Simple Harmonic Motion (SHM) Lecture Notes

Introduction

• Topic: Simple Harmonic Motion (SHM)
• Previous Lesson Recap: Reviewed Springs, sine and cosine functions
• Goals: Introduction to equations for position, velocity, and acceleration of SHM. Discussion on frequency, period, amplitude, and frequency factor.

Key Concepts in SHM

Oscillation Parameters

• Frequency of Oscillation: How often the oscillation repeats per unit time (measured in Hertz).
• Period of Oscillation: Time taken for one complete cycle of oscillation.
• Amplitude of Oscillation: Maximum displacement from the equilibrium position.
• Frequency Factor (Omega, ω): Related to frequency and period.

Simple Harmonic Motion in Springs

Review of Springs (Previous Lesson)

• Hook’s Law: F = -kx
• Elastic Potential Energy: U = 1/2 k x^2
• Ideal Spring: No internal friction or non-conservative forces.

SHM Equations for Springs

• Position Function: x(t) = A cos(ωt)
  • A: Amplitude
  • ω: Frequency factor (angular frequency)
• Velocity Function: v(t) = -Aω sin(ωt)
• Acceleration Function: a(t) = -Aω² cos(ωt)

Derivation of SHM Equations

• From Calculus (Conceptual Explanation): Velocity (v) is the derivative of the position (x) function, and acceleration (a) is the derivative of the velocity (v) function.
• Velocity: v = dx/dt = -Aω sin(ωt)
• Acceleration: a = dv/dt = -Aω² cos(ωt)

Peak Values and Amplitudes

• Amplitude of Velocity: Aω
• Amplitude of Acceleration: Aω²

Relationship between Frequency, Period, and Omega

• Period (T): T = 2π / ω
• Frequency (f): f = 1/T = ω / 2π
• Frequency Factor: ω = 2πf

Pendulums and SHM

Pendulums Overview

• Restoring Force: Proportional to sin(θ)
• Approximate SHM: Valid for small angles (θ < 15°)

SHM Equations for Pendulums

• Restoring Force: F = -mg sin(θ)
• Angular Displacement: θ ≈ s / L (for small angles)
• Frequency Factor for Pendulums: ω = √(g/L)

Comparison of Springs and Pendulums

• Springs: ω = √(k/m)
• Pendulums: ω = √(g/L)
• Equations for position, velocity, and acceleration are similarly structured but differ in ω.

Practice Problems

Problem 1: Spring System

• Given: x(t) = 0.20 m cos(4.0 t)
• Amplitude: 0.20 m
• Frequency Factor (ω): 4.0 s⁻¹

Calculations

• Period (T): T = 2π / 4.0 = π/2 seconds
• Frequency (f): f = 1/T = 2 / π Hz
• Max Velocity (v_max): Aω = 0.20 * 4.0 = 0.80 m/s
• Max Acceleration (a_max): Aω² = 0.20 * 16 = 3.2 m/s²
• Max Restoring Force (F_max):
  • Using Newton's Second Law: F = ma = 2.0 kg * 3.2 m/s² = 6.4 N
  • Can also use Hook’s Law: F = -kx, with k derived from ω² = k/m.

Problem 2: Pendulum System

• Given: Length of pendulum = 4 times initial, mass doubled
• Effect on Frequency: Mass doesn’t affect frequency; frequency factor ω is proportional to √(1/L).
• Conclusion: Increasing length by 4 decreases ω (and f) by √4 = 2.

Summary

• Amplitude affects maximum values for position, velocity, and acceleration.
• Frequency Factor (ω) is critical in determining the period and frequency of oscillation.
• Springs and pendulums both exhibit SHM, but their equations differ due to different restoring forces.
• Practice Problems reinforce understanding of key concepts and formula applications in SHM.