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Undergraduate · Harmonic oscillators

Harmonic oscillators: Positive starting displacement

Harmonic oscillators: investigate positive starting displacement with angular frequency ω = 1; initial displacement b = 2.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Harmonic oscillators: Positive starting displacement. Time: 0. Displacement: 2. Velocity: 1. Conserved energy: 2.5Displacement varies; energy is conserved-2.242.2403.146.28displacementtime t → · labeled axes rescale to this model
There is no damping or external force; the mass normalization is one and ω>0. Time and position use consistent arbitrary units. The graph shows displacement versus time, with energy computed from the exact derivative.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Time
0
Displacement
2
Velocity
1
Conserved energy
2.5

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Understand what you are seeing

The idea behind the motion.

A second-order equation needs both initial position and initial velocity. Here the velocity starts at one, so changing frequency also changes the sine coefficient needed to preserve that initial condition. The sum of kinetic and potential quadratic energies stays constant throughout the exact motion. This investigation starts with Angular frequency ω = 1; Initial displacement b = 2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

y″+ω²y=0; y=b cos(ωt)+sin(ωt)/ω

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Verify y(0)=b and y′(0)=1 from the displayed solution. The starting case is “Positive starting displacement.”

  2. STEP 2

    Follow the changing quantity

    Trace displacement over 0≤t≤2π and watch velocity change sign at turning points.

  3. STEP 3

    Explain and test the result

    Calculate (y′²+ω²y²)/2 at several times. Compare different parameter settings without expecting the energy itself to be identical across different initial conditions.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Angular frequency ω = 1; Initial displacement b = 2. Pause the timeline at 40%. Given time = 2.513, calculate displacement, velocity, conserved energy. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

At t=2.513274, y=2cos(1t)+sin(1t)/1=-1.030249. Differentiate to obtain y′=-1.984587; energy is [(-1.984587)²+(1)²(-1.030249)²]/2=2.5. Results: Displacement: -1.03; Velocity: -1.985; Conserved energy: 2.5. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Undergraduate · Calculus

Use two initial conditions in a second-order motion model

Solve an undamped oscillator and verify conserved energy.