Unit 7: Oscillations

Introduction

Simple Harmonic Motion (SHM) describes periodic motion where a restoring force is proportional to displacement. Many physical systems, such as springs, pendulums, and waves, exhibit SHM. Understanding SHM is fundamental to analyzing oscillatory motion in physics.

Key Concepts

• Periodic Motion: Motion that repeats at regular time intervals.
• Restoring Force: The force that pulls an object back toward equilibrium (e.g., Hooke’s Law for springs: \( F = -kx \)).
• Conditions for SHM:
  • The restoring force is proportional to displacement.
  • The force is always directed toward equilibrium.
• Equation of Motion: The acceleration of an object in SHM follows: \[ a = -\frac{k}{m} x \] where k is the spring constant and m is mass.

Mathematical Routines

• The displacement function for SHM is: \[ x(t) = A \cos(\omega t + \phi) \] where:
  • A = amplitude (maximum displacement)
  • ω = angular frequency (\(\omega = \sqrt{k/m}\))
  • ϕ = phase constant
• Velocity and acceleration functions: \[ v(t) = -A \omega \sin(\omega t + \phi) \] \[ a(t) = -A \omega^2 \cos(\omega t + \phi) \]

Creating Representations

• Diagrams: Draw force diagrams for a mass-spring system at different points in its motion.
• Graphs: Sketch **x vs. t**, **v vs. t**, and **a vs. t** to see how these values change over time.

Scientific Questioning & Argumentation

• How does the motion of a pendulum compare to that of a mass on a spring?
• Why is acceleration always directed toward the equilibrium position in SHM?

Support your reasoning using equations and graphical representations.

Practice Activities

Summary & Exam Preparation Tips

• SHM occurs when the restoring force is proportional to displacement.
• Displacement, velocity, and acceleration are sinusoidal functions of time.
• Use **F = -kx** to analyze SHM in springs and relate motion to energy and forces.