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OscillationsIntroStarter track

Concept module

Oscillation Energy

Watch kinetic and potential energy trade places in simple harmonic motion while the total stays fixed by amplitude and spring stiffness.

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

Oscillation Energy

Drag the mass to set the starting displacement.

Graphs

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

Energy balance

Kinetic and potential energy trade places while the total stays flat in the ideal model.

time (s): 0 to 8energy (J): 0 to 4

KineticPotentialTotal energy

Hover or scrub to link the graph back to the stage.time (s) / energy (J)

Controls

Adjust the physical parameters and watch the motion respond.

Amplitude

A

1.4 m

Changes the turning points and the total-energy scale.

Spring constant

k

3.2 N/m

Changes how stiff the spring is, so it affects both the energy scale and the cycle speed.

Mass

m

1 kg

Changes how quickly the oscillator responds without changing the total energy when amplitude and spring constant stay fixed.

Phase

ϕ

0.9 rad

Changes where the cycle begins.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

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

Graph readingPrompt 1 of 2

Notice that the mass pauses at the turning points even though the total-energy line stays flat.

Try this

Freeze the motion near

+ A

- A

and compare the displacement marker with the potential-energy peak.

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 turning points and the total-energy scale. Doubling amplitude makes the swing wider and raises total energy by a square-law effect.

Graph: Energy balanceGraph: Displacement over timeOverlay: Turning-point markersOverlay: Energy bars

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 cue at a time. The prompt changes when the current graph, mode, or control change makes a different energy pattern worth following.

Graph readingPrompt 1 of 2

Graph: Energy balance

Notice that the mass pauses at the turning points even though the total-energy line stays flat.

Try this

Freeze the motion near

+ A

- A

and compare the displacement marker with the potential-energy peak.

Why it matters

A flat total-energy line means the energy has moved into the spring rather than disappearing.

Graph: Energy balanceGraph: Displacement over timeOverlay: Turning-point markersOverlay: Energy barsEquation

A

Equation

k

Guided overlays

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

4 visible

Overlay focus

Equilibrium line

Marks the zero-displacement position where spring stretch is minimal.

What to notice

Kinetic energy peaks as the mass passes this line in the ideal model.

Why it matters

It anchors the moment where the energy balance flips most strongly toward kinetic energy.

Control: PhaseGraph: Energy balanceGraph: Displacement over timeGraph: Velocity over timeEquation

ϕ

Challenge mode

Use the live energy graph and inspect-time controls to turn the oscillator into a small energy-targeting task.

0/2 solved

TargetCore

2 of 3 checks

Build five joules

From the mixed-energy baseline, raise the stored energy to about

5 J

without making the oscillator heavier than about

1.2 kg

Graph-linkedGuided start2 hints

Suggested start

Use the energy graph as the target readout while you adjust amplitude or spring stiffness.

Matched

Keep the energy graph active.

Energy balance

Matched

Keep mass between

0.8

and

1.2 kg

1 kg

Pending

Bring the total energy into the

4.8

5.3 J

window.

3.14 J

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

At t = 0 s, the oscillator is 0.87 m from equilibrium, moving at -1.96 m/s, with acceleration -2.78 m/s². The phase angle is 0.9 rad and the period is about 3.51 s.

Equation detailsDeeper interpretation, notes, and worked variable context.

Displacement

The oscillator's position in the cycle.

x (t) = A cos (ω t + ϕ)

Turning points occur at plus or minus the amplitude.

Equilibrium is the moment when displacement is zero.

A

Amplitude 1.4 m

ϕ

Phase 0.9 rad

Natural angular frequency

The spring constant and mass together set how quickly the energy exchange repeats.

ω = \frac{k}{m}

k

Spring constant 3.2 N/m

m

Mass 1 kg

Kinetic energy

Energy carried by the moving mass.

K = \frac{1}{2} m v^{2}

m

Mass 1 kg

Potential energy

Energy stored in the stretched or compressed spring.

U = \frac{1}{2} k x^{2}

k

Spring constant 3.2 N/m

Total energy

In ideal SHM the total stays constant while kinetic and potential energy trade places.

E = K + U = \frac{1}{2} k A^{2}

A

Amplitude 1.4 m

k

Spring constant 3.2 N/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 2 of 30 / 3 complete

Oscillations and Energy

Earlier steps still set up Oscillation Energy.

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

Previous step: Simple Harmonic Motion.

Start from track beginning

Short explanation

What the system is doing

Oscillation energy is the same simple harmonic motion you already know, but viewed through what the system stores and releases. The mass never loses total energy in the ideal model; it keeps trading that energy between spring potential energy and kinetic energy.

That exchange is easiest to trust when the motion, the bars, and the graph stay locked together. The goal here is to see why turning points are pure potential-energy moments, why equilibrium is the kinetic-energy peak, and how amplitude and spring stiffness set the total-energy scale.

Key ideas

01At the turning points, velocity is zero, so kinetic energy is zero and the spring stores all of the energy as potential energy.

02At equilibrium, displacement is zero, so spring potential energy is minimal and the motion carries almost all of the energy as kinetic energy.

03In ideal SHM, the total-energy line stays flat while kinetic and potential energy trade places.

04Changing amplitude or spring constant changes the total stored energy, while changing phase only changes where the cycle starts.

Live worked example

Solve the exact state on screen.

Solve the energy state that is actually on screen. Freeze it if you want a stable snapshot, or keep it live and let the example follow the motion.

Live valuesLive at t = 0.00 s

At $t = 0 s$ , how is the oscillator's total energy split between kinetic and potential energy?

t

Time

0 s

m

Mass

1 kg

k

Spring constant

3.2 N/m

x

Current displacement

0.87 m

v

Current velocity

-1.96 m/s

1. Write the two energy relations

Use

K = \frac{1}{2} m v^{2}

and

U = \frac{1}{2} k x^{2}

for the same instant.

2. Substitute the live values

K = \frac{1}{2} (1) (- 1.96)^{2}

and

U = \frac{1}{2} (3.2) (0.87)^{2}

3. Compare the two parts

That gives

K = 1.92 J

and

U = 1.21 J

, so the total stays

E = 3.14 J

Energy at this instant

K = 1.92 J, U = 1.21 J, E = 3.14 J

Kinetic energy is larger here, so the mass is in the faster middle part of the swing rather than near an edge.

Common misconception

If the mass stops for an instant at a turning point, the oscillator has no energy there.

The motion stops only because the energy is temporarily stored in the spring instead of in the mass's motion.

At the turning point, kinetic energy is zero but potential energy is at its maximum, so the total energy is still present and ready to pull the system back.

Mini challenge

Two oscillators have the same amplitude, mass, and spring constant. One starts at equilibrium and the other starts at a turning point. Which one has more total energy?

Prediction prompt

Decide whether starting position changes the total energy or only changes how that same total is split between kinetic and potential energy.

Check your reasoning

They have the same total energy. The starting position only changes the split between kinetic and potential energy.

Amplitude and spring constant set the total-energy scale in ideal SHM. Phase only changes where the cycle begins, so one oscillator can start as pure kinetic while the other starts as pure potential even though the total is the same.

Quick test

Graph reading

Question 1 of 4

Answer from the motion and the energy meaning together. These checks are about what must be true, not about plugging numbers into a formula by habit.

At the instant the mass crosses equilibrium in this ideal oscillator, what must the energy graph show?

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. Optional markers show the turning points, and an energy card shows how the total energy is split between kinetic and spring potential energy at the current instant.

Changing amplitude, spring constant, mass, or phase updates the same motion, the same energy readout, and the same graph so the energy story stays tied to one physical state.

Graph summary

The energy graph shows kinetic, potential, and total energy as synchronized views of the same oscillation. The displacement and velocity graphs sit beside it so you can compare where the mass is, how fast it is moving, and which energy form is dominating.

Pausing or scrubbing the graph keeps the stage, the energy bars, and the marker locked to the same instant.

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.

Boundary patternOscillations

Oscillation Energy

See each variable before you move it.

Equilibrium line

Build five joules

Oscillations and Energy

What the system is doing

Solve the exact state on screen.

At $t = 0 s$ , how is the oscillator's total energy split between kinetic and potential energy?

At the instant the mass crosses equilibrium in this ideal oscillator, what must the energy graph show?

Keep moving through the concept map.

Standing Waves

Wave Interference

Damping / Resonance

Oscillation Energy

See each variable before you move it.

Equilibrium line

Build five joules

Oscillations and Energy

What the system is doing

Solve the exact state on screen.

At t=0s, how is the oscillator's total energy split between kinetic and potential energy?

At the instant the mass crosses equilibrium in this ideal oscillator, what must the energy graph show?

Keep moving through the concept map.

Standing Waves

Wave Interference

Damping / Resonance

At $t = 0 s$ , how is the oscillator's total energy split between kinetic and potential energy?