Calculator Inputs
Example data table
This sample uses A = 2, ω = 3, φ = 0, and the cosine model.
| Time t | Displacement x(t) | Velocity v(t) | Acceleration a(t) |
|---|---|---|---|
| 0 | 2 | 0 | -18 |
| 0.25 | 1.463378 | -4.089833 | -13.1704 |
| 0.5 | 0.141474 | -5.98497 | -1.27327 |
| 0.75 | -1.256347 | -4.668439 | 11.307125 |
| 1 | -1.979985 | -0.84672 | 17.819865 |
| 1.25 | -1.641119 | 3.429368 | 14.770068 |
Formula used
Simple harmonic motion follows a repeating position equation built from amplitude, angular frequency, time, and phase angle.
- Cosine model: x(t) = A cos(ωt + φ)
- Sine model: x(t) = A sin(ωt + φ)
- Velocity for cosine: v(t) = -Aω sin(ωt + φ)
- Velocity for sine: v(t) = Aω cos(ωt + φ)
- Acceleration: a(t) = -ω²x(t)
- Angular frequency from period: ω = 2π / T
- Angular frequency from frequency: ω = 2πf
- Period: T = 2π / ω
- Frequency: f = 1 / T
- Spring-mass relation: ω = √(k / m)
- Maximum speed: vmax = Aω
- Maximum acceleration: amax = Aω²
- Total energy: E = 1/2 mω²A²
These equations let you analyze instantaneous state values and overall motion characteristics in one pass.
How to use this calculator
- Select the sine or cosine model that matches your equation setup.
- Enter amplitude, time, phase, and the phase unit.
- Choose how angular frequency will be determined.
- Provide ω directly, or use period, frequency, or spring-mass inputs.
- Optionally enter mass to estimate force and energy terms.
- Press the calculate button to display results above the form.
- Review the generated graph, table values, and derived motion quantities.
- Use the CSV and PDF buttons to export the current results.
About this SHM equation calculator
This calculator helps evaluate a full simple harmonic motion state from a compact group of inputs. It supports both sine and cosine displacement models, which makes it useful for classwork, lab interpretation, and general oscillator analysis.
You can enter angular frequency directly when the motion equation is already known. You can also derive angular frequency from period, ordinary frequency, or the spring-mass relation. That flexibility makes the page helpful across physics problems involving oscillations, springs, and cyclic motion.
The result block shows the quantities most people need during SHM work: displacement, velocity, acceleration, period, frequency, phase angle, and maximum values. If mass is supplied, the page also estimates restoring force, kinetic energy, potential energy, total energy, and the implied spring constant.
The graph helps compare how position, velocity, and acceleration evolve over time. Since SHM variables are phase shifted, the chart makes those relationships easier to inspect than a single equation line. That is especially useful when checking sign changes and peak values.
The sample table and formula section provide a quick reference for practical use. Together they make the page suitable for revision, teaching support, and repeated calculations where a clean exported record is useful.
FAQs
1) What does this calculator solve?
It solves core SHM quantities from the motion equation. You can compute displacement, velocity, acceleration, frequency, period, phase angle, maxima, and optional force and energy values.
2) Should I choose sine or cosine?
Choose the form that matches your original equation or initial condition setup. Both describe SHM, but the phase angle and starting position differ between them.
3) Can I enter phase in degrees?
Yes. Select degrees in the phase unit field. The calculator converts that value to radians before applying the SHM equations.
4) Why is acceleration negative relative to displacement?
In SHM, acceleration always points toward equilibrium. That is why the equation uses a negative sign, showing acceleration opposes displacement from the center.
5) When should I enter mass?
Enter mass when you want force, energy, or derived spring constant outputs. For displacement, velocity, acceleration, period, and frequency, mass is not always required.
6) What is the difference between frequency and angular frequency?
Frequency counts cycles per second. Angular frequency measures the same oscillation rate in radians per second. They are related by ω = 2πf.
7) Does the graph show one point or many points?
The graph plots many time samples across multiple periods. That lets you inspect the continuous variation of displacement, velocity, and acceleration.
8) Can I export the results?
Yes. After calculation, use the CSV button for spreadsheet-ready data or the PDF button for a compact report of the current result table.