Energy of Simple Harmonic Oscillators · 简谐振子的能量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| damping/ˈdæmpɪŋ/ | 阻尼 | zǔ ní |
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.
秋千一次次地用高度换速度
- 在弧顶,秋千静止,但很高;在弧底,它很低,却在飞驰。
- 它在无休止地把势能换成动能,再换回来。
- SHM 振子做着完全相同的事,把能量在两种形式之间来回倾倒。
- 把它们加起来,总量从不改变。
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).
两种能量互换位置
- 在两端,振子瞬间静止:全部能量是势能($\tfrac12 kx^2$)。
- 在中间,它动得最快:全部能量是动能($\tfrac12 mv^2$)。
- 在两者之间,能量在两者间分配。
- 这种交换每半个周期重复一次,永远持续(无摩擦时)。
Watch PE and KE swap · 观察势能与动能的交换
Step through the oscillation and watch potential and kinetic energy trade off while the total holds. · 逐步展示振动过程，观察势能和动能如何交替变化，同时总能量保持不变。
Where in the swing is all the energy kinetic? · 在摆动的哪个位置，所有能量均为动能？
At the middle the speed is greatest and $x = 0$, so all the energy is kinetic. · 在中间速度最大且$x = 0$，因此所有能量都是动能。
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.
总量恒定
- 无摩擦振子的总机械能固定为 $E = \tfrac12 kA^2$。
- 它取决于振幅的平方——更大的摆动储存更多能量。
- 在任意点,$\tfrac12 kx^2 + \tfrac12 mv^2 = \tfrac12 kA^2$。
- 能量守恒应用于振动,把整个运动联系在一起。
A spring with $k = 200\ \tfrac{\text{N}}{\text{m}}$ oscillates with amplitude $0.1\ \text{m}$. What is the total energy, in joules? · 劲度系数为$k = 200\ \tfrac{\text{N}}{\text{m}}$的弹簧以振幅$0.1\ \text{m}$振动。其总能量是多少焦耳？
$E = \tfrac12 kA^2 = \tfrac12 (200)(0.01) = 1\ \text{J}$.
The total energy of a frictionless oscillator is $\tfrac12 k \_\_$ (fill in the missing part). · 无摩擦振子的总能量为$\tfrac12 k \_\_$（填写缺失部分）。
$E = \tfrac12 kA^2$ — set by the amplitude squared. · $E = \tfrac12 kA^2$——由振幅的平方决定。
Select all · 所有 true statements about energy in undamped SHM. · 选择所有关于无阻尼简谐运动能量的正确陈述。
The total stays fixed while PE (ends) and KE (middle) trade off at opposite points. · 总能量保持不变，而势能（两端）和动能（中间）在相反位置相互转化。
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.
阻尼把能量泄漏掉
- 真实振子把能量损失给摩擦和空气阻力——这就是阻尼。
- 随着机械能变成热,振幅慢慢缩小。
- 轻微阻尼的秋千在许多周期里逐渐停下;重阻尼的几乎不摆。
- 无阻尼的 SHM(振幅恒定)是我们用来比较的理想。
If you double the amplitude of an oscillator, its total energy becomes: · 如果将振子的振幅加倍，其总能量变为：
$E = \tfrac12 kA^2 \propto A^2$, so doubling $A$ gives four times the energy. · $E = \tfrac12 kA^2 \propto A^2$，因此将$A$加倍会使能量变为四倍。
Damping causes the amplitude of a real oscillator to slowly shrink over time. · 阻尼会导致真实振子的振幅随时间缓慢衰减。
Damping converts mechanical energy to heat, so the amplitude decays cycle by cycle. · 阻尼将机械能转化为热能，因此振幅逐周期衰减。
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.
总能量随振幅的平方变化($E = \tfrac12 kA^2$),而非振幅。振幅加倍,能量就变为四倍。而且 KE 和 PE 各自在摆动的相反点最大——绝不在同一处。
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 $1\ \text{J}$.
一根 $k = 200\ \tfrac{\text{N}}{\text{m}}$ 的弹簧以振幅 $A = 0.1\ \text{m}$ 振动。
- 总能量 $E = \tfrac12 kA^2 = \tfrac12 (200)(0.1^2) = \tfrac12 (200)(0.01) = 1\ \text{J}$。
在中间这全是动能;在两端全是势能——但始终是 $1\ \text{J}$。
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.
在 SHM 中,能量在动能($\tfrac12 mv^2$,中间最大)和势能($\tfrac12 kx^2$,两端最大)之间互换。总量恒定,$E = \tfrac12 kA^2$——随振幅的平方变化。真实振子通过阻尼损失能量。