Unit 7: Oscillations

Introduction

The frequency and period of simple harmonic motion (SHM) describe how quickly an oscillating system completes one full cycle. These properties are determined by the system’s physical parameters, such as mass and spring constant for a mass-spring system, or length and gravity for a pendulum.

Key Concepts

• Period (T): The time required for one full oscillation. Measured in seconds (s).
• Frequency (f): The number of oscillations per second. Measured in Hertz (Hz).
• Relationship Between Period and Frequency: \[ f = \frac{1}{T} \]
• Angular Frequency (ω): Describes how quickly the system oscillates in radians per second: \[ \omega = 2\pi f = \frac{2\pi}{T} \]

Mathematical Routines

Different types of oscillators have unique equations for period:

• For a **mass-spring system**: \[ T = 2\pi \sqrt{\frac{m}{k}} \]
• For a **simple pendulum** (small angles): \[ T = 2\pi \sqrt{\frac{L}{g}} \]

Key steps for solving problems:

• Identify the type of oscillator (mass-spring or pendulum).
• Use the appropriate formula to calculate **T** or **f**.
• Convert between period, frequency, and angular frequency as needed.

Creating Representations

• Graphs: Sketch **position vs. time** for a simple harmonic oscillator to visualize the period.
• Diagrams: Label forces acting on a pendulum and a mass-spring system at different points in their motion.

Scientific Questioning & Argumentation

• Why does increasing mass **increase** the period of a mass-spring system but **not** affect a pendulum’s period?
• How does the restoring force affect the frequency of oscillation?

Support your reasoning using equations and graphical representations.

Practice Activities

Summary & Exam Preparation Tips

• Use **T = 2π√(m/k)** for a **mass-spring system** and **T = 2π√(L/g)** for a **pendulum**.
• Period and frequency are **inversely related**.
• Angular frequency **ω = 2πf** links linear and angular oscillations.
• For pendulums, **mass does not affect the period**—only length and gravity do.