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

A pendulum swinging back and forth shows a type of motion that is simple to observe but complex to describe mathematically.The equation governing this system relates angular acceleration to acceleration due to gravity, the length of the pendulum, and the sine of the angle, theta.The inclusion of the sine term makes this a nonlinear differential equation, making it challenging to solve analytically.To resolve this, the sine of theta is expanded using a Maclaurin series, a power series centered at zero. Because theta varies constantly, the expansion replaces the trigonometric function with an infinite sum of powers.When the pendulum swing is small, the higher-order terms&#x2014;such as theta cubed or theta to the fifth&#x2014;become negligible in magnitude, which allows the series to be shortened, approximating the sine of theta as simply theta itself.This substitution converts the difficult nonlinear equation into a solvable, linear second-order differential equation.Such mathematical simplifications are vital for precision in mechanical clocks, because for small angles the pendulum&#x2019;s period depends mainly on its length and gravitational acceleration.

real-world applications of power series

The motion of a simple pendulum is governed by Newton&#x2019;s Second Law in its rotational form, which relates the net torque on the bob to its angular acceleration. This physical law gives rise to a second-order differential equation in which the angular acceleration is proportional to the sine of the displacement angle.Because of the sin(&#x1D703;) term, the governing equation is a nonlinear differential equation, which is difficult to solve analytically. To simplify the mathematical model, the sine function is expanded using a Maclaurin series, which is a specific type of power series centered at zero:\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*}When the pendulum&#x27;s swing is small, the higher-order terms, such as the cube or fifth power of the angle, become negligible in magnitude.This allows for the small-angle approximation, where\begin{equation*}\sin(\theta)\approx\theta\end{equation*}This mathematical manipulation transforms the complex nonlinear equation into a solvable linear second-order differential equation:\begin{equation*}\jfrac{d^2\theta}{dt^2}+\jfrac{g}{L}\,\theta=0\end{equation*}This simplification is critical for the precision of mechanical clocks. For small angles, the period of the pendulum depends primarily on its length (L) and the acceleration due to gravity (g), rather than the amplitude of the swing.In physics and engineering, the study of angular displacement determines how a system handles energy and time. In precision devices like mechanical clocks, angular displacement is intentionally kept small. This ensures that the period of each swing remains constant, a principle known as isochronism. If the angle stays within a narrow range, the clock remains accurate regardless of slight changes in the swing&#x27;s width. In seismic and safety devices, the study of the angle is used to measure force. In a seismometer, the maximum angle reached by the pendulum indicates the intensity of ground vibrations. Similarly, in industrial machinery, monitoring the angular displacement is essential for safety limits. Engineers calculate the maximum allowable angle to ensure the pendulum does not strike the protective casing or exceed the structural strength of the suspension cable.

Real-World Applications of Power Series

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JoVE Core

Calculus

حساب التفاضل والتكامل

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Sequences and Series

المتتاليات والمتسلسلات

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

سلسلة التطبيقات الواقعية للطاقة

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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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

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calculus

Pendulum Motion and the Small-Angle Approximation

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