Energy in simple harmonic motion
| English | Chinese | Pinyin |
|---|---|---|
| equilibrium | 平衡 | píng héng |
| potential energy | 势能 | shì néng |
| damping | 阻尼 | zǔ ní |
| amplitude | 振幅 | zhèn fú |
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 平衡: speed is greatest → KE is maximum, PE is minimum.
- At the extremes: at rest → KE is zero, PE is maximum.
Kinetic and potential energy 势能 swap over a cycle while the total energy stays constant
Energy in SHM
KE + PE = constant
Energy trades between kinetic (fastest at the centre) and potential (max at the ends).
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 阻尼, 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.
A plucked guitar string vibrates at its resonant frequencies
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 振幅
- Total energy is proportional to the square of the amplitude.
- Double the amplitude → four times the energy.
The acceleration-displacement graph is a straight line through the origin with gradient minus omega squared
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