Simple Harmonic Motion in Trig (Precalculus - Trigonometry 35)

Overview

Professor Leonard introduces simple harmonic motion, explaining it as vibratory motion that doesn't lessen over time and can be modeled by sinusoidal functions (sine and cosine). He details how to derive displacement functions from given information, focusing on amplitude, period (2π/ω), frequency, and initial conditions. The explanation covers choosing between sine (starting at rest) and cosine (starting away from rest) and interpreting the sign of the amplitude to determine initial motion direction.

Key takeaways

Simple harmonic motion is characterized by constant amplitude oscillations, modeled by sine or cosine functions.
The choice between sine and cosine depends on the initial condition: sine for starting at rest (displacement = 0), cosine for starting away from rest.
The sign of the amplitude determines the initial direction of motion: positive sine/cosine typically means upward/downward initial motion, respectively, while negative sine/cosine reverses this.
The period (T) of oscillation is related to the angular frequency (ω) by T = 2π/ω, and frequency (f) is its reciprocal (f = 1/T).
Vertical shifts in the displacement function (e.g., `6 + ...`) change the resting position but do not affect the amplitude or period of the oscillation.

Chapters

0:00 Introduction to Simple Harmonic Motion

Simple harmonic motion is vibratory motion that does not lessen over time.
It can be modeled using sinusoidal functions like sine and cosine.
Key parameters include displacement (d), amplitude (A), period (T), and frequency (f).

2:21 Understanding Amplitude, Period, and Frequency

Amplitude is the maximum distance from the resting position.
The period is the time for one complete oscillation, calculated as 2π/ω.
Frequency is the reciprocal of the period (1/T), representing oscillations per unit time.

4:05 Choosing Between Sine and Cosine Functions

Use cosine when starting motion away from the resting position.
Use sine when starting motion at the resting position (displacement = 0).
The sign of the amplitude (positive or negative) indicates the initial direction of motion.

6:47 Example 1: Modeling a Pulled-Down Spring

A sign pulled down 7 inches and released with a 4-second period.
Amplitude is 7 inches (initial displacement is -7).
Period of 4 seconds yields ω = 2π/4 = π/2.
The function is modeled as d(t) = -7 cos(π/2 * t).

17:04 Example 2: Modeling a Pushed-Up Spring

An object pushed up 4 feet, with a period of π/2 seconds.
Amplitude is 4 (initial displacement is +4).
Period of π/2 yields ω = 2π/(π/2) = 4.
The function is modeled as d(t) = 4 cos(4t).

26:52 Example 3: Modeling an Object Hit from Rest

An object starting at rest and moving upward, amplitude 4, period π/2.
Starting at rest indicates using a sine function.
Moving upward initially implies a positive sine function.
The function is modeled as d(t) = 4 sin(4t).

31:49 Analyzing Displacement Functions

For d(t) = 5 sin(3t): amplitude is 5, period is 2π/3, frequency is 3/(2π), starts at rest (d(0)=0), initial motion is upward.
For d(t) = -3 sin(1/2 * t): amplitude is 3, period is 4π, frequency is 1/(4π), starts at rest (d(0)=0), initial motion is downward.
For d(t) = -2 cos(2t): amplitude is 2, period is π, frequency is 1/π, starts at -2 (away from rest), initial motion is upward.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Leonard.