Unit 7: Oscillations

Introduction

Simple Harmonic Motion (SHM) can be represented through equations, motion graphs, and physical models. Understanding these representations allows us to analyze oscillatory motion quantitatively and qualitatively.

Key Concepts

• Displacement as a Function of Time: The position of an object in SHM follows: \[ x(t) = A \cos(\omega t + \phi) \] where:
  • A = amplitude (maximum displacement)
  • ω = angular frequency (\(\omega = \sqrt{k/m}\))
  • ϕ = phase constant (determined by initial conditions)
• Velocity and Acceleration: \[ v(t) = -A \omega \sin(\omega t + \phi) \] \[ a(t) = -A \omega^2 \cos(\omega t + \phi) \]
• Phase Relationships:
  • Velocity leads displacement by **π/2 radians (90°)**.
  • Acceleration is **opposite in phase** to displacement.

Mathematical Routines

• Use trigonometric functions to describe motion at any time **t**.
• Analyze graphs of **x vs. t**, **v vs. t**, and **a vs. t** to determine relationships.
• Relate **velocity and acceleration** to **displacement** using derivatives.

Creating Representations

• Graphs: Plot displacement, velocity, and acceleration as functions of time.
• Diagrams: Draw a mass-spring system and pendulum, showing forces at different points.

Scientific Questioning & Argumentation

• Why does velocity reach a maximum when displacement is zero?
• How does acceleration depend on displacement in SHM?

Support your answers with graphs and equations.

Practice Activities

Summary & Exam Preparation Tips

• Displacement, velocity, and acceleration in SHM follow **sinusoidal equations**.
• Acceleration is **opposite in phase** to displacement.
• Graphs provide valuable insights into SHM properties like **amplitude and period**.