Periodic and Simple Harmonic Motion

Overview

This lecture covers the fundamentals of periodic motion and simple harmonic motion (SHM), describing their characteristics, equations, and energy principles, with real-world examples and practical problem solving.

Periodic Motion Basics

• Periodic motion repeats itself at regular intervals, examples include pendulums, pistons, and musical vibrations.
• Key terms: amplitude (maximum displacement), period (time for one cycle, T), frequency (cycles per second, f), and angular frequency (rate of phase change, ω).
• Frequency f = 1/T, period T = 1/f; SI unit of frequency is hertz (Hz).
• Angular frequency ω = 2πf; unit is radians per second (rad/s).

The Spring-Mass System

• A mass attached to a spring on a frictionless surface exhibits periodic motion when displaced.
• The restoring force by the spring obeys Hooke’s Law: Fₓ = -kx, where k is the spring constant and x is the displacement.
• At equilibrium (x=0), force and acceleration are zero; force is proportional and opposite to displacement elsewhere.

Simple Harmonic Motion (SHM)

• SHM occurs when the restoring force is directly proportional to the displacement and opposite in direction.
• Equation: Fₓ = -kx leads to acceleration aₓ = -k/m x; acceleration is not constant but depends on x.
• Harmonic oscillator: any system undergoing SHM.

Characteristics & Equations of SHM

• Displacement as a function of time: x(t) = A cos(ωt + φ), where φ is the phase angle.
• Velocity: vₓ = -ωA sin(ωt + φ); Acceleration: aₓ = -ω²x.
• Angular frequency ω = √(k/m), frequency f = (1/2π)√(k/m), period T = 2π√(m/k).
• Frequency and period are independent of amplitude.

Graphical Representation of SHM

• Displacement, velocity, and acceleration graphs are sinusoidal and phase-shifted relative to each other.
• Changing mass (m) or spring constant (k) affects frequency and period, not amplitude.
• Changing amplitude (A) affects displacement but not period or frequency.

SHM & Circular Motion Connection

• Projection of uniform circular motion onto one axis is equivalent to SHM.
• The reference circle’s radius is the amplitude; position given by x = A cos(θ), θ = ωt + φ.

Energy in SHM

• Total mechanical energy (E) is conserved: E = Kinetic Energy + Potential (Elastic) Energy.
• Kinetic Energy: (1/2)mvₓ²; Elastic Potential Energy: (1/2)kx².
• At maximum displacement, all energy is potential; at equilibrium, all is kinetic.

Problem Examples

• Calculations included determining period, frequency, amplitude, phase angle, and energy for different spring-mass and collision scenarios using the above formulas.

Key Terms & Definitions

• Amplitude (A) — Maximum displacement from equilibrium.
• Period (T) — Time for one complete cycle.
• Frequency (f) — Number of cycles per second (Hz).
• Angular Frequency (ω) — 2π times the frequency, in rad/s.
• Restoring Force — Force directing the system back to equilibrium, F = -kx.
• Simple Harmonic Motion — Oscillation with restoring force proportional and opposite to displacement.
• Harmonic Oscillator — System exhibiting SHM.
• Phase Angle (φ) — Initial angle, determines starting position in the cycle.
• Elastic Potential Energy — Energy stored in a stretched/compressed spring, (1/2)kx².

Action Items / Next Steps

• Review readings on damped and forced oscillations for next lecture.
• Practice problems calculating SHM properties from given initial conditions.
• Prepare questions about SHM applications and energy conversions.