📚 Analysing Displacement-Time Graphs for Simple Harmonic Motion | A-Level 物理：简谐运动的位移-时间图像分析
In A-Level Physics, the displacement-time (x-t) graph is one of the most direct ways to describe simple harmonic motion (SHM). It shows how the position of an oscillator changes with time and encodes information about amplitude, period, phase, velocity and acceleration.
在 A-Level 物理中，位移-时间（x-t）图像是描述简谐运动（SHM）最直接的方式之一。它展示振动物体的位置如何随时间变化，并包含振幅、周期、相位、速度和加速度等信息。
1. Definition and Characteristics of SHM | 简谐运动的定义与特征
An object performs simple harmonic motion when its acceleration is proportional to its displacement from a fixed equilibrium position and is always directed towards that position.
当物体的加速度与其相对固定平衡位置的位移成正比，且方向始终指向平衡位置时，物体做简谐运动。
a = -ω²x
Here, a is the acceleration, x is the displacement and ω is the angular frequency.
其中 a 是加速度，x 是位移，ω 是角频率。
This linear restoring force leads to sinusoidal displacement-time graphs.
这种线性回复力使位移-时间图呈现正弦曲线。
2. General Form of the Displacement-Time Graph | 位移-时间图像的一般形式
The displacement x of an oscillator can be written as:
振子的位移 x 可写为：
x = A cos(ωt + φ₀)
where A is the amplitude, ω = 2π/T is the angular frequency, and φ₀ is the initial phase.
其中 A 为振幅，ω = 2π/T 为角频率，φ₀ 为初相位。
ph, the curve is a cosine (or sine) wave whose peaks and troughs are at x = +A and x = −A.
在 x-t 图像上，曲线为余弦（或正弦）波，波峰和波谷分别位于 x = +A 和 x = −A。
m positive displacement, φ₀ = 0; if it starts from equilibrium moving positive, φ₀ = −π/2.
若从最大正位移开始运动，则 φ₀ = 0；若从平衡位置向正方向运动开始，则 φ₀ = −π/2。
3. Reading Amplitude and Period from the Graph | 从图像读取振幅与周期
Amplitude: A is the maximum distance from the equilibrium position. Measure the vertical distance from the centre line to a peak or trough.
振幅：A 是距平衡位置的最大距离。测量中心线到波峰或波谷的垂直距离即可。
Period: T is the time for one complete cycle. Measure the time between two successive peaks, two successive troughs, or any two successive points with the same displacement and same velocity direction.
周期：T 是完成一次全振动所需的时间。测量连续两个波峰、连续两个波谷，或任意两个位移相同且速度方向也相同的相邻点之间的时间。
Once T is found, the frequency and angular frequency follow:
求出 T 后，频率和角频率为：
f = 1/T, ω = 2π/T = 2πf
Use at least 10 cycles to reduce timing errors in experiments.
实验中至少测量 10 个周期以减小计时误差。
4. Initial Phase and the t = 0 Intercept | 初相位与 t = 0 截距
The value of x at t = 0 is x₀ = A cos φ₀.
t = 0 时的位移为 x₀ = A cos φ₀。
Because cos φ₀ can range from −1 to 1, x₀ tells you how far from equilibrium the oscillator starts, but not the direction of motion.
由于 cos φ₀ 的取值范围为 −1 到 1，x₀ 只能告诉你振子的起始位置离平衡点有多远，并不能确定运动方向。
To determine φ₀ uniquely, also look at the initial slope of the x-t graph: a positive slope means the oscillator is moving in the positive direction.
要唯一确定 φ₀，还需观察 x-t 图在 t = 0 处的斜率：斜率为正则说明振子向正方向运动。
₀ = 0 and the slope is positive, then φ₀ = −π/2; if the slope is negative, then φ₀ = +π/2.
例如，若 x₀ = 0 且斜率为正，则 φ₀ = −π/2；若斜率为负，则 φ₀ = +π/2。
5. Reference Circle and the Origin of the Sine Curve | 参考圆与正弦曲线的来源
SHM can be treated as the projection of uniform circular motion onto a diameter.
简谐运动可以看作匀速圆周运动在某条直径上的投影。
tant angular speed ω. Its projection along the x-axis has displacement x = A cos(ωt + φ₀).
想象一个质点半径为 A 的圆上以恒定角速度 ω 运动，它沿 x 轴的投影位移为 x = A cos(ωt + φ₀)。
This model explains why the x-t graph has a trigonometric shape: the x-coordinate of a point rotating uniformly varies cosinusoidally with time.
该模型解释了 x-t 图为何呈三角函数形状：做匀速旋转的点的 x 坐标随时间按余弦规律变化。
rs when the particle passes through the centre (x = 0), and the speed is zero at the ends.
它也说明，当质点经过中心（x = 0）时速率最大，而在两端速率为零。
6. Velocity from the Gradient of
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