Unwrapper Graphing Worksheet Unwrapper Guide
Modeling Rotational Dynamics • Lesson 1
Name: __________________________
Date: ___________________________
The Mission
As the wheel turns at a constant speed, the rider's height changes. Your job is to "unwrap" this circular motion and see what shape it creates on a height-vs-time graph.
0° 90° 180° 270°
A wheel with radius 1 unit, centered at (0,0)
Angle (θ) Height (sin θ) 0° 30° 45° 60° 90° 120° 180° 270° 360°
Height vs. Angle Graph
0°45°90°135°180°225°270°315°360°
1.00.50.0-0.5-1.0
Instructions: Use your table to plot the points. Connect them with a smooth curve. What is the name of this function shape?
Connecting the Dots
1. The Ferris Wheel Scenario:
Imagine a wheel with a radius of 50 feet. It rotates at a constant speed. The bottom of the wheel is 5 feet off the ground.
Visual Prompt
Ground
What is the highest point the rider reaches?
What is the lowest point?
Where is the center (hub) of the wheel relative to the ground?
2. Wave Components:
Match the "Ferris Wheel term" on the left with its "Trigonometric graph term" on the right.
A. The Radius of the wheel
B. The Hub Height (Center)
C. One full revolution
____ Midline / Vertical Shift
____ Amplitude
____ Period
3. Sketching the Real Wheel:
Using the wheel from Question 1 (Radius 50ft, Hub 55ft), sketch what the height graph would look like over two full rotations (0 to 720 degrees).
0°360°720°
The Big Reveal
If the wheel is spinning at a constant rate , the angle θ is actually proportional to time . This means we can model the height of a passenger as a function of time: \[ h(t) = a \sin(b(t - c)) + d \]
Unwrapper Intro Slides Unwrapping
the Circle
From Rotational Motion to Sine Waves
Lesson 1 Trigonometry
Visualizing the Ride
Watch the rider as the wheel rotates. Focus only on their vertical height above the ground.
Sketch Challenge:
Draw a rough sketch of the rider's height vs. time. Don't worry about numbers—just the shape of the path.
Ground Level
Physical to Mathematical
Hub Height
The center of the wheel is our Midline (\(d\)). It's the average height.
Radius
The radius is our Amplitude (\(a\)). It's the max distance from center.
Revolution
One full lap is our Period . How long it takes to repeat.
The Big Reveal
"As the wheel turns through an angle, the sine function tracks the vertical displacement."
Height(angle) = radius ⋅ sin(angle) + hub
\( h(\theta) = a \sin(\theta) + d \)
Wheel Dimensions Worksheet Wheel Specs
Lesson 2: Midline & Amplitude
Name: __________________________
Date: ___________________________
1
Case Study: The London Eye
The London Eye is one of the world's most famous Ferris wheels. It has a diameter of 120 meters . The boarding platform is 15 meters above the ground.
Ground
A. What is the radius (Amplitude) of the wheel?
B. What is the Hub Height (Midline) above the ground?
Hint: Boarding height + Radius
C. Calculate the Maximum Height reached by a rider.
The Height Equation
In the equation \( h(\theta) = \mathbf{a} \sin(\theta) + \mathbf{d} \):
a Amplitude: The radius of the wheel. Measures vertical distance from center.
d Midline: The height of the center (hub). Calculated as: \( \text{Boarding Height} + \text{Radius} \).
Engineering Lab
Complete the table for the following Ferris wheels:
Wheel Name Radius Boarding Ht. Midline (d) Max Ht. Sky High 40 ft 8 ft Star Spinner 75 m 10 m Junior Wheel 15 ft 3 ft
Graphing from Stats
2. Draw the Midline and Amplitude for a wheel with radius 30ft and hub height 40ft.
0°90°180°270°360°
Label your midline, max height, and min height on the Y-axis.
3. Reverse Engineering: Write the function from the graph.
1007550250
Identify Midline (\(d\)):
Identify Amplitude (\(a\)):
Write Final Equation:
\( h(\theta) = a \sin(\theta) + d \)
The Designer's Dilemma
A new theme park wants a Ferris wheel where the rider reaches a max height of 210 feet and the boarding platform is exactly 10 feet off the ground.
Calculate: What must the radius of this wheel be?
Work Space
Final Answer
Timing the Turn Slides Timing
the Turn
RPM, Period, and the Frequency Parameter
The Speed Limit
Ferris wheels don't just sit there—they spin! Some spin fast, some spin slow.
"The Speed Trap"
If a wheel makes 2 revolutions per minute (RPM) , how many seconds does one full turn take?
Definitions
Period (P)
The time for one full revolution.
Frequency
How often it repeats in a unit of time (RPM).
Finding the B-Value
To turn time into angle for our function, we use \( b \).
\( h(t) = a \sin(\mathbf{b}t) + d \)
Degrees Mode
\( b = \frac{360}{\text{Period}} \)
How many degrees the wheel turns per second.
Radians Mode
\( b = \frac{2\pi}{\text{Period}} \)
How many radians the wheel turns per second.
Mental Gym
Wheel A
"Takes 120 seconds to rotate once."
Period = 120s
b = 3°/s
Wheel B
"Spins at 1.5 RPM."
Period = 40s
b = 9°/s
Wheel C
"A complete ride is 60 seconds."
Period = 60s
b = 6°/s
Timing the Turn Worksheet Timing Lab
Lesson 3: Speed & Period
Name: __________________________
Date: ___________________________
The Frequency Formula
To model height over time (\(t\)) , we must find the multiplier \(b\).
Step 1: Convert Speed to Period
Calculate how many seconds it takes for one full revolution based on the given speed.
A. Speed: 3 RPM
3 revolutions every 60 seconds
Period = ________ seconds
B. Speed: 0.5 RPM
Half a revolution every 60 seconds
Period = ________ seconds
Step 2: Calculate the B-Value (Degrees)
For each wheel below, find the period and then find the \(b\) value for the equation \( h(t) = a \sin(bt) + d \).
Wheel Name Total Ride Time (1 Rev) Period (sec) B-Value (\(360/P\)) Fast Tracker 30 seconds Scenic Glider 4 minutes Turbo Wheel 15 seconds
Full Model Synthesis
Design Specs: A wheel has a radius of 40ft, boarding height of 5ft, and completes one revolution every 120 seconds.
Amplitude (a):
Midline (d):
Period (P):
B-Value:
Final Height Equation:
h(t) = ________________________________
Launch Point Logic Worksheet Launch Point
Lesson 4: Phase Shifts & Base Functions
Name: __________________________
Date: ___________________________
Where do we start?
Most Ferris wheels load passengers at the bottom . However, some load in the middle or at the top . Depending on where you start, you might want to use a specific base function to make your equation simpler.
Start Middle
Best Function:
sin(bt)
Start Top
Best Function:
cos(bt)
Start Bottom
Best Function:
-cos(bt)
The Shift Debate
You can model any starting position using a Sine function if you include a phase shift \( c \): \[ h(t) = a \sin(b(t - c)) + d \]
Question: If a rider starts at the BOTTOM of a wheel, which equation is easier to write? Explain why.
h(t) = -a cos(bt) + d
...or a shifted sine?
Identify the Function Type
Choose the most efficient base function (sin, -sin, cos, -cos) for each starting scenario:
1
A rider boards at the lowest point of the wheel.
____________
2
A rider boards halfway up as the wheel moves upward .
____________
3
A rider is placed at the top of the wheel by a crane.
____________
Synthesis: The Full Equation
Scenario: The Sunset Spinner
The Sunset Spinner has a radius of 60 feet . The center of the wheel is 70 feet above the ground. It rotates once every 90 seconds . A passenger boards at the bottom (t = 0).
Amplitude (a):
Midline (d):
B-Value (360/90):
Base Function Selection
sin
-sin
cos
-cos
The Model: h(t) =
Graphing the Ride
Sketch one full cycle of the "Sunset Spinner" below.
0s45s90s
130ft70ft10ft
Self-Check: Does your graph start at the bottom? Does it reach the correct max height? Does one cycle take exactly 90 seconds?
Ride Design Proposal Project Ride Proposal
DEPARTMENT OF THEME PARK ENGINEERING
Project ID: FW-2026-X
Status: Draft Phase
The Brief
A new downtown entertainment district has issued a Request for Proposals (RFP) for a signature Ferris wheel. As lead engineers, your team must design a wheel that maximizes rider experience while adhering to strict zoning and safety constraints.
Zoning Constraints
Max Height: The highest point cannot exceed 180 feet .Safety Clearance: The wheel must be at least 10 feet above the ground at all times.Ride Duration: One full revolution must take between 60 and 180 seconds .
ENGINEERING TEAM
Phase 1: Dimensions
Choose your wheel's specifications. Stay within the zoning constraints!
Boarding Height (ft)
Must be ≥ 10
Maximum Height (ft)
Must be ≤ 180
Ride Duration (seconds)
Between 60 - 180
Calculated Specifications
Radius (Amplitude):
Hub Height (Midline):
B-Value (Speed):
Phase 2: Mathematical Model
Write the full mathematical equation that describes the height of a rider over time, assuming they board at the BOTTOM.
h(t) = ___________________________
The Launch Check
Calculate the rider's height at the following times to verify your model:
h(0) = _______ ft
h(Period/4) = _______ ft
h(Period/2) = _______ ft
Ride Performance Graph (One Cycle)
0P/4P/23P/4P
MaxHubMin
Pitch Your Ride
Briefly explain why your design is the best choice for the entertainment district. How did you balance ride excitement (height) with safety and time constraints?
