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Premium experiment • phase-space oscillation

PHYSICS • OSCILLATIONS

Simple Harmonic Motion & Energy

Watch an ideal spring-mass oscillator exchange kinetic and potential energy while total mechanical energy remains constant.

● ENERGY EXCHANGEIdeal spring-mass oscillator
Position
Velocity
Period
Total energy

Reasoning inspector

live model state
Angular frequency ω—
Period T—
Position x—
Velocity v—
Phase angle—
Kinetic energy—
Potential energy—
Total energy—
Maximum speed—
Maximum acceleration—

Energy moves, total stays fixed

In ideal simple harmonic motion, spring potential energy is greatest at the turning points and kinetic energy is greatest at equilibrium.

LEARN WITH THE SIMULATION

Understand Simple Harmonic Motion & Energy by experimenting.

START HERE

Simple harmonic motion is the cleanest example of an oscillator whose force, timing and energy can all be described exactly. In this mass-spring model, ω = √(k/m) sets the time scale, x = A cos(ωt) and v = −Aω sin(ωt) trace a closed ellipse in phase space, and kinetic plus spring potential energy stays constant. The premium lab adds a period-target challenge and a live x-v phase portrait so timing and energy reasoning can be tested together.

BUILD THE IDEA

What you should understand first

  • Relate angular frequency and period to spring constant and mass.
  • Use the x-v phase-space ellipse to connect position and velocity during a cycle.
  • Track kinetic and potential energy exchange while total mechanical energy remains constant.
  • Explain why amplitude changes total energy and maximum speed but not the ideal period of a linear spring-mass oscillator.

TRY IT YOURSELF

Can you change oscillator energy without changing its period?

  1. Choose a mass and spring constant, then reach a selected period target.
  2. Record the period, angular frequency and total energy.
  3. Increase amplitude while leaving mass and spring constant fixed.
  4. Compare total energy, maximum speed and the phase-space ellipse with the unchanged period.
Read the visual:

For ideal linear SHM, T = 2π√(m/k) contains no amplitude term, so changing amplitude leaves the period unchanged. Energy grows as ½kA², maximum speed grows as Aω and the phase-space ellipse expands even though the oscillation timing is identical.

WHY IT WORKS 01

The period is controlled by the ratio m/k

Hooke's law gives m x¨ = −kx, or x¨ + (k/m)x = 0. Comparing this with the standard SHM equation identifies ω² = k/m, so T = 2π/ω = 2π√(m/k). Increasing mass adds inertia and lengthens the period, while increasing spring stiffness strengthens the restoring acceleration and shortens it.

WHY IT WORKS 02

Phase space turns a cycle into a geometric ellipse

Using x = A cos θ and v = −Aω sin θ gives (x/A)² + (v/Aω)² = 1. Every ideal trajectory therefore lies on a closed ellipse in the x-v plane. The moving point in the phase-space inset shows where the oscillator is in its cycle without needing the clock explicitly, and the ellipse size encodes amplitude and maximum speed.

WHY IT WORKS 03

Energy exchange follows the same phase relation

Spring potential energy is ½kx² and kinetic energy is ½mv². Substituting the SHM solution shows their sum is always ½kA². At maximum displacement all ideal mechanical energy is spring potential; at equilibrium all of it is kinetic. The energy bars and phase portrait therefore tell the same story in two complementary representations.

DO NOT MIX THESE UP

Common misconception

At equilibrium the oscillator does not have zero energy; spring potential energy is minimum there while kinetic energy is maximum. At the turning points velocity is zero but acceleration magnitude is maximum. The amplitude-independent period is a property of the ideal linear spring model and does not extend automatically to nonlinear springs or large-angle pendulums.

PRACTICE CHALLENGE

Can you hit a requested period using mass and stiffness, then prove that amplitude changes energy rather than timing?

  • Choose a target period.
  • Adjust mass and spring constant until the period target is reached.
  • Change amplitude only and verify that the period and angular frequency remain fixed.
  • Use total energy, maximum speed and the phase-space ellipse to explain what amplitude changed instead.
Reveal the reasoning

The target period depends only on m/k through T = 2π√(m/k). Once that ratio is correct, changing amplitude does not change T in the ideal linear model. Instead, total energy scales with A², maximum speed scales with A and the x-v phase-space ellipse expands proportionally.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • The oscillator is an ideal one-dimensional mass-spring system obeying Hooke's law.
  • Spring constant and mass remain constant, with no damping or external driving force.
  • Mechanical energy is conserved and the equilibrium position is fixed at x equals zero.

Model limitations

  • Friction, air resistance, nonlinear springs, finite spring mass and resonance are omitted.
  • The visual spring geometry is schematic and does not encode coil mechanics.

CONNECT IT

Where this matters

  • Mass-spring vibration systems
  • Mechanical and electrical oscillator analogies
  • Phase-space reasoning
  • Foundations for resonance and normal modes

WORDS THAT MATTER

Key vocabulary

Simple harmonic motion
Oscillation governed by a restoring acceleration proportional to and opposite displacement.
Angular frequency
Rate ω = √(k/m) in radians per second for the ideal spring-mass oscillator.
Period
Time for one complete oscillation, T = 2π/ω.
Phase space
Representation using position and velocity coordinates rather than position and time.
Mechanical energy
Constant sum of kinetic and spring potential energy in the undamped ideal model.