Name: pendulum motion and the small-angle approximation
Uploaded: 2026-06-08T09:38:33.041Z
Duration: 1 min 27 s
Channel: JoVE

来回摆动的单摆展示了一种易于观察但用数学描述较为复杂的运动形式。描述该系统的方程将角加速度与重力加速度、单摆长度以及角度 theta 的正弦值关联起来。正弦项的引入使得该方程成为非线性微分方程，难以进行解析求解。为解决此问题，将 theta 的正弦函数用麦克劳林级数展开，即在零点处展开的幂级数。由于 theta 持续变化，该展开式用无穷幂级数之和替代了三角函数。当单摆摆角较小时，高阶项（如 θ 的三次方或五次方）的量值变得可忽略不计，因此可将级数截断，将 sin θ 近似为 θ 本身。这种代换将难以求解的非线性方程转化为可求解的线性二阶微分方程。此类数学简化对机械钟的精确性至关重要，因为在小角度下，摆的周期主要取决于其长度和重力加速度。

幂级数的实际应用

Real-World Applications of Power Series

单摆的运动由牛顿第二定律的转动形式所决定，该物理定律将摆锤所受的合外力矩与其角加速度相关联。此物理定律导出一个二阶微分方程，其中角加速度与位移角的正弦值成正比。由于含有 sin(𝜃) 项，控制方程是一个非线性微分方程，难以进行解析求解。为了简化数学模型，使用麦克劳林级数（一种在零点展开的特殊幂级数）对正弦函数进行展开：\begin{equation*}\sin(\theta) = \Sigma_{n=0}^{\infty} (-1)^n \frac{\theta^{2n+1}}{(2n + 1)!} = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \dots\end{equation*}当摆的摆动角度较小时，角度的高次项（如立方项或五次方项）在量值上变得可以忽略不计。这允许采用小角度近似，其中\begin{equation*}\sin(\theta)\approx\theta\end{equation*}这种数学处理将复杂的非线性方程转化为可求解的线性二阶微分方程：\begin{equation*}\jfrac{d^2\theta}{dt^2}+\jfrac{g}{L}\,\theta=0\end{equation*}这种简化对于机械钟的精确性至关重要。在小角度情况下，摆的周期主要取决于其长度（L）和重力加速度（g），而非摆动的幅度。在物理学和工程学中，角位移的研究用于确定系统如何响应能量和时间的变化。在机械钟等精密装置中，角位移被有意保持在较小范围内，以确保每次摆动的周期恒定，这一原理称为等时性。只要摆动角度保持在狭窄范围内，即使摆幅有轻微变化，钟表仍能保持准确。在地震监测和安全装置中，角度研究被用于测量受力情况。在地震仪中，摆锤达到的最大角度反映了地面振动的强度。类似地，在工业机械中，监测角位移对于确定安全限值至关重要。工程师会计算允许的最大角度，以确保摆锤不会撞击保护外壳，也不会超过悬挂缆线的结构强度。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Sequences and Series

数列与级数

KEY TERMS AND CONCEPTS

关键术语与概念

Power 系列的实际应用

pendulum motion and the small-angle approximation

Pendulum motion can be simplified with the small-angle approximation and a power series. A simple pendulum follows Newton’s Second Law in rotational form, which links the net torque on the bob to its angular acceleration. That relationship leads to a second-order differential equation. The equation is nonlinear because the angular acceleration depends on the sine of the displacement angle.

To make the model easier to solve, the sine function is expanded with a Maclaurin series. A Maclaurin series is a power series centered at zero. For sine, the expansion is sin(θ) = θ - θ^3/3! + θ^5/5! - ... . When the swing angle is small, the higher-order terms become very small. In that case, the sine function can be approximated by sin(θ) ≈ θ.

This approximation turns the nonlinear pendulum equation into a linear second-order differential equation. The simplified form is d^2θ/dt^2 + (g/L)θ = 0. That result is important in precision mechanical clocks. For small angles, the pendulum period depends mainly on the length L and the acceleration due to gravity g, not on the swing amplitude.

The angle of swing also matters in real devices that use pendulums to measure motion or set safety limits. In a mechanical clock, the angle is kept small so each swing takes the same amount of time. This is called isochronism. If the angle stays within a narrow range, the clock stays accurate even when the swing width changes slightly.

Angular displacement is also useful in seismometers and industrial machinery. In a seismometer, the maximum angle reached by the pendulum shows the intensity of ground vibrations. In industrial systems, engineers monitor the angle to stay within safe limits. They calculate the maximum allowable angle so the pendulum does not hit the protective casing or exceed the strength of the suspension cable.

Pendulum Motion and the Small-Angle Approximation

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real-world applications of power series

Pendulum motion can be simplified with the small-angle approximation and a power series. A simple pendulum follows Newton’s Second Law in rotational form, which links the net torque on the bob to its angular acceleration. That relationship leads to a second-order differential equation. The equation is nonlinear because the angular acceleration depends on the sine of the displacement angle. To make the model easier to solve, the sine function is expanded with a Maclaurin series. A Maclaurin...

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See how a pendulum becomes easier to model with a Maclaurin series, the small-angle approximation, and a linear equation used in clocks and sensors.

Pendulum Motion and the Small-Angle Approximation | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Pendulum Motion and the Small-Angle Approximation

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