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Classical Mechanics

Simple Harmonic Motion

The one differential equation that shows up everywhere, and the approximation hiding inside every pendulum clock.

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Simple Harmonic Motion

The one differential equation that shows up everywhere, and the approximation hiding inside every pendulum clock.

17 min2 checkpoints4 simulation tasks

Introduction

A mass on a spring, a pendulum, a molecule and a building in an earthquake all oscillate, and near equilibrium they obey the same equation.

Predict

You pull a mass on a spring twice as far from equilibrium and release it. What happens to the time for one complete oscillation?

Same period, any swing

Try it

The mass starts 0.30 m from equilibrium and takes 0.99 s per swing. Pull it twice as far out, run it, and read the measured period.

GoalRun the mass from twice as far out and time its swing.

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Whenever the restoring force is proportional to the displacement and points back toward equilibrium, F=−kxF = -kx, the motion is a sinusoid with period

T=2πmkT = 2\pi\sqrt{\frac{m}{k}}

How you start it sets the amplitude; the system itself sets the period.

What sets the period

Try it

The spring constant is 20 N/m. Change the mass until the period is 2.0 s.

GoalMake the period 2.0 s.

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Practice

A kg mass hangs on a spring with k=k = N/m. How long does one full oscillation take?

s

Predict

What does the phase portrait (velocity plotted against displacement) look like for the mass on a spring?

The pendulum's small print

A pendulum's restoring torque is −mgLsin⁡θ-mgL\sin\theta, not −mgLθ-mgL\theta. Its motion is simple harmonic only when sin⁡θ≈θ\sin\theta \approx \theta, which is excellent for small swings and quietly fails for large ones.

Read the graph

This pendulum starts at 60°, beside a faint small-angle ghost. Run it and time one full swing from the graph. How long does it take?

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s

Run the simulation first, then read the value off it.

Try it

Lower the start angle until the small-angle period is off by just 1%.

GoalFind the start angle where small-angle theory is 1% off.

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What comes next

You doubled a swing without changing its period, tuned a spring to 2 s, and found where a pendulum stops being simple. Real oscillators lose energy and eventually stop; push them at the right rhythm, though, and something dramatic happens.

Ideas this lesson took on

Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.

  • “The period depends on the amplitude”
  • “Everything is zero at the turning point”
  • “The pendulum is exactly isochronous”

Finished with this lesson?

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Research behind this lesson

  • Trowbridge & McDermott 1981: The companion interviews on acceleration: students routinely read acceleration off the velocity, momentarily at rest means no acceleration, moving faster means accelerating, rather than off its rate of change.
  • Viennot 1979: Students spontaneously supply a force in the direction of motion and treat it as proportional to the speed: a pattern that appears across countries, languages and curricula.
  • Van Heuvelen 1991: Experts move fluently between words, pictures, diagrams, graphs and equations; novices tend to jump straight to equations. Explicit practice moving between representations is what closes the gap.