Calculator Inputs
Choose a spring or pendulum model, then evaluate motion at any time.
Example Data Table
Sample spring case: amplitude 0.08 m, mass 1.50 kg, spring constant 24 N/m, phase 0.30 rad.
| Time (s) | Displacement (m) | Velocity (m/s) | Acceleration (m/s²) | Potential Energy (J) | Kinetic Energy (J) |
|---|---|---|---|---|---|
| 0.00 | 0.076427 | -0.094566 | -1.222831 | 0.070093 | 0.006707 |
| 0.25 | 0.021400 | -0.308339 | -0.342399 | 0.005495 | 0.071305 |
| 0.50 | -0.053302 | -0.238626 | 0.852833 | 0.034093 | 0.042707 |
| 0.75 | -0.078998 | 0.050479 | 1.263974 | 0.074889 | 0.001911 |
| 1.00 | -0.032064 | 0.293173 | 0.513023 | 0.012337 | 0.064463 |
Formula Used
Displacement: x(t) = A cos(ωt + φ)
Velocity: v(t) = -Aω sin(ωt + φ)
Acceleration: a(t) = -ω²x(t)
Period: T = 2π / ω
Frequency: f = 1 / T
Spring angular frequency: ω = √(k / m)
Pendulum angular frequency: ω = √(g / L)
Total energy: E = 1/2 keffA²
Effective stiffness: keff = k for springs, and keff = mg / L for pendulums.
These relationships describe ideal simple harmonic motion. Pendulum mode is most accurate for small angular displacements.
How to Use This Calculator
1. Select a model
Choose a mass-spring oscillator or a simple pendulum. The calculator automatically switches required inputs for the selected system.
2. Enter physical values
Add amplitude, mass, time, and initial phase. Then enter spring constant for springs, or length and gravity for pendulums.
3. Run the calculation
Press the calculate button. The report appears immediately below the header and above the form, matching your requested layout.
4. Review and export
Check state variables, timing, force, and energy. Then download the result table as a CSV file or PDF report.
Frequently Asked Questions
1. What systems does this calculator support?
It supports an ideal mass-spring oscillator and a simple pendulum. Both models calculate timing, displacement, velocity, acceleration, energy, and force at a chosen instant.
2. What amplitude should I enter for a pendulum?
Enter linear arc displacement in meters. The tool converts that to an approximate angular amplitude using length, then applies the small-angle pendulum model.
3. Why does the pendulum result mention small angles?
The classic simple harmonic pendulum formula assumes small oscillation angles. As amplitude increases, the true period becomes longer than the approximation predicts.
4. Why is acceleration opposite to displacement?
In simple harmonic motion, acceleration always points toward equilibrium. That restoring behavior is why the sign of acceleration is opposite the sign of displacement.
5. Does the calculator include damping?
No. This version models ideal undamped motion only. Energy stays constant, and no external driving force or frictional loss is included.
6. What does the phase input control?
Initial phase shifts the waveform horizontally. It sets the starting position inside the oscillation cycle when time equals zero.
7. Can I use this for classroom examples?
Yes. The built-in example table helps compare changing displacement, velocity, acceleration, and energy over time, making it useful for demonstrations and homework checks.
8. What is included in the exported files?
The CSV export saves the result table values. The PDF export captures the motion report section, including the equation, result table, and calculation notes.