Energy is the other way to read an oscillation. It is conserved overall, but it shuttles between kinetic and potential in a way fixed entirely by the amplitude.
In SHM kinetic and potential energy interchange while the total stays constant. KE is greatest at the centre, PE at the extremes, and the total is E = ½mω²x₀².
As an oscillator moves, energy sloshes between kinetic and potential. At the centre it is all kinetic, at the extremes all potential, and at every point in between the two add up to the same total. The simulation shows the bars trading height while the total bar stands still.
The total energy of an undamped oscillation is E = ½mω²x₀². The potential energy is ½mω²x² and the kinetic energy is the remainder, ½mω²(x₀² − x²). Because E depends on the square of the amplitude, doubling the amplitude makes the energy four times as large.
Four quick checks on the energy interchange and the total energy. Each correct answer earns XP and lights this skill on your star map.
For an undamped oscillator, the total energy during the motion is:
The kinetic energy of an oscillator is greatest:
The total energy of an oscillation is E = ½mω²x₀². Doubling the amplitude changes the total energy to:
At the extremes of the motion (x = ±x₀), the energy of the oscillator is:
The total energy is proportional to the amplitude squared (and to ω²), so doubling the amplitude quadruples the energy. Kinetic energy is greatest at the centre and potential energy greatest at the extremes; their sum is constant only while there is no damping to drain energy away.
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