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OscillationsIntroStarter tracks

Concept module

Simple Harmonic Motion

See one repeating system from displacement to acceleration and back again, with the math tied directly to the motion on screen.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 8.00 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s8.00 s

Simple Harmonic Motion

Drag the mass to set the starting displacement.

Graphs

Switch graph views without breaking the live stage and time link.

Displacement over time

Shows the position cycle repeating around equilibrium.

time (s): 0 to 8displacement (m): -3.2 to 3.2

Displacement

Hover or scrub to link the graph back to the stage.time (s) / displacement (m)

Controls

Adjust the physical parameters and watch the motion respond.

Amplitude

A

1.4 m

Controls how far the oscillator moves from equilibrium.

Angular frequency

ω

1.8 rad/s

Controls how quickly the motion repeats.

Phase

ϕ

0 rad

Sets the starting point in the oscillation cycle.

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 1

Notice that the mass is fastest as it passes equilibrium and slowest near the turning points.

Try this

Watch the block cross the center, then compare that moment to the peak of the velocity graph.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

A

Amplitude

1.4 m

Sets the maximum displacement from equilibrium, so it changes the height of the motion and the scale of the curves.

Graph: Displacement over timeGraph: Velocity over timeGraph: Energy balanceOverlay: Motion trail

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use one live cue at a time while you manipulate the oscillator. The prompt changes when the graph, mode, or current state makes a different pattern more useful to watch.

ObservationPrompt 1 of 1

Graph: Displacement over time

Notice that the mass is fastest as it passes equilibrium and slowest near the turning points.

Try this

Watch the block cross the center, then compare that moment to the peak of the velocity graph.

Why it matters

It separates displacement from speed. The object is not fastest where it is farthest from equilibrium.

Control: AmplitudeControl: Angular frequencyGraph: Displacement over timeGraph: Velocity over timeOverlay: Motion trailOverlay: Velocity vector

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Equilibrium line

Marks the zero-displacement reference line.

What to notice

The oscillator crosses this line as it moves through the center.

Why it matters

It anchors the motion so the sign of displacement stays obvious.

Graph: Displacement over timeGraph: Velocity over timeGraph: Acceleration over time

Challenge mode

Solve compact motion targets with the same live controls, compare state, and inspect-time tools that power the concept.

0/2 solved

ConditionStretch

1 of 5 checks

Compare the timing

Use compare mode to keep a calm baseline in Setup A and make Setup B cycle faster while both setups keep about the same swing size.

Compare modeGraph-linkedGuided start2 hints

Suggested start

Open compare mode from the calm start, then edit Setup B only.

Matched

Open the Displacement over time graph.

Displacement over time

Pending

Stay in compare mode while editing Setup B.

Explore

Pending

Keep Setup A amplitude near

1.2 m

Pending

Keep Setup B amplitude near

1.2 m

Pending

Raise Setup B angular frequency above

2.6 rad/s

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the oscillator is 1.4 m from equilibrium, moving at 0 m/s, with acceleration -4.54 m/s². The phase angle is 0 rad and the period is about 3.49 s.

Equation detailsDeeper interpretation, notes, and worked variable context.

Displacement

The position of the oscillator as a function of time.

x (t) = A cos (ω t + ϕ)

A sets the size of the motion.

\omega sets how quickly the cycle repeats.

\phi shifts the starting point.

A

Amplitude 1.4 m

ω

Angular frequency 1.8 rad/s

ϕ

Phase 0 rad

Velocity

The rate at which displacement changes.

v (t) = - A ω sin (ω t + ϕ)

Velocity leads displacement by a quarter cycle.

A

Amplitude 1.4 m

ω

Angular frequency 1.8 rad/s

ϕ

Phase 0 rad

Acceleration

The restoring acceleration that pulls the system back toward equilibrium.

a (t) = - ω^{2} x (t)

Acceleration always points opposite the displacement.

ω

Angular frequency 1.8 rad/s

Energy scale

In the idealized case, the total energy remains constant and shifts between kinetic and potential forms.

E = \frac{1}{2} k A^{2}

A

Amplitude 1.4 m

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 1 of 30 / 3 complete

Oscillations and Energy

Next after this: Oscillation Energy.

1. Simple Harmonic Motion2. Oscillation Energy3. Damping / Resonance

This concept is the track start.

See next concept

Starter track

Step 1 of 90 / 9 complete

Waves

Next after this: Wave Speed and Wavelength.

1. Simple Harmonic Motion2. Wave Speed and Wavelength3. Sound Waves and Longitudinal Motion4. Pitch, Frequency, and Loudness / Intensity+5 more steps

This concept is the track start.

See next concept

Short explanation

What the system is doing

Simple harmonic motion is the cleanest place to see how a restoring effect creates repeating motion. The system keeps trying to return to equilibrium, but it overshoots, so the pattern loops instead of settling immediately.

In Open Model Lab, the goal is not to memorize a formula first. The simulation shows the object, the graph, and the derived quantities together so you can connect amplitude, phase, and angular frequency to what the system is actually doing.

Key ideas

01The object moves fastest near equilibrium and slows near the turning points.

02Acceleration always points back toward equilibrium, not toward where the object is already moving.

03Changing phase shifts the whole pattern without changing the overall shape.

Live worked example

Solve the exact state on screen.

Solve the exact state currently on screen. The steps update from the real controls and the current inspected time unless you freeze them.

Live valuesLive at t = 0.00 s

Using the current state, what is the displacement of the oscillator at $t = 0 s$ ?

t

Time

0 s

A

Amplitude

1.4 m

ω

Angular frequency

1.8 rad/s

ϕ

Phase

0 rad

1. Identify the relation

Use the live displacement relation

x (t) = A cos (ω t + ϕ)

2. Substitute the current values

x (0) = 1.4 cos (1.8 \cdot 0 + 0)

3. Compute the phase angle

The current phase angle is

ω t + ϕ = 0

, so the displacement becomes

x = 1.4 m

Current displacement

x (0) = 1.4 m

The displacement is positive, so the oscillator is on the positive side of equilibrium and the restoring acceleration points back toward the center.

Common misconception

A larger amplitude means the object moves faster at every point.

Amplitude changes the size of the oscillation, but speed still depends on position within the cycle.

Near equilibrium the speed is greatest, while the turning points are where the speed drops to zero.

Mini challenge

If you keep the amplitude the same but double the angular frequency, what changes most clearly on the graphs?

Prediction prompt

Predict whether the object travels farther or just cycles faster.

Check your reasoning

The oscillation cycles faster, but the amplitude stays the same.

A larger angular frequency means more cycles per second, so the period shrinks. The turning points stay at the same displacement because amplitude did not change.

Quick test

Variable effect

Question 1 of 4

Use the motion, graph, and equation meaning together. Each question checks whether you can explain what the system must be doing.

If amplitude stays fixed and angular frequency increases, what must change first in the stage and displacement graph?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a single oscillator moving back and forth across an equilibrium point. The object position, velocity direction, and acceleration direction are linked to the same motion so the relationship stays readable without needing to decode the graph alone.

When the user changes amplitude, angular frequency, or phase, the motion updates immediately and the graph follows the same cycle.

Graph summary

The graphs show displacement, velocity, acceleration, and energy as separate views of the same oscillation.

Each plot makes one part of the cycle obvious so the user can compare phase shifts, turning points, and equilibrium crossings.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Energy viewOscillations

Simple Harmonic Motion

See each variable before you move it.

Equilibrium line

Compare the timing

Oscillations and Energy

Waves

What the system is doing

Solve the exact state on screen.

Using the current state, what is the displacement of the oscillator at $t = 0 s$ ?

If amplitude stays fixed and angular frequency increases, what must change first in the stage and displacement graph?

Keep moving through the concept map.

Oscillation Energy

Wave Interference

Standing Waves

Simple Harmonic Motion

See each variable before you move it.

Equilibrium line

Compare the timing

Oscillations and Energy

Waves

What the system is doing

Solve the exact state on screen.

Using the current state, what is the displacement of the oscillator at t=0s?

If amplitude stays fixed and angular frequency increases, what must change first in the stage and displacement graph?

Keep moving through the concept map.

Oscillation Energy

Wave Interference

Standing Waves

Using the current state, what is the displacement of the oscillator at $t = 0 s$ ?