Energy in simple harmonic motion
Sloshing energy
- A swing rises, slows, stops, then races back through the bottom.
- Energy keeps sloshing between kinetic and potential forms.
- Add them up and the total never changes (with no friction).
The swap
- At equilibrium: speed is greatest → KE is maximum, PE is minimum.
- At the extremes: at rest → KE is zero, PE is maximum.
The kinetic energy of an oscillator is greatest:
Speed is greatest at the middle, so KE peaks there while PE is at its lowest.
At the extreme positions of the motion, all the energy is potential.
The oscillator is momentarily at rest there, so KE = 0 and all the energy is potential.
Constant total
- With no damping, the total energy stays constant (conservation of energy).
- $E_{\text{total}} = \tfrac{1}{2}m\omega^{2}x_0^{2}$ — the value at the moment of maximum speed.
With no damping, the total energy of an oscillator stays constant.
Energy just swaps between KE and PE; their sum is conserved.
Energy and amplitude
- Total energy is proportional to the square of the amplitude.
- Double the amplitude → four times the energy.
If the amplitude doubles, the total energy of an oscillator becomes:
$E \propto x_0^{2}$, so doubling $x_0$ gives $2^{2} = 4$ times the energy.
An oscillator has $m = 0.20\ \text{kg}$, $\omega = 10\ \dfrac{\text{rad}}{\text{s}}$ and amplitude $0.10\ \text{m}$. What is its total energy?
$E = \tfrac{1}{2}m\omega^{2}x_0^{2} = \tfrac{1}{2} \times 0.20 \times 10^{2} \times 0.10^{2} = 0.10\ \text{J}$.
You've got it
- KE and PE swap: KE max at the middle, PE max at the ends
- total energy is constant (no damping): $E = \tfrac{1}{2}m\omega^{2}x_0^{2}$
- $E \propto x_0^{2}$ — double the amplitude, four times the energy