Energy of Simple Harmonic Oscillators
Introduced| English |
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| damping/ˈdæmpɪŋ/ |
A swing trades height for speed, over and over
- At the top of its arc a swing is still, but high up; at the bottom it is low, but racing.
- It is endlessly swapping potential energy for kinetic energy and back.
- An SHM oscillator does exactly the same, sloshing energy between two forms.
- Add them up and the total never changes.
Two energies that trade places
- At the extremes, the oscillator is momentarily still: all energy is potential ($\tfrac12 kx^2$).
- At the middle, it moves fastest: all energy is kinetic ($\tfrac12 mv^2$).
- In between, energy is shared between the two.
- The exchange repeats every half-cycle, forever (with no friction).
Watch PE and KE swap
Step through the oscillation and watch potential and kinetic energy trade off while the total holds.
Where in the swing is all the energy kinetic?
At the middle the speed is greatest and $x = 0$, so all the energy is kinetic.
The total is constant
- The total mechanical energy of a frictionless oscillator is fixed at $E = \tfrac12 kA^2$.
- It depends on the amplitude squared — a bigger swing stores more energy.
- At any point, $\tfrac12 kx^2 + \tfrac12 mv^2 = \tfrac12 kA^2$.
- Energy conservation, applied to oscillation, ties the whole motion together.
A spring with $k = 200\ \tfrac{\text{N}}{\text{m}}$ oscillates with amplitude $0.1\ \text{m}$. What is the total energy, in joules?
$E = \tfrac12 kA^2 = \tfrac12 (200)(0.01) = 1\ \text{J}$.
The total energy of a frictionless oscillator is $\tfrac12 k \_\_$ (fill in the missing part).
$E = \tfrac12 kA^2$ — set by the amplitude squared.
Select all true statements about energy in undamped SHM.
The total stays fixed while PE (ends) and KE (middle) trade off at opposite points.
Damping bleeds energy away
- Real oscillators lose energy to friction and air resistance — this is damping 阻尼.
- The amplitude slowly shrinks as mechanical energy turns to heat.
- A gently damped swing dies down over many cycles; a heavily damped one barely swings.
- Undamped SHM (constant amplitude) is the ideal we compare against.
If you double the amplitude of an oscillator, its total energy becomes:
$E = \tfrac12 kA^2 \propto A^2$, so doubling $A$ gives four times the energy.
Damping causes the amplitude of a real oscillator to slowly shrink over time.
Damping converts mechanical energy to heat, so the amplitude decays cycle by cycle.
The total energy scales with amplitude squared ($E = \tfrac12 kA^2$), not amplitude. Double the amplitude and you quadruple the energy. And KE and PE are each largest at opposite points of the swing — never at the same place.
A spring with $k = 200\ \tfrac{\text{N}}{\text{m}}$ oscillates with amplitude $A = 0.1\ \text{m}$.
- Total energy $E = \tfrac12 kA^2 = \tfrac12 (200)(0.1^2) = \tfrac12 (200)(0.01) = 1\ \text{J}$.
At the middle this is all kinetic; at the extremes it is all potential — but always $1\ \text{J}$.
In SHM, energy trades between kinetic ($\tfrac12 mv^2$, max at the middle) and potential ($\tfrac12 kx^2$, max at the ends). The total is constant, $E = \tfrac12 kA^2$ — scaling with amplitude squared. Real oscillators lose energy through damping.