NovaMath

Spring oscillator

Smooth-track SHM: F=−kx; T=2π√(m/k) depends only on m, k (isochronism); phase, energy, orbit, damping.

Vibration & waves (senior)SHM, resonance, wave formation and interference.

Start here

Spring oscillator

A smooth-track spring–mass is classic SHM: F=−kx, T=2π√(m/k) independent of amplitude (isochronism), and kinetic/elastic energy exchange. Use phase, energy, T–m, and orbit charts together.

Memory cue: F=−kx · T=2π√(m/k) independent of A · endpoints: v=0, |a| max · equilibrium: |v| max, a=0.

Key formulas

  • F = −kx
  • T = 2π√(m/k) (independent of amplitude)
  • E = ½kA² = Ek + Ep

How to use

  1. Read Start here, then Related knowledge §§1–3.
  2. Tune m, k, A (or drag the cart), press Play, and compare measured vs theoretical T.
  3. Switch Phase / Energy / T–m / Orbit; enable F, v, a overlays.

Read next

Related knowledge

Start from F=−kx and isochronism, then check phase, energy, and period — and verify the SHM conclusions in the sim.

1. Basic ideas

  • Equilibrium: where the net force is zero; take x = 0 there.
  • Displacement x is the departure from equilibrium; amplitude A is the maximum |x|.
  • Period T is the time for one full cycle; frequency f = 1/T.
  • Initial phase φ sets the release position/velocity (0, π/2, or π in this sim).

2. Restoring force & equation

On a smooth track the spring provides the restoring force toward equilibrium. Within the elastic limit it is proportional to displacement and opposite in direction.

  • F = −kxSHM criterion: the minus sign means F always points to equilibrium; k is stiffness.
  • a = −(k/m)x = −ω²xFrom Newton’s second law; acceleration ∝ −x.

3. Period & isochronism

  • Larger m → longer T; larger k → shorter T.
  • Larger A means a bigger swing, not a slower cycle.
  • T = 2π√(m/k)Period depends only on m and k, not on A or φ — isochronism.

4. Energy

  • Endpoints: v = 0, potential max, kinetic min.
  • Equilibrium: v max, kinetic max, potential min.
  • With light damping, amplitude and total energy E decay over time.
  • E = ½kA² = Ek + EpWithout damping, mechanical energy is constant: Ek and Ep exchange.

5. Phase relations

  • Velocity v leads displacement x by about a quarter period (π/2).
  • Acceleration a is opposite to x (a = −ω²x).
  • Compare x, v, and a on one time axis in the Phase chart.

6. What to watch

  • Endpoints: v = 0, |a| and |F| max; equilibrium: |v| max, a and F zero.
  • The green T–m curve is theory at the current k; changing m moves the point along it, changing k reshapes the curve.
  • The (x, v) portrait is roughly a closed ellipse undamped; with damping it spirals inward.

7. Common pitfalls

  • Do not assume larger A means larger T — not for ideal SHM.
  • Restoring force is the net force toward equilibrium, not an extra spring.
  • Max speed is where a = 0; max |a| is where v = 0 — they are out of phase.
  • Default is undamped (constant A); turn on light damping to see decay.

8. Practice in this sim

  1. Run the guided demo first, then tune m, k, and A freely.
  2. Vary only A to check isochronism; then change m or k and watch T.
  3. Enable overlay vectors; switch Phase / Energy / T–m / Orbit and match charts to readouts.

9. Textbook links

  • Selective Compulsory Vol. 1 · Ch. 2 Mechanical vibration
  • SHM · restoring force · A / T / f

Applications

  • Car suspensions and force meters use spring–mass models.
  • Seismographs and dampers need period, damping, and energy ideas.

FAQ

Does period depend on amplitude?

Not in the ideal model. T=2π√(m/k) depends only on m and k — changing A barely changes T (isochronism).

Why doesn’t the T–m green curve move when I change m?

The green curve is T(m)=2π√(m/k) at the current k. Changing m slides the current point along the curve; change k to reshape the whole curve.

How are v and a related to x on the phase chart?

In SHM, v leads x by about π/2, and a is opposite in phase to x (a=−ω²x). Compare them on one timeline in Phase view.

What happens to energy with light damping?

Without damping, total E stays roughly constant while Ek and Ep trade places. With light damping, amplitude shrinks and E decays over time.

Does a larger amplitude change the period?

Ideally no — SHM period is independent of amplitude.

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