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# Chapter 16: Periodic Motion

Any motion that repeats (think of a position vs. time graph), no matter how complex, is called periodic. This type of motion is important to study since many natural systems are periodic.

When the cause of motion is a linear restoring force, the periodic motion is particularly simple and is called simple harmonic motion. This motion has the remarkable property that the period of oscillation is independent of the amplitude of the motion.

Complicated periodic motion is rather remarkable as well, but for a different reason. Complicated periodic motion can always be described in terms of a sum of sines and/or cosines. This is Fourier's theorem.

# Table of Contents

## Illustrations

- Illustration 16.1: Representations of Simple Harmonic Motion.
- Illustration 16.2: The Simple Pendulum and Spring Motion.
- Illustration 16.3: Energy and Simple Harmonic Motion.
- Illustration 16.4: Damped and Forced Motion.
- Illustration 16.5: Fourier Series 1, Qualitative Features.
- Illustration 16.6: Fourier Series 2, Quantitative Features.

## Explorations

- Exploration 16.1: Spring and Pendulum Motion.
- Exploration 16.2: Pendulum Motion and Energy.
- Exploration 16.3: Simple Harmonic Motion With and Without Damping.
- Exploration 16.4: Pendulum Motion, Forces, and Phase Space.
- Exploration 16.5: Driven Motion and Resonance.
- Exploration 16.6: Damped and Forced Motion.
- Exploration 16.7: A Chain of Oscillators.

## Problems

- Problem 16.1: Hooke's law and simple harmonic motion.
- Problem 16.2: A ball attached to a spring: position vs. time graph.
- Problem 16.3: A ball attached to a spring: velocity vs. time graph.
- Problem 16.4: A ball attached to a spring: simple harmonic motion.
- Problem 16.5: Determine the spring constant of the spring.
- Problem 16.6: Determine several properties of the mass-spring system.
- Problem 16.7: Which graph properly denotes position versus time?
- Problem 16.8: What is the maximum speed of the hanging mass?
- Problem 16.9: Which graph properly shows the position/velocity/acceleration?
- Problem 16.10: Determine the effective acceleration due to gravity.
- Problem 16.11: Dig a hole through the center of the Earth.
- Problem 16.12: A block floating in water is displaced from equilibrium.