Representing and Analyzing SHM · 简谐运动的表示与分析
SHM draws a perfect sine wave in time
- Track an oscillating mass and plot its position against time — you get a smooth, repeating wave.
- Not a jagged zig-zag, but the graceful curve of a sine or cosine.
- That shape is the fingerprint of simple harmonic motion, and it lets us predict every instant.
- Reading it tells you position, velocity and acceleration at a glance.
SHM 随时间画出完美的正弦波
- 追踪一个振动的质量,把它的位置对时间作图——你得到一条平滑、重复的波。
- 不是锯齿状的曲折,而是正弦或余弦优雅的曲线。
- 那个形状是简谐运动的指纹,它让我们能预测每一瞬间。
- 读它就能一眼看出位置、速度和加速度。
The equation of motion
- Starting from the maximum, position follows $x(t) = A\cos(\omega t)$.
- $A$ is the amplitude (the maximum displacement) and $\omega = 2\pi f$ is the angular frequency.
- After one period $T$ the curve repeats exactly.
- Feed in a time and the formula gives the exact position.
运动方程
- 从最大处开始,位置遵循 $x(t) = A\cos(\omega t)$。
- $A$ 是振幅(最大位移),$\omega = 2\pi f$ 是角频率。
- 经过一个周期 $T$,曲线精确重复。
- 代入一个时间,公式就给出精确位置。
Position around a cycle · 振动周期中的位置
Step through the phase of the oscillation and see how position and energy vary across a cycle. · 逐步展示振动的相位，观察一个周期内位置和能量的变化。
An oscillator is $x(t) = 0.1\cos(4t)$ (metres). What is its position at $t = 0$, in metres? · 某振子的振幅为$x(t) = 0.1\cos(4t)$（米）。它在时刻$t = 0$时的位置是多少米？
$x(0) = 0.1\cos 0 = 0.1\ \text{m}$ — the maximum displacement (amplitude). · $x(0) = 0.1\cos 0 = 0.1\ \text{m}$——最大位移（振幅）。
In $x(t) = A\cos(\omega t)$, the symbol $A$ is the . · 在公式$x(t) = A\cos(\omega t)$中，符号$A$代表。
$A$ is the amplitude — the maximum displacement from equilibrium. · $A$是振幅——即偏离平衡位置的最大位移。
Where it's fastest and where it accelerates
- Speed is greatest at the middle (equilibrium) and zero at the ends (turning points).
- Acceleration is greatest at the ends and zero at the middle.
- This is because acceleration follows the force: $a = -\dfrac{k}{m}x$, biggest where $x$ is biggest.
- So velocity and acceleration are "out of step" — one peaks where the other is zero.
哪里最快、哪里加速最大
- 速率在中间(平衡位置)最大,在两端(折返点)为零。
- 加速度在两端最大,在中间为零。
- 这是因为加速度跟随力:$a = -\dfrac{k}{m}x$,在 $x$ 最大处最大。
- 所以速度和加速度"错开步调"——一个的峰值在另一个为零处。
Where is an SHM oscillator moving fastest? · 简谐运动振子在何处速度最快？
Speed is greatest at the middle, where the acceleration (and force) is zero. · 速度在中间处最大，此时加速度（和力）为零。
Where is the acceleration of an SHM oscillator greatest? · 简谐运动振子在何处加速度最大？
$a = -\tfrac{k}{m}x$ is largest where $x$ is largest — at the extremes. · $a = -\tfrac{k}{m}x$在$x$最大处达到最大——即在极值点。
A mass starts at the far right. Order what happens next, first at the top. · 一物体从最右端开始运动。按顺序排列接下来发生的情况，首先列出顶部选项。
It accelerates toward the middle (max speed there), then decelerates to a stop at the other extreme. · 它向中间加速（在中间速度最大），然后减速至另一极值点停止。
Position, velocity, acceleration
- If position is a cosine, velocity is a (negative) sine, and acceleration is a (negative) cosine.
- Each is shifted a quarter-cycle from the last.
- The acceleration is always opposite the displacement — that is the SHM condition again.
- Together the three graphs tell the whole story of the motion.
位置、速度、加速度
- 如果位置是余弦,速度就是(负的)正弦,加速度就是(负的)余弦。
- 每一个都比前一个错开四分之一周期。
- 加速度总是与位移相反——这又是 SHM 的条件。
- 三张图合起来讲述了运动的全部故事。
In SHM the acceleration always points opposite to the displacement. · 在简谐运动中，加速度方向始终与位移方向相反。
$a = -\tfrac{k}{m}x$: the minus sign means acceleration opposes displacement. · $a = -\tfrac{k}{m}x$：负号表示加速度与位移方向相反。
In SHM, velocity and acceleration do not peak at the same place. Velocity is largest at the centre (where acceleration is zero); acceleration is largest at the extremes (where velocity is zero). Mixing these up is a classic error.
在 SHM 中,速度和加速度不在同一处达到峰值。速度在中心最大(那里加速度为零);加速度在两端最大(那里速度为零)。把它们弄混是一个经典错误。
An oscillator has amplitude $A = 0.1\ \text{m}$ and angular frequency $\omega = 4\ \tfrac{\text{rad}}{\text{s}}$, starting at the maximum.
- Position: $x(t) = 0.1\cos(4t)$.
- At $t = 0$: $x = 0.1\cos 0 = 0.1\ \text{m}$ (the maximum), where the speed is zero.
一个振子振幅 $A = 0.1\ \text{m}$、角频率 $\omega = 4\ \tfrac{\text{rad}}{\text{s}}$,从最大处开始。
- 位置:$x(t) = 0.1\cos(4t)$。
- 在 $t = 0$:$x = 0.1\cos 0 = 0.1\ \text{m}$(最大值),此处速率为零。
SHM position follows $x(t) = A\cos(\omega t)$ — a sinusoid of amplitude $A$ and angular frequency $\omega = 2\pi f$. Speed peaks at the centre, acceleration peaks at the extremes, and acceleration always points opposite the displacement.
SHM 位置遵循 $x(t) = A\cos(\omega t)$——一条振幅 $A$、角频率 $\omega = 2\pi f$ 的正弦曲线。速率在中心最大,加速度在两端最大,而加速度总是指向与位移相反的方向。