Defining Simple Harmonic Motion · 定义简谐运动
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| simple harmonic motion/ˈsɪmpl hɑːˈmɒnɪk ˈməʊʃn/ | 简谐运动 | jiǎn xié yùn dòng |
| restoring force/rɪˈstɔːrɪŋ fɔːs/ | 回复力 | huí fù lì |
The back-and-forth that nature loves
- A mass on a spring bobs up and down, over and over.
- A guitar string, a swing, a tuning fork -- all repeat the same dance.
- Pull them aside and something always tugs them back.
- One rule governs every smooth, repeating wobble.
大自然钟爱的来回摆动
- 弹簧上的物块上下起伏,一次又一次。
- 吉他弦、秋千、音叉——都重复着同样的舞蹈。
- 把它们拉到一旁,总有某种东西把它们拉回来。
- 一条规则支配着每一次平滑、重复的晃动。
The restoring rule
- Simple harmonic motion 简谐运动 happens when the restoring force 回复力 is proportional to displacement:
- The minus sign means the force always points back toward equilibrium.
- A bigger pull the farther you go -- that is the signature of SHM.
回复规则
- 当回复力与位移成正比时,就发生简谐运动:
- 负号表示力总是指回平衡位置。
- 走得越远拉力越大——这就是简谐运动的标志。
A spring with $k = 20\ \text{N/m}$ is stretched $x = 0.15\ \text{m}$. The magnitude of the restoring force (in N)? · 弹簧劲度系数为$k = 20\ \text{N/m}$时被拉伸了$x = 0.15\ \text{m}$。回复力的大小（单位：N）是多少？
$|F| = kx = 20 \times 0.15 = 3\ \text{N}$.
A motion is simple harmonic when the restoring force is... · 当回复力满足以下条件时，运动即为简谐运动：...
$F = -kx$ -- proportional to $x$ and pointing back toward equilibrium. · $F = -kx$ -- 与$x$成正比并指向平衡位置。
The minus sign in $F = -kx$ means the force always points back toward equilibrium. · $F = -kx$中的负号表示力始终指向平衡位置。
The force opposes the displacement, restoring the mass toward $x = 0$. · 该力与位移方向相反，将质量拉回$x = 0$。
Acceleration points home
- Dividing by mass, the acceleration obeys:
- Acceleration is proportional to displacement but opposite in direction.
- The constant $\omega^2 = k/m$ sets how quickly it oscillates.
加速度指向家的方向
- 除以质量,加速度满足:
- 加速度与位移成正比,但方向相反。
- 常数 $\omega^2 = k/m$ 决定它振荡得多快。
For SHM with $\omega^2 = 25\ \text{rad}^2/\text{s}^2$ at displacement $x = 0.2\ \text{m}$, the acceleration magnitude (in m/s²)? · 对于位移为 $\omega^2 = 25\ \text{rad}^2/\text{s}^2$ 的简谐运动 (SHM)，加速度大小（单位为 m/s $x = 0.2\ \text{m}$，加速度大小（以 m/s 为单位）²)?
$|a| = \omega^2 x = 25 \times 0.2 = 5\ \text{m/s}^2$.
The usual suspects
- A mass on a spring: $F = -kx$ directly.
- A pendulum at small angles: gravity gives a near-linear pull back.
- Any system with a linear restoring force wobbles as SHM.
常见的例子
- 弹簧上的物块:直接就是 $F = -kx$。
- 小角度的单摆:重力提供近似线性的回拉。
- 任何具有线性回复力的系统都以简谐运动晃动。
The restoring force of SHM · 简谐运动的回复力
Simple harmonic motion needs a restoring force proportional to the displacement and pointing back to the centre. · 简谐运动需要一个与位移成正比且指向中心的回复力。
A $0.5\ \text{kg}$ mass on a spring with $k = 8\ \text{N/m}$ is pulled $0.1\ \text{m}$ and released.
- Restoring force at release: $F = -kx = -8 \times 0.1 = -0.8\ \text{N}$.
- Acceleration: $\omega^2 = k/m = 16$, so $a = -\omega^2 x = -16 \times 0.1 = -1.6\ \text{m/s}^2$.
一个 $0.5\ \text{kg}$ 的物块挂在 $k = 8\ \text{N/m}$ 的弹簧上,被拉 $0.1\ \text{m}$ 后释放。
- 释放时的回复力:$F = -kx = -8 \times 0.1 = -0.8\ \text{N}$。
- 加速度:$\omega^2 = k/m = 16$,所以 $a = -\omega^2 x = -16 \times 0.1 = -1.6\ \text{m/s}^2$。
In SHM, where is the acceleration largest? · 在简谐运动中，加速度在哪里最大？
$a = -\omega^2 x$ is largest where $x$ is largest -- the extremes. · $a = -\omega^2 x$在$x$最大处最大——即极值点。
A pendulum swings as simple harmonic motion only for small angles. · 单摆仅在角度较小时才作简谐运动。
Only at small angles is the restoring pull proportional to displacement. · 只有在小角度下，回复拉力才与位移成正比。
A pendulum is only SHM for small angles (roughly under $15^\circ$); at large angles the pull-back is no longer proportional to displacement. The minus sign is essential -- without it the force would push the mass away and nothing would oscillate. And acceleration is largest at the extremes (biggest $x$) and zero at equilibrium, the opposite of speed.
单摆只有在小角度(大约小于 $15^\circ$)时才是简谐运动;大角度时回拉不再与位移成正比。负号至关重要——没有它,力就会把物块推开,什么也不会振荡。而且加速度在两端最大($x$ 最大)、在平衡位置为零,与速度正好相反。
Simple harmonic motion is the smooth repeat you get when the restoring force is proportional to displacement and points back home: $F = -kx$, so $a = -\omega^2 x$ with $\omega^2 = k/m$. A spring shows it exactly; a small-angle pendulum shows it approximately. Acceleration peaks at the extremes and vanishes at equilibrium.
简谐运动是当回复力与位移成正比并指回原处时得到的平滑重复:$F = -kx$,于是 $a = -\omega^2 x$,其中 $\omega^2 = k/m$。弹簧精确地展示它;小角度单摆近似地展示它。加速度在两端最大,在平衡位置消失。