Unit 7: Oscillations

Introduction

A simple harmonic oscillator (SHO) continuously converts energy between **kinetic energy (KE)** and **potential energy (PE)**. Understanding this energy transformation is essential for analyzing oscillatory motion and solving AP Physics problems.

Key Concepts

• Energy in SHM: The total mechanical energy of a simple harmonic oscillator remains constant (assuming no energy loss).
• Potential Energy (Elastic Energy): Stored in the system when displaced from equilibrium: \[ PE = \frac{1}{2} k x^2 \] where **k** is the spring constant and **x** is displacement.
• Kinetic Energy: Energy of motion: \[ KE = \frac{1}{2} m v^2 \]
• Total Energy in SHM: \[ E_{\text{total}} = KE + PE = \frac{1}{2} k A^2 \] The total energy depends only on the **amplitude (A)** and **spring constant (k)**.
• Energy Distribution:
  • At maximum displacement (\(|x| = A\)): **PE is maximum, KE is zero**.
  • At equilibrium (\(x = 0\)): **KE is maximum, PE is zero**.

Mathematical Routines

• Use **PE + KE = constant** to analyze motion at different positions.
• Find velocity using: \[ v = \pm \sqrt{\frac{k}{m} (A^2 - x^2)} \]
• Recognize how **energy graphs** relate to motion graphs.

Creating Representations

• Graphs: Plot **KE, PE, and total energy vs. time** or **vs. displacement**.
• Diagrams: Illustrate where PE and KE are maximized in a **mass-spring system**.

Scientific Questioning & Argumentation

• Why does increasing amplitude increase total energy in SHM?
• How does the energy transformation in SHM compare to a free-fall motion?

Support your answers using energy equations and graphs.

Practice Activities

Summary & Exam Preparation Tips

• SHM energy continuously converts between **KE and PE**, with total energy remaining constant.
• At **maximum displacement**, PE is maximum, KE is zero.
• At **equilibrium**, KE is maximum, PE is zero.
• Use **E = ½ kA²** to find total energy.