Library / Foundations: Physics / FPH-62
Foundations: Physics
Displacement of a simple harmonic oscillator
Position of a body in simple harmonic motion at time t, given its amplitude A, angular frequency ω and starting phase φ.
- Ref
- FPH-62
- Rev
- v1
- Field
- Foundations: Physics
- Result
- x [m]
A is the amplitude, the largest displacement; ω is the angular frequency in rad/s (ω = 2πf = 2π/T); t is time in seconds; φ is the phase angle that sets where in the cycle the motion starts. The angle ωt + φ is in radians inside the sine; φ is entered here in degrees for convenience.
Position of a body in simple harmonic motion at time t, given its amplitude A, angular frequency ω and starting phase φ. The cosine form is the same curve with φ shifted by 90°.
Calculator
- Period, 2π/ω0.6283 s
- Frequency, ω/2π1.592 Hz
- Velocity, Aω cos(ωt + φ)0.2702 m/s
- Acceleration, −ω²x−4.207 m/s²
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.
- At t = 0.1 s the body is at 0.04207 m, 84.1 % of the way to the end of its swing.
- The motion repeats every 0.6283 s (1.592 Hz). The velocity is largest as the body passes the centre, where the displacement is zero.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Amplitude | 0.001 to 1 m | 8.415e-4 | 0.8415 | 0.8406 | +10% | +10 % |
| φ Phase angle | −180 to 180 ° | −0.04992 | 0.04992 | 0.09983 | +7.2 ° | +7.26 % |
| t Time | 0 to 10 s | −0.04969 | 0.04997 | 0.09966 | +10% | +5.91 % |
| ω Angular frequency | 0.1 to 1,000 rad/s | −0.04987 | 0.04889 | 0.09876 | +10% | +5.91 % |
Sweep of Time
| Time (s) | Displacement (m) |
|---|---|
| 0 | 0 |
| 1.25 | −0.003316 |
| 2.5 | −0.006618 |
| 3.75 | −0.00989 |
| 5 | −0.01312 |
| 6.25 | −0.01629 |
| 7.5 | −0.01939 |
| 8.75 | −0.0224 |
| 10 | −0.02532 |
Sweep one variable
Variables and how each one contributes
Displacement
What the equation gives you. Everything else feeds it.
Amplitude
The amplitude, the largest displacement, in m. Raises the displacement in direct proportion at any fixed time. Raising it by 10 % (to 0.055) moves the result by +10 %, from 0.04207 to 0.04628 m.
Angular frequency
The angular frequency, in rad/s (2π times the frequency). Changes the phase ωt, so the effect on x depends on where in the cycle the body is. Raising it by 10 % (to 11) moves the result by +5.91 %, from 0.04207 to 0.04456 m.
Time
The time since t = 0, in s. Moves the body along its cycle; the effect depends on the phase and can be positive or negative. Raising it by 10 % (to 0.11) moves the result by +5.91 %, from 0.04207 to 0.04456 m.
Phase angle
The starting phase angle, in degrees. Shifts the whole motion in time; the effect on x depends on the phase at that instant. Raising it by 10 % (to 0) moves the result by +0 %, from 0.04207 to 0.04207 m.
Worked example with the values above
- Angle: ωt + φ = 10 × 0.1 + 0 = 1 rad.
- Sine: sin(1) = 0.841471.
- x = 0.05 × 0.841471 = 0.0420735 m.
Assumptions and limits
- Pure simple harmonic motion with no damping, so the amplitude stays constant.
- Constant angular frequency.
Common mistakes
- Setting a calculator to degrees while ω is in rad/s.
- Entering f in hertz in the ω box (ω = 2πf).
- Forgetting that a cosine form needs φ shifted by 90°.
Statistics and uncertainty
The displacement is the amplitude times a sine, so it depends on phase: near the centre a small timing error gives a large relative error in x, while near the ends of the swing the same timing error changes x very little.
Where it is used
- Predicting the position of a mass on a spring or a pendulum at a given time.
- Modelling vibrating structures and AC waveforms.
Maths you need
- Radians: angle measure for ωt
- Sine wave: amplitude, frequency, phase