4. The Pendulum Equation | 摆方程

📚 4. The Pendulum Equation | 摆方程

The pendulum is one of the most iconic systems in physics and applied mathematics. Its motion, governed by a second-order differential equation, provides a rich context for exploring trigonometric functions, small-angle approximations, and the nature of oscillatory solutions. In IB Mathematics, the pendulum equation bridges algebra, calculus, and modelling, making it a powerful revision topic.

摆是物理学和应用数学中最具标志性的系统之一。其运动由一个二阶微分方程控制，为探索三角函数、小角度近似以及振荡解的本质提供了丰富的背景。在 IB 数学中，摆方程将代数、微积分与建模连接起来，使其成为一个极具价值的复习主题。

1. The Simple Pendulum Model | 单摆模型

A simple pendulum consists of a point mass m suspended from a light, inextensible string of length L, swinging under gravity. The displacement from the vertical is measured by the angle θ (in radians). We assume no air resistance and that the string remains taut throughout the motion.

单摆模型由一个挂在轻质、不可伸长的长度为 L 的细绳上的质点 m 组成，在重力作用下摆动。偏离竖直方向的位移用角度 θ（以弧度为单位）量度。我们假设没有空气阻力，且绳子在整个运动过程中始终紧绷。

2. Forces and Torque Analysis | 力与力矩分析

The weight mg has a tangential component −mg sin θ that acts to restore the bob towards equilibrium. Taking the moment about the pivot, the restoring torque is −mgL sin θ. By Newton’s second law for rotation, the torque equals moment of inertia I times angular acceleration. For a point mass, I = mL².

重力 mg 的切向分量为 −mg sin θ，它将摆锤拉回平衡位置。对悬点取矩，回复力矩为 −mgL sin θ。根据转动形式的牛顿第二定律，力矩等于转动惯量 I 乘以角加速度。对于质点，I = mL²。

The resulting equation is:

由此得到方程：

mL² · d²θ/dt² = −mgL sin θ

Cancelling common factors and simplifying gives the fundamental equation of motion.

约去公因子并化简，就得到了基本的运动方程。

3. The Nonlinear Pendulum Equation | 非线性摆方程

After simplification, we obtain the exact pendulum equation:

化简后，我们得到精确的摆方程：

d²θ/dt² + (g/L) sin θ = 0

This is a second-order nonlinear ordinary differential equation because of the sine term. Unlike linear ODEs, it cannot be solved using simple characteristic equations. Its solutions involve elliptic functions for arbitrary amplitudes.

由于含有正弦项，这是一个二阶非线性常微分方程。与线性常微分方程不同，它不能用简单的特征方程求解。对于任意振幅，其解涉及椭圆函数。

4. Small-Angle Approximation | 小角度近似

When the maximum angular displacement is small (typically θ₀ < 0.2 rad or about 11°), we can use the approximation sin θ ≈ θ. This linearises the pendulum equation, making it solvable with elementary functions.

当最大角位移很小时（通常 θ₀ < 0.2 弧度，约 11°），我们可以使用近似 sin θ ≈ θ。这将摆方程线性化，可以用初等函数求解。

Applying this approximation yields the simple harmonic oscillator equation:

应用该近似，得到简谐振子方程：

d²θ/dt² + (g/L) θ = 0

Comparing with standard form d²x/dt² + ω²x = 0, we identify the angular frequency ω = √(g/L). This linearised model is the foundation of most IB pendulum problems.

与标准形式 d²x/dt² + ω²x = 0 比较，可知角频率 ω = √(g/L)。这个线性化模型是大多数 IB 摆问题的基础。

5. General Solution for Small Oscillations | 小振荡的通解

The linearised equation is homogeneous with constant coefficients. Its auxiliary equation is m² + ω² = 0, giving roots m = ±iω. The general solution is a combination of sine and cosine functions:

线性化后的方程是常系数齐次微分方程。其辅助方程为 m² + ω² = 0，根为 m = ±iω。通解是正弦与余弦函数的组合：

θ(t) = A cos(ωt) + B sin(ωt)

Alternatively, this can be expressed in amplitude-phase form θ(t) = θ₀ cos(ωt + φ) or θ₀ sin(ωt + δ), where θ₀ is the amplitude and φ is the phase constant, determined by initial conditions.

或者，这可以表示为振幅-相位形式 θ(t) = θ₀ cos(ωt + φ) 或 θ₀ sin(ωt + δ)，其中 θ₀ 为振幅，φ 为相位常数，由初始条件确定。

6. Period of a Simple Pendulum | 单摆的周期

Using ω = √(g/L) and the relation T = 2π/ω, the period of small oscillations is:

利用 ω = √(g/L) 以及关系式 T = 2π/ω，小角度振荡的周期为：

T = 2π √(L/g)

This famous result is independent of the mass and amplitude (for small angles). In IB problems, you may be asked to derive this from the differential equation or to use it in experimental data analysis where a graph of T² against L gives a straight line of slope 4π²/g.

这一著名公式与质量和（小角度下）振幅均无关。在 IB 试题中，你可能需要从微分方程推导出该式，或在实验数据分析中使用它：画出 T²-L 图，应得到一条斜率为 4π²/g 的直线。

7. Energy in the Pendulum System | 摆系统的能量

The pendulum offers a clear illustration of energy conservation. Choosing the lowest point as reference, the total mechanical energy E = ½ mL² (dθ/dt)² + mgL(1 − cos θ). For small angles, 1 − cos θ ≈ θ²/2, and the energy simplifies to that of a simple harmonic oscillator.

摆清晰地展示了能量守恒。若以最低点为参考点，总机械能 E = ½ mL² (dθ/dt)² + mgL(1 − cos θ)。对于小角度，1 − cos θ ≈ θ²/2，能量简化为简谐振子的能量形式。

By differentiating the energy expression with respect to time and setting dE/dt = 0, one can re-derive the pendulum equation — an exercise often seen in IB Higher Level calculus tasks.

对能量表达式求时间导数，并令 dE/dt = 0，可以重新推导出摆方程——这是 IB 高等级微积分题目中常见的练习。

8. Large-Angle Oscillations and Corrections | 大角度振荡与修正

When the small-angle approximation fails, the period depends on amplitude. The exact period is given by an elliptic integral or, more commonly, by a series expansion:

当小角度近似不成立时，周期依赖于振幅。精确周期由椭圆积分给出，更常见的是用级数展开表示：

T ≈ 2π √(L/g) [1 + (1/16)θ₀² + (11/3072)θ₀⁴ + … ]

This shows that the period increases with amplitude. IB explorations may involve verifying this approximation with simulations or comparing it against data collected from a physical pendulum.

这表明周期随振幅增大而增大。IB 探索作业可能涉及通过模拟验证该近似，或将其与从实际摆收集的数据进行对比。

9. Damped and Driven Pendulum Extensions | 阻尼摆与驱动摆的扩展

In more realistic models, a damping term proportional to angular velocity (−b dθ/dt) and a periodic driving force F₀ cos(ω_d t) may be included. The equation becomes:

在更现实的模型中，可以加上与角速度成正比的阻尼项 (−b dθ/dt) 和周期性驱动力 F₀ cos(ω_d t)。方程变为：

d²θ/dt² + β dθ/dt + (g/L) sin θ = F₀ cos(ω_d t)

Although the full nonlinear driven pendulum exhibits chaotic behaviour, the small-angle damped case leads to standard second-order linear ODEs with particular and complementary functions — a topic firmly within the IB HL calculus option.

虽然完整的非线性驱动摆会表现出混沌行为，但小角度阻尼情况可化为标准的二阶线性常微分方程，具有特解和补解——这无疑是 IB 高等级微积分选修中的内容。

10. IB Exam Strategies and Modelling | IB 考试策略与建模

In IB Mathematics: Analysis and Approaches and Applications and Interpretation, pendulum questions may appear in calculus papers, exploration prompts, or modelling tasks. Key steps include: setting up the differential equation from torque or energy, applying sin θ ≈ θ, solving to find θ(t), and interpreting constants.

在 IB 数学：分析与方法和应用与解释中，摆的问题可能出现在微积分试卷、探索课题提示或建模任务中。关键步骤包括：从力矩或能量出发建立微分方程，应用 sin θ ≈ θ，求解得到 θ(t)，并对常数做出解释。

Always remember to state assumptions (small angle, no damping, point mass) and use radian measure when differentiating trigonometric functions. Linking calculus results with physical behaviour is highly rewarded in IB assessments.

务必记住陈述假设（小角度、无阻尼、质点），并在对三角函数求导时使用弧度制。将微积分结果与物理行为联系起来，在 IB 评估中会得到很高的评价。

11. Connecting to the Real World | 联系实际世界

Pendulums are not just theoretical; they appear in clocks, seismometers, and even in the measurement of gravitational acceleration g. By timing oscillations and using the period formula, students can estimate g to a surprising level of accuracy, reinforcing the link between pure mathematics and experimental science.

摆不仅是理论上的，它出现在钟表、地震仪，甚至用于测量重力加速度 g 中。通过计时振荡并运用周期公式，学生可以以惊人的精度估算 g，从而强化纯数学与实验科学之间的联系。

12. Summary of the Pendulum Equation Journey | 摆方程之旅总结

Beginning from Newton’s laws, we derived d²θ/dt² + (g/L) sin θ = 0. With small angles this becomes a linear ODE solved by θ(t) = θ₀ cos(ωt + φ) with period T = 2π√(L/g). The analysis extends naturally to energy methods, series corrections, and damped-driven systems. Mastery of these steps equips you with versatile tools for IB exams and beyond.

从牛顿定律出发，我们推导出 d²θ/dt² + (g/L) sin θ = 0。在小角度下，它化为线性常微分方程，解为 θ(t) = θ₀ cos(ωt + φ)，周期 T = 2π√(L/g)。分析自然延伸到能量方法、级数修正和阻尼驱动系统。掌握这些步骤，你便拥有了应对 IB 考试及更高层次学习的多用途工具。

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