Defining Simple Harmonic Motion (SHM) · 定义简谐运动 (SHM)
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| simple harmonic motion/ˈsɪmpl hɑːˈmɒnɪk ˈməʊʃn/ | 简谐运动 | jiǎn xié yùn dòng |
Pull a swing back and it always returns
- Pull a child's swing to one side and let go — it rushes back, overshoots, and swings the other way.
- The farther you pull it, the harder it is tugged back toward the middle.
- Any motion with this "always pulled back, in proportion" rule is simple harmonic motion 简谐运动 (SHM).
- It is the physics of springs, pendulums, guitar strings and atoms in a solid.
把秋千拉回来,它总是荡回来
- 把孩子的秋千拉向一侧再松手——它急速荡回,冲过头,再荡向另一侧。
- 你拉得越远,它被拉回中间的力就越强。
- 任何具有这种"总是按比例被拉回"规则的运动,就是简谐运动(SHM)。
- 它是弹簧、摆、吉他弦以及固体中原子的物理。
The defining rule
- SHM happens when the restoring force is proportional to the displacement from equilibrium, and points back toward it.
- In symbols: $F = -kx$ — the minus sign means "back toward the middle".
- Push it further out (bigger $x$) and the pull-back force grows in step.
- This single condition produces the smooth back-and-forth of every oscillator.
定义性规则
- 当回复力与相对平衡位置的位移成正比、并指向平衡位置时,就发生 SHM。
- 用符号:$F = -kx$——负号表示"朝中间回"。
- 把它推得更远(更大的 $x$),回拉的力就同步增大。
- 这一个条件就产生了每个振子平滑的来回运动。
Restoring force grows with displacement · 恢复力随位移增大
Stretch the spring and see the restoring force grow in proportion — the heart of SHM. · 拉伸弹簧并观察恢复力按比例增长——这是SHM的核心。
A spring has $k = 50\ \tfrac{\text{N}}{\text{m}}$. What is the size of the restoring force at a displacement of $0.2\ \text{m}$, in $\text{N}$? · 一个弹簧具有$k = 50\ \tfrac{\text{N}}{\text{m}}$。在位移为$0.2\ \text{m}$时的恢复力大小是多少，单位为$\text{N}$？
$|F| = kx = 50 \times 0.2 = 10\ \text{N}$, pointing back toward equilibrium. · $|F| = kx = 50 \times 0.2 = 10\ \text{N}$，指向平衡位置方向。
What defines simple harmonic motion? · 什么定义了简谐运动？
SHM needs $F = -kx$: a restoring force proportional to displacement, pointing back to equilibrium. · SHM需要$F = -kx$：与位移成正比且指向平衡位置的恢复力。
In $F = -kx$, the minus sign means the force points ____ toward equilibrium. · 在$F = -kx$中，负号意味着力指向____平衡位置。
The force always opposes the displacement — it points back · 反向 toward the middle. · 力总是与位移相反——它指向返回中间方向。
Equilibrium and overshoot
- The oscillator has an equilibrium point where the net force is zero.
- Displaced, it is pulled back — but it arrives at equilibrium moving, so it overshoots.
- Then the restoring force slows it, stops it, and pulls it back again.
- The result is an endless, smooth oscillation about the middle.
平衡与冲过头
- 振子有一个平衡点,那里合力为零。
- 被移开后,它被拉回——但它到达平衡时还在运动,所以冲过头。
- 然后回复力使它变慢、停下,再把它拉回来。
- 结果是绕中间无休止、平滑的振动。
Why does an SHM oscillator not simply stop at the equilibrium point? · 为什么SHM振荡器不在平衡点简单地停止？
At equilibrium the force is zero but the speed is greatest, so the oscillator overshoots and swings on. · 在平衡点力为零但速度最大，因此振荡器会冲过并继续摆动。
Where SHM shows up
- A mass on a spring is the textbook example: $F = -kx$ exactly.
- A pendulum swings with SHM too, as long as the angle stays small.
- Atoms vibrating in a crystal, and a plucked string, are SHM as well.
- Whenever a system is nudged from a stable balance, SHM tends to appear.
SHM 出现在哪里
- 弹簧上的质量是教科书例子:精确的 $F = -kx$。
- 摆也做 SHM,只要角度保持很小。
- 晶体中振动的原子、被拨动的弦,也都是 SHM。
- 每当一个系统被从稳定平衡处轻推,SHM 往往就出现。
A pendulum is simple harmonic motion only for small swing angles. · 只有在小摆角时，单摆才是简谐运动。
The restoring force is proportional to displacement only at small angles; large swings are not SHM. · 恢复力仅在小角度时与位移成正比；大幅摆动不是SHM。
Select all · 所有 systems that undergo simple harmonic motion (at least approximately). · 选择所有经历简谐运动（至少近似地）的系统。
Springs, small-angle pendulums and strings all have restoring forces $\propto$ displacement. Constant-speed driving does not. · 弹簧、小角度单摆和弦都具有与$\propto$位移成正比的恢复力。匀速行驶不符合。
A pendulum is only approximately SHM — the restoring force is proportional to displacement only for small angles. Swing it too far and the motion is no longer simple harmonic. The mass-on-a-spring is the cleaner ideal.
摆只是近似的 SHM——回复力只在小角度下才与位移成正比。摆得太远,运动就不再是简谐的。弹簧上的质量是更纯粹的理想。
A spring pulls back with $F = -kx$, where $k = 50\ \tfrac{\text{N}}{\text{m}}$. Find the restoring force at a displacement of $0.2\ \text{m}$.
- $F = -kx = -50 \times 0.2 = -10\ \text{N}$.
The $10\ \text{N}$ force points back toward equilibrium, opposite the displacement.
一根弹簧以 $F = -kx$ 回拉,其中 $k = 50\ \tfrac{\text{N}}{\text{m}}$。求位移 $0.2\ \text{m}$ 处的回复力。
- $F = -kx = -50 \times 0.2 = -10\ \text{N}$。
这 $10\ \text{N}$ 的力指向平衡位置,与位移相反。
Simple harmonic motion is oscillation where the restoring force is proportional to displacement and points back to equilibrium: $F = -kx$. The object overshoots equilibrium and swings endlessly. Springs are exact SHM; pendulums are SHM only for small angles.
简谐运动是这样一种振动:回复力与位移成正比并指向平衡位置:$F = -kx$。物体冲过平衡位置并无休止地摆动。弹簧是精确的 SHM;摆只在小角度下才是 SHM。