SHM and the Reference Circle

Also called

simple harmonic motion · SHM · oscillations · reference circle · phase · amplitude · angular frequency · time period · spring-mass system · restoring force · kinetic and potential energy in SHM · damped oscillations · forced oscillations · resonance · Barton's pendulums · projection of circular motion · x = A cos ωt · सरल आवर्त गति · दोलन · अनुनाद · आयाम

Topics

• In SHM the restoring force is proportional to the displacement, F = −kx, so the motion repeats with a fixed period.
• Displacement x = A cos(ωt + φ): A is the amplitude, ω = 2π/T the angular frequency and ωt + φ the phase.
• SHM is the projection of uniform circular motion: the shadow of a point moving round the reference circle.
• The radius of the reference circle is the amplitude; the angle the arm has turned is the phase.
• For a spring of constant k and a mass m, T = 2π√(m/k); the period does not depend on the amplitude.
• Energy swaps between PE = ½kx² and KE = ½mv²; their sum ½kA² stays constant in undamped SHM.
• Friction or a viscous fluid damps the motion: the amplitude dies away exponentially while the period hardly changes.
• A driven oscillator responds most when the driving frequency equals its natural frequency: resonance.
• In Barton's pendulums the one as long as the driver swings most; shorter ones move with it, longer ones against it.

About this model

A GLB file of about 353 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

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