Simple Harmonic Slides RHYTHM OF THE MACHINE
Modeling Simple Harmonic Motion
The Watcher's Sway
Imagine a hypnotist's pocket watch swinging back and forth.
It starts at its furthest point to the right (5 cm from center).
It takes 2 seconds to swing back to that same spot.
The center point is our midline .
Task: Write the equation for the watch's position over time.
ANATOMY OF A WAVE
Amplitude (A)
The maximum displacement from equilibrium. Half the distance between high and low points.
Period (P)
The time taken for one complete cycle.
\[ B = \frac{2\pi}{\text{Period}} \]
Midline (D)
The vertical center of the motion. The average of the maximum and minimum values.
THE GENERAL FORM
y = A cos(B(t - C)) + D
Why Cosine?
Great for systems that start at a maximum displacement (like releasing a spring or a pendulum).
Why Sine?
Great for systems that start at the midline moving upward.
PRACTICE: THE SPRING
Live Modeling
A mass is attached to a spring hanging from the ceiling. Its resting height is 1.5 meters.
It is pulled down 0.5 meters and released. It takes 3 seconds to return to its lowest point.
Find the Parameters:
Amplitude (A) = ?
Midline (D) = ?
Period (P) = ?
B Value = ?
Midline: 1.5m
Simple Harmonic Worksheet OSCILLATION ANALYSIS
Unit: Applied Trig Modeling | Lesson 1
Student Name
The Goal
Your mission is to convert physical movement into mathematical functions. For every scenario, identify the Midline (D) , the Amplitude (A) , and the Period (P) to construct the general equation:
\( y = A \cos(B(t - C)) + D \)
1
The Heavy Lifter
A 5kg weight is attached to a vertical spring. At rest, the weight hangs 20 cm above the floor. A student pulls the weight down to a height of 5 cm and releases it at \( t = 0 \). It takes exactly 1.2 seconds for the weight to return to its lowest point.
A. Determine Midline (D)
B. Determine Amplitude (A)
C. Determine Period (P) & B-Value
D. Sketch the Graph (1.2 second interval)
E. Write the Equation:
2
The Classic Metronome
A mechanical metronome arm swings from side to side. The tip of the arm moves between a horizontal position of -8 cm and +8 cm. It is set to 120 beats per minute (BPM), where each beat represents the arm reaching one of its maximum side positions.
Amplitude
Period (sec)
B Value
Construct the model assuming the arm starts at the far right (+8 cm) at \( t = 0 \):
Critical Thinking: If we speed up the metronome to 240 BPM, which variable in the equation changes? How does it change?
3
The Ocean Buoy
A navigation buoy bobs up and down in the water. Its height relative to sea level follows a sine curve. The distance between its highest point (2m) and lowest point (-2m) is its total vertical travel. One full oscillation takes 6 seconds.
Assuming the buoy is at sea level and moving upward at \( t = 0 \), write the sine function \( h(t) \):
What will the height of the buoy be at \( t = 4.5 \) seconds?
Simple Harmonic Teacher Guide Teaching Guide
Simple Harmonic Motion (Lesson 1)
Lesson 1
Learning Objectives
Identify midline, amplitude, and period from a verbal description of physical motion.
Construct a trigonometric function (sine or cosine) to model harmonic movement.
Apply initial conditions (\(t=0\)) to choose between sine and cosine models.
Key Discussion Prompts
"Why might we choose cosine over sine if we release a spring from its maximum stretched position? What does that do to our phase shift?"
Materials
Simple Harmonic Slides
Harmonic Worksheet
Metric Rulers
Stopwatches
Pacing & Flow
10 min
The Hook: Pocket Watch
Show Slide 2. Have students turn-and-talk about how they would track the 'distance from center' over time. Draw their collective graph on the board—identify it as a cosine wave.
15 min
Direct Instruction
Use Slides 3-4 to define terms. Focus heavily on the relationship between Period and \(B\). Students often mistake the period for the \(B\) value itself. Remind them: \(B = 2\pi / P\).
20 min
Guided Practice
Slide 5: Work through the spring problem together. Emphasize that 'starting 0.5m below equilibrium' at \(t=0\) suggests a negative cosine model if you don't want to use a phase shift.
30 min
Independent Workshop
Distribute the worksheet. Circulate and check for the common error of mixing up 'half-distance' (Amplitude) with 'total-distance' (Peak-to-Peak).
Watch Out!
Students often calculate frequency (\(1/P\)) instead of the angular coefficient (\(B\)). In the metronome problem, 120 BPM means 2 beats per second, but one "cycle" is two beats (right to left and back). Period = 1 second.
Simple Harmonic Answer Key Answer Key
Simple Harmonic Motion (Lesson 1)
1
The Heavy Lifter
A. Midline (D)
D = 20
B. Amplitude (A)
A = 15 (20 - 5 = 15)
C. Period & B-Value
P = 1.2s; B = 2π / 1.2 = 5π/3 ≈ 5.24
E. The Equation
\( y = -15 \cos(\frac{5\pi}{3}t) + 20 \)
*Note: Using negative cosine models the release from the lowest point (5cm) at t=0.
2
The Classic Metronome
Amplitude
8
Period
1.0s
B Value
2π
Model (t=0 at +8)
\( y = 8 \cos(2\pi t) \)
Critical Thinking Answer
The B-value changes. If BPM doubles, the period is halved (0.5s). Thus, B doubles to \( 4\pi \).
3
The Ocean Buoy
Equation:
\( h(t) = 2 \sin(\frac{\pi}{3}t) \)
Height at t=4.5:
\( h(4.5) = 2 \sin(\frac{4.5\pi}{3}) = 2 \sin(1.5\pi) = 2(-1) = -2\text{ meters} \)
Biological Rhythms Slides Bio-Systems Analysis
THE BREATHING CURVE
Biological Rhythms & Trig Modeling
Your Personal Rhythm
Your lungs aren't just organs; they're oscillating pumps.
Mini-Lab Task:
Count your breaths for 60 seconds.
Calculate the Period of one breath.
Assume you inhale 0.5 Liters of air.
Normal Lung Volume: 2L to 2.5L
We will model this volume change over time.
FROM RATE TO PERIOD
The Math Link
In biology, we often get a Rate (beats per min, breaths per min).
Period = 60 / Rate
Example: 12 breaths/min = 5 sec/breath
1
Identify Midline
The average volume in the lungs.
2
Identify Amplitude
Half of the "Tidal Volume" (amount inhaled/exhaled).
CASE STUDY: BLOOD PRESSURE
The Data:
High (Systolic): 120 mmHgLow (Diastolic): 80 mmHgPulse Rate: 60 BPM
Model Goal:
Find \( P \), \( A \), and \( D \) to write the function.
Visualization
100 mmHg (Midline) 120 mmHg (Peak)
Biometric Data Worksheet PULSE & PATH
Biometric Data Workshop
Session ID: BIO-MODEL-02
Name: ______________________
PHASE 1: THE RESPIRATORY CYCLE
A healthy teenager at rest has a Functional Residual Capacity (the volume of air left in the lungs after a normal exhale) of 2.4 Liters. During a normal breath, they inhale 0.5 Liters of air (Tidal Volume ). One full breath takes 4 seconds.
Your Data Points
Min Vol
2.4 L
Max Vol
2.9 L
1. Find Midline (D)
2. Find Amplitude (A)
3. Find B-Value (P=4s)
Write the model \( V(t) \). Assume the lungs are at their minimum volume at \( t = 0 \).
PHASE 2: BLOOD PRESSURE DYNAMICS
Blood pressure is not constant; it pulses. An athlete has a blood pressure of 110/70 (Systolic/Diastolic) and a resting heart rate of 50 beats per minute.
A. Calculate Period (sec per beat)
B. Midline & Amplitude
Graph the Model (2 cycles)
Predictive Equation \( P(t) \):
Medical Interpretation
How would the trigonometric function for heart rate change if the athlete began sprinting? Specifically, identify which coefficients change and why.
Write your response here...
Biometric Data Answer Key Answer Key
Biological Rhythms (Lesson 2)
Phase 1: Lung Cycle
Midline (D)
2.65 L
(2.4 + 2.9) / 2
Amplitude (A)
0.25 L
(2.9 - 2.4) / 2
B-Value
π/2
2π / 4
Equation (Starting at min volume):
\( V(t) = -0.25 \cos(\frac{\pi}{2}t) + 2.65 \)
Phase 2: Blood Pressure
Period (P)
1.2s
60 / 50 BPM
B-Value
5π/3 ≈ 5.24
2π / 1.2
Midline (D)
90 mmHg
(110 + 70) / 2
Amplitude (A)
20 mmHg
(110 - 70) / 2
Equation:
\( P(t) = 20 \sin(\frac{5\pi}{3}t) + 90 \)
*Note: Using sine assumes pulse starts at midline. Cosine or negative cosine could also be used depending on phase start.
Medical Interpretation Answer
If sprinting, the Period decreases because heart rate increases. This means the B-value increases (the waves are compressed). Additionally, the systolic pressure might increase more than diastolic, potentially increasing the Amplitude and shifting the Midline upward .
Biological Rhythms Teacher Guide Teaching Guide
Case Study: Biological Rhythms
Lesson 2
Strategy: Case Study Method
This lesson moves away from abstract physics (Lesson 1) into human physiology. The goal is to show students that "math is happening inside them." Use the "Mini-Lab" in Slide 2 to ground the math in their own bodies.
Teaching Tip
Remind students that unlike a spring, which might start at a max/min point, biological rhythms often have no "official" start. Choosing between sine and cosine is often a matter of convenience or defining t=0.
Discussion Questions
"If you take a deep breath and hold it, what happens to the graph? Is it still trigonometric?"
"What do the values between the peak and trough represent in terms of blood flow?"
Pacing & Flow
Hook: Pulse Count (10 min)
Students measure their own respiration. Calculate the period immediately to establish the connection.
Concept: Rate vs. Period (15 min)
Direct instruction on the 60/Rate conversion. This is the primary point of failure for students.
Workshop (30 min)
Students work on the Biometric Data Worksheet. Part 1 focuses on volume, Part 2 on pressure.
Synthesis (5 min)
Discuss the "Medical Interpretation" question at the end of the worksheet.
Orbital Mechanics Slides Navigation: Sector 03
COSMIC CYCLES
Orbital Mechanics and Position Modeling
Shadows of Jupiter
Galileo tracked the moon Io orbiting Jupiter. From Earth, Io looks like it's simply moving back and forth across Jupiter.
The Observation:
Max distance: 422,000 km right of center
Min distance: 422,000 km left of center
One full orbit: 42.5 hours
Scientific Notation Alert
In space, numbers get big. We often use \( 4.22 \times 10^5 \) instead of \( 422,000 \).
A = 4.22e5
HANDLING LARGE SCALES
The Distance
Distances are our Amplitude. Midline is typically the center of the primary body (0).
D = 0
The Time
Orbits take days, months, or years. Ensure your t and Period use the same units.
P = Orbtial Time
The Coefficient
Calculate B as usual, but keep it in terms of \(\pi\) for precision.
B = 2\pi / P
THE EXOPLANET TEST
LEVEL: EXPERT
A planet orbits a distant star. The telescope records its maximum displacement as \( 1.5 \times 10^8 \) km.
It takes the planet 360 Earth days to complete one orbit.
Task: Write the Equation
Model the horizontal distance \( x(t) \) from the star if the planet is at its maximum distance at \( t = 0 \).
r = 1.5e8 km
Orbital Position Worksheet Mission Log: Orbitals
Planetary Position Modeling
Star Date: 2026.01
Unit: TR-03
DATASET ALPHA: LEO SATELLITE
ENCRYPTED // CLEARANCE LEVEL 2
A Low Earth Orbit (LEO) satellite is tracked relative to a ground station directly below its path. The satellite orbits at an altitude of \( 2,000 \) km. Because it orbits the Earth, its horizontal distance \( x \) from the station varies periodically as it passes over. We model this as a wave with an amplitude of \( 2 \times 10^3 \) km and a period of \( 90 \) minutes.
A. Parameters
Amp (A)
Midline (D)
B. Period & B-Value
C. Final Model \( x(t) \)
Assume satellite is directly over the station (x=0) and moving east at t=0.
DATASET BETA: EUROPA'S ORBIT
TELESCOPE DATA: G-400
"The moon Europa orbits Jupiter at a distance of approximately \( 6.7 \times 10^5 \) km. It completes one full revolution every \( 3.5 \) Earth days."
Mission Tasks:
1
Define the model \( E(t) \)
Represent Europa's horizontal distance from Jupiter (t in days, Europa starts at max distance).
2
Calculate Position at \( t = 1.75 \) days
Show all work using your model.
Predicted Orbit Graph (One Period)
Time (Days) vs. Distance (10^5 km)
Astronomy Note: Scale Errors
If a student uses \( 3.5 \) as the \( B \) value in their equation, what physical error are they making in their model of Europa's orbit? Explain the difference between "Period" and the "Coefficient B".
Orbital Position Answer Key Answer Key
Orbital Mechanics (Lesson 3)
Dataset Alpha: LEO Satellite
Parameters
A = \( 2,000 \) or \( 2 \times 10^3 \)
D = 0
B-Value
B = \( 2\pi / 90 = \pi/45 \approx 0.0698 \)
Final Model:
\( x(t) = 2000 \sin(\frac{\pi}{45}t) \)
*Sine is used because it starts at x=0 moving upward/east.
Dataset Beta: Europa
Task 1: Model \( E(t) \)
\( E(t) = 6.7 \times 10^5 \cos(\frac{2\pi}{3.5}t) \) or \( E(t) = 6.7 \times 10^5 \cos(\frac{4\pi}{7}t) \)
Task 2: Position at \( t = 1.75 \)
At \( t = 1.75 \), we are at exactly half the period (\( 3.5 / 2 = 1.75 \)).
Extension Answer
If they use \( 3.5 \) as the \( B \) value, they are saying the satellite completes \( 3.5 \) oscillations in \( 2\pi \) units of time (about 6.28 units), rather than completing one oscillation every \( 3.5 \) units. The physical error is failing to scale the unit circle rotation to the actual orbital period.
Orbital Position Teacher Guide Teaching Guide
Astronomy: Orbital Mechanics
Lesson 3
Strategy: Skill Transfer & Scale
This lesson shifts the scale from centimeters (Lesson 1) to hundreds of thousands of kilometers. The mathematics remains identical, but students often feel intimidated by scientific notation. Focus on the fact that \( 10^5 \) is just a unit, like "meters".
Concept Focus
Midline is always zero in these orbital distance problems because we are measuring distance from the center of the orbit (the star or planet).
Teacher Prep
Review scientific notation entry on calculators.
Prepare Jupiter moon diagrams (Io, Europa).
Ensure students are in Radian mode.
Pacing & Flow
10 min
The Hook: Jupiter Moons
Slide 2. Discuss how a circular orbit looks like back-and-forth motion (a wave) when viewed edge-on from Earth. This is a crucial conceptual leap.
15 min
Modeling Large Numbers
Slide 3. Demonstrate how to keep \( 10^5 \) as a coefficient. Model Europa's orbit on the board before handing out the worksheet.
35 min
Independent Exploration
Worksheet: Dataset Alpha and Beta. Part 1 (Satellite) uses minutes; Part 2 (Europa) uses days. Emphasize unit consistency.
Inverse Time Slides TARGETING TIME
Inverse Trigonometric Operations in Context
The Security Sweep
A security laser sweeps back and forth across a hallway. Its position \( y \) is modeled by:
y = 3 cos(πt)
The "danger zone" is when the laser is between 1m and 2m.
The Problem: WHEN does the laser hit 2 meters?
Active Laser Zone
THE INVERSE PATHWAY
STEP 1
Isolate the Trig
Get sin/cos by itself
STEP 2
Inverse Function
Use \(\arccos\) or \(\arcsin\)
STEP 3
Solve for \(t\)
Divide by coefficient B
STEP 4
Find Cycles
Add Period (P) to find all times
THE PERIODIC TRAP
A calculator only gives ONE answer.
But periodic functions hit the same value twice per cycle.
Symmetry: \( \pi - \theta \) for Sine
Symmetry: \( 2\pi - \theta \) for Cosine
t1 t2
Inverse Time Worksheet CHRONOS TARGET
Inverse Trigonometric Modeling
Time Analysis Log
Identity: ______________________
MISSION 1: THE SECURITY BEAM
A rotating security scanner emits a light beam. Its horizontal position \( x \) (in meters) from the center of the vault door is modeled by: \( x(t) = 4 \cos(\frac{\pi}{6}t) \) Where \( t \) is in seconds. An alarm triggers if the beam hits the vault handle at \( x = 2 \) meters.
A. Find the first time \( t \) (where \( t > 0 \)) when the alarm triggers.
B. What is the period of the scanner?
C. Find the second time the alarm triggers.
MISSION 2: THE MIDNIGHT VIEW
You are on a Ferris wheel. Your height \( h \) (in meters) after \( t \) minutes is modeled by: \( h(t) = 15 \sin(\frac{\pi}{5}t) + 18 \) You can see the city skyline only when you are at a height of 25.5 meters or higher .
Task: Solve for the exact time window.
1. Isolate the Sine Function:
2. Apply Inverse Sine:
3. Solve for \( t_1 \) and \( t_2 \) (first two solutions):
4. How long (in minutes) are you above the "skyline view" height during each rotation?
Mathematical Logic
Explain why a calculator's \(\arcsin\) value might result in a negative time value in some problems, and how you would find a physically meaningful positive solution from that negative one.
Inverse Time Teacher Guide Teaching Guide
Inverse Trig Problems: Finding Time
Lesson 4
Strategy: Working Backward
Students are comfortable plugging in time (\(t\)) to find a value (\(y\)). This lesson reverses that cognitive flow. The major hurdle is not the algebra, but the periodicity —the fact that there are infinite solutions and two unique solutions per cycle.
Teaching Tip
Always encourage students to sketch the graph and draw a horizontal line at the target value. This visually confirms that there are multiple intersection points.
Required Prior Knowledge
Mastery of isolating variables in equations.
Familiarity with \(\sin^{-1}\) and \(\cos^{-1}\).
Understanding of wave symmetry.
Pacing & Flow
10 min
The Hook: Laser Sweep
Slide 2. Present the security laser scenario. Ask: "If the laser moves at a constant speed across the Hallway, why is the math so hard?" Answer: "Because it's rotating, not moving linearly."
20 min
Inverse Pathway
Slides 3-4. Walk through the steps of isolating the trig function. Focus heavily on symmetry . Show how to find the second solution using \(\pi - \theta\) or \(2\pi - \theta\).
30 min
Workshop: Chronos Target
Independent work on the worksheet. Circulate and assist with Mission 2 (Ferris Wheel), as it includes a midline shift which adds an extra isolation step.
The "Stuck at Inverse" Error
Students often calculate \(\cos^{-1}(2)\) instead of \(\cos^{-1}(2/4)\). Remind them: the inverse function must be applied to the ratio , not the raw value.
Inverse Time Answer Key Answer Key
Inverse Time Problems (Lesson 4)
Mission 1: The Security Beam
A. First Time Alarm Triggers
\( 2 = 4 \cos(\frac{\pi}{6}t) \implies \frac{1}{2} = \cos(\frac{\pi}{6}t) \)
B. Period
P = 2\pi / (\pi/6) = 12s
C. Second Time
By symmetry of cosine: \( t = 12 - 2 \)
Mission 2: Ferris Wheel
1. Isolate & 2. Inverse
\( 25.5 = 15 \sin(\frac{\pi}{5}t) + 18 \implies 7.5 = 15 \sin(\frac{\pi}{5}t) \)
3. Solve for \(t_1, t_2\)
\( t_1 = 5/6 \approx \mathbf{0.833\text{ minutes}} \)
4. Time Window Duration
\( 4.167 - 0.833 = 3.334\text{ minutes} \)
Mathematical Logic Answer
Calculators return values in the range \( [-\pi/2, \pi/2] \) for \(\arcsin\). A negative time suggests an event that "happened before" \( t=0 \). To find a positive solution, simply add the Period of the function to the negative result.
Modeling Gauntlet Slides LEVEL: MASTERY
THE GAUNTLET
Rapid Trig Modeling Rotation
SPEED
PRECISION
MASTERY
RULES OF THE RUN
1
Rotate through 4 Combat Stations every 10 minutes.
2
Each station presents a unique oscillating phenomenon.
3
Capture the Parameters and write the Model .
The Winning Strategy:
Is it Sine or Cosine?
P = Distance between repeats.
A = Half the range.
D = Average value.
STATION 1: THE TIDES
ENVIRONMENTAL
At Bay of Fundy, the water is 2m deep at low tide and 16m deep at high tide. The time between high tides is 12 hours.
Challenge:
Write a model for height h(t) starting at high tide (t=0).
STATION 2: THE COOLING FAN
MECHANICAL
A blade on a ceiling fan is 50cm long. The center of the fan is 250cm from the floor. It spins at 60 RPM.
Challenge:
Model the height h(t) of the blade tip if it starts at the lowest point.
Gauntlet Record Sheet GAUNTLET RECORD
Mastery Level Verification
Station Pass No.
Operative: ______________________
Instructions:
Rotate to each station. You have 10 minutes to analyze the data, determine your coefficients (A, B, C, D), and construct the model. Show your work in the grid provided. Accuracy and speed are your primary objectives.
STATION 1: THE TIDES Environment
Amp (A)
Mid (D)
Per (P)
B-Val
Equation \( h(t) \):
STATION 2: THE FAN Mechanical
Amp (A)
Mid (D)
Per (P)
B-Val
Equation \( h(t) \):
STATION 3: SOUND WAVE Acoustic
Amp (A)
Mid (D)
Per (P)
B-Val
Equation \( p(t) \):
STATION 4: THE SWING Physics
Amp (A)
Mid (D)
Per (P)
B-Val
Equation \( y(t) \):
The Final Verification
Once all four equations are written, calculate the sum of all Midlines (D values) . This sum is your team's verification code.
CODE: ________
Report to the Command Center immediately.
Modeling Gauntlet Answer Key Answer Key
The Modeling Gauntlet (Lesson 5)
Station 1: The Tides
A = 7
D = 9
P = 12
B = π/6
Model: \( h(t) = 7 \cos(\frac{\pi}{6}t) + 9 \)
Station 2: The Fan
A = 50
D = 250
P = 1.0s
B = 2π
Model: \( h(t) = -50 \cos(2\pi t) + 250 \)
Station 3: Sound Wave (Data: 440 Hz / Peak 0.02 Pa)
A = 0.02
D = 0
P = 1/440
B = 880π
Model: \( p(t) = 0.02 \sin(880\pi t) \)
Station 4: The Swing (Data: Max 3m, Min 0.5m, cycle 4s)
A = 1.25
D = 1.75
P = 4
B = π/2
Model: \( y(t) = 1.25 \sin(\frac{\pi}{2}t) + 1.75 \)
Final Verification Code
Sum of all D values: \( 9 + 250 + 0 + 1.75 \)
CODE: 260.75
Modeling Gauntlet Teacher Guide Teaching Guide
Mastery Workshop: The Gauntlet
Lesson 5
Strategy: Mastery & Rotation
This is the capstone lesson for the sequence. Instead of a standard test, students demonstrate mastery through a high-energy rotation. The "Escape Room" element (The Final Code) creates urgency and a sense of shared purpose within groups.
Teacher Role
Act as the "Command Center." Do not give answers. If a group is stuck, ask: "Where is the midline on your sketch?" or "How long does one full loop take?"
Station Setup
Station 1: Data on high/low tides.Station 2: A small desk fan or diagram with RPM data.Station 3: Audio clip of a 440Hz tone (A4) or frequency data.Station 4: Video/GIF of a child on a swing.
The Rotation (60 min)
0-10
Briefing
Show Slides 1-2. Explain the Gauntlet Record Sheet. Assign groups to their starting stations.
10-50
The Run (4 Rounds)
Rotate every 10 minutes. Use a loud timer. Students must complete the parameters and equation for their current station before moving.
50-60
Verification
Groups calculate their Final Code (Sum of D values). If the code is correct (260.75), they "pass" the gauntlet.
The Sound Challenge (Station 3)
Frequency (Hz) is \( 1/P \). For sound at 440 Hz, the period is \( 1/440 \) seconds. Students may struggle with this small fraction; encourage them to write \( B = 2\pi / (1/440) \), which simplifies to \( 880\pi \).
