PHYSICS-301: CLASSICAL MECHANICS

PENDULUM
PRECISION

Deriving the simple harmonic motion of pendulums and the limits of the small-angle approximation.

Lesson 01

Simple Pendulums

Topic

Taylor Series & Torque

The Simple Pendulum Model

Assumptions

Point mass \(m\) at the end of the string.
Massless, inextensible string of length \(L\).
Frictionless pivot; No air resistance.

"The simplest rotational system that exhibits periodic motion, yet its exact solution is surprisingly complex."

Free Body Diagram

Forces: Tension \(T\), Weight \(mg\)

TORQUE & DYNAMICS

\( \sum \tau = I \alpha \)

STEP 1: Identify Torque

The restoring torque is provided by the tangential component of gravity:

\( \tau = -mgL \sin\theta \)

STEP 2: Moment of Inertia

For a point mass \(m\) at distance \(L\) from the pivot:

\( I = mL^2 \)

STEP 3: Equation of Motion

Substitute into \( I\ddot{\theta} = \tau \):

\( mL^2 \frac{d^2\theta}{dt^2} = -mgL \sin\theta \)

Resultant ODE

\[ \frac{d^2\theta}{dt^2} + \frac{g}{L} \sin\theta = 0 \]

This is a non-linear second-order differential equation. It does not have a solution in terms of elementary functions.

Small Angle Approximation

To linearize the equation, we utilize the Taylor Series expansion of \( \sin\theta \) about \( \theta = 0 \):

\( \sin\theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \dots \)

The Assumption:

If \( \theta \ll 1 \) radian, then terms of \( \mathcal{O}(\theta^3) \) and higher are negligible.

\( \sin\theta \approx \theta \)

Error Analysis

Angle (deg)	\( \theta \) (rad)	\( \sin\theta \)	Error %
5°	0.0872	0.0871	0.13%
10°	0.1745	0.1736	0.51%
15°	0.2618	0.2588	1.14%
20°	0.3491	0.3420	2.02%
30°	0.5236	0.5000	4.72%

The "1% rule" typically limits our model to angles under 15°.

THE LINEARIZED SOLUTION

Approximated Equation

\[ \ddot{\theta} + \frac{g}{L} \theta = 0 \]

Angular Frequency

\( \omega = \sqrt{\frac{g}{L}} \)

Period

\( T = 2\pi\sqrt{\frac{L}{g}} \)

Key Insights

1 The period is independent of the mass \(m\).
2 The period is independent of the amplitude (Isochronism).
! Warning: These only hold true while \( \sin\theta \approx \theta \).

Rotational Dynamics and Approximation Theory

Pendulum Precision

Rigid Body Rhythms

Twist and Torque

The Sweet Spot

Beyond Small Angles

Pendulum Precision Slides

Approximation Analysis Worksheet

Pendulum Mastery Teacher Guide

Rigid Body Rhythms Slides

Inertia Impact Worksheet

Rigid Body Rhythms Teacher Guide

Twist and Torque Slides

Torsional Tech Worksheet

Torsional Tech Teacher Guide

The Sweet Spot Slides

The Sweet Spot Worksheet

The Sweet Spot Teacher Guide

Beyond Small Angles Slides

Elliptic Error Worksheet

Beyond Small Angles Teacher Guide

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