Simple Harmonic Motion Displacement Calculator

Calculator Inputs

Wave model

Amplitude

Displacement unit

Time t

Use seconds.

Equilibrium offset x₀

Phase angle φ

Phase unit

Frequency input type

Angular frequency ω

Used when angular frequency is selected.

Frequency f

Used when frequency is selected.

Period T

Used when period is selected.

Mass, optional

Use kg for force and energy estimates.

Spring constant, optional

Use N/m for spring force estimates.

Decimal places

Formula Used

Cosine model: x(t) = x₀ + A cos(ωt + φ)

Sine model: x(t) = x₀ + A sin(ωt + φ)

Angular frequency from frequency: ω = 2πf

Angular frequency from period: ω = 2π / T

Velocity: derivative of displacement with respect to time.

Acceleration: a(t) = -ω²[x(t) - x₀]

Spring energy: PE = 0.5kx². Mass model energy uses E = 0.5mω²A².

How to Use This Calculator

Choose the sine or cosine model.
Enter the amplitude and unit.
Enter the time value in seconds.
Add the phase angle in degrees or radians.
Select angular frequency, frequency, or period.
Enter optional mass or spring constant for force and energy estimates.
Press Calculate to view the result above the form.
Use CSV or PDF to save the current calculation.

Example Data Table

Case	A	ω	t	φ	Model	x(t)	v(t)	a(t)
Spring start at peak	0.05 m	2 rad/s	0 s	0	Cosine	0.05 m	0 m/s	-0.20 m/s²
After half second	0.05 m	2 rad/s	0.5 s	0	Cosine	0.02702 m	-0.08415 m/s	-0.10806 m/s²
Sine model crossing	10 cm	6.283 rad/s	0 s	0	Sine	0 cm	62.83 cm/s	0 cm/s²
Offset motion	4 cm	3 rad/s	0.8 s	30°	Cosine	-2.17 cm plus offset	-10.04 cm/s	19.53 cm/s²

Understanding Simple Harmonic Motion Displacement

Motion Around Equilibrium

Simple harmonic motion describes repeated motion around a stable center. A spring mass system is the classic example. A small pendulum also follows this model when its angle is small. The displacement changes smoothly because the restoring force always points back toward equilibrium. This calculator helps you model that changing position at any selected time.

What Displacement Means

Displacement is the signed distance from the center point. Positive values show one side of equilibrium. Negative values show the other side. Zero means the object is crossing the center. The amplitude sets the largest possible displacement. Angular frequency controls how quickly the cycle repeats. Phase angle shifts the wave left or right. An offset lets you model a center that is not zero.

Flexible Frequency Options

The tool can use angular frequency, frequency, or period. That makes it useful for lab work and homework. You can choose a sine or cosine model. Cosine is often used when the object starts at maximum displacement. Sine is useful when the object starts at the center. The calculator also reports velocity, acceleration, restoring-force factor, and a simple energy index.

Unit Consistency

Good results need consistent units. Use meters for amplitude when studying a spring. Use seconds for time and period. Frequency should be entered in hertz. Phase can be entered in degrees or radians. If you add an offset, use the same unit as amplitude. The graph keeps the same unit for displacement.

Graph and Exports

The chart shows how position changes around your selected time. It also highlights the current point. This is helpful because a single value can hide the full motion pattern. Export buttons let you save results as a CSV file or a simple PDF report. The example table gives quick test values. You can compare them with your own inputs to check the model.

Best Use

Use this calculator as a study guide, not as a replacement for careful measurements. Real systems may have friction, air resistance, large angles, or nonlinear springs. Those effects can change the motion. For ideal simple harmonic motion, the formulas here give a clear and dependable estimate. When values look unusual, review each input. Phase changes can move peaks fast. This often matters during quick lab checks.

FAQs

1. What is displacement in simple harmonic motion?

Displacement is the signed distance from equilibrium at a selected time. It can be positive, negative, or zero. The sign shows which side of the center the object is on.

2. Should I choose sine or cosine?

Choose cosine when motion starts at maximum displacement. Choose sine when motion starts at the equilibrium point. A phase angle can shift either model to match many starting positions.

3. What is angular frequency?

Angular frequency measures how fast the phase changes in radians per second. It equals 2π times frequency. A larger angular frequency means the object completes cycles faster.

4. Why does phase angle matter?

Phase angle shifts the wave left or right. It changes the starting position of the motion. This helps model systems that do not begin at a peak or center point.

5. Can this calculator find velocity?

Yes. It calculates velocity by differentiating the selected displacement equation. Velocity is largest near equilibrium and zero at the extreme points in ideal motion.

6. Can this calculator find acceleration?

Yes. It uses the simple harmonic relation a = -ω²x relative to equilibrium. Acceleration always points back toward the center in the ideal model.

7. Why are mass and spring constant optional?

Displacement does not require mass or spring constant. They are added for advanced force and energy estimates. Use meters, kilograms, and newtons per meter for standard results.

8. Are real systems always this exact?

No. Friction, damping, air resistance, and nonlinear springs can change real motion. This calculator models ideal simple harmonic motion, which is best for basic physics analysis.