affects the amplitude when the body is near the centre of its swing.
Where it is used
- Finding how far an oscillator will travel from a measured position and speed.
- Checking amplitude from a sensor reading in vibration tests.
Maths you need
- Energy in SHM: ½mv² + ½kx² = ½kA²
- Angular frequency: ω = √(k/m)