Spatial Geometry & Functions · SAT Math Historical Problems, AP Physics 1 (Rotational Kinematics & Rolling Motion), High School Geometry & Kinematics.
Coin Rolling Paradox & Epicycloid Simulator
Why does rolling a coin around an equal coin yield 2 rotations instead of 1? Because orbital revolution around the center inherently adds 1 full rotation! Interactive arrow tracking reveals the 720° spin in real time.
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- Live Derivation
- Projector Mode
- Print Worksheet
3D Simulation Stage● 60 FPS
Parameter Controls
mm
mm
°
on/off
Live Derivations & Metrics
total_turns
--
N = 1 + R/r
current_spin_deg
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Absolute spin angle relative to ground
orbit_radius
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Distance from center: R + r
swept_ring_area
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π(R + 2r)² - πR²
Real-Time Formula Derivation
Governing Formula:
--Substitution:
--Regime: --
Worked Examples & Exam Question Deep Dive
Historical Case: The 1982 May US SAT Question #17 Debacle
Problem: Circle A with radius r rolls without slipping around fixed Circle B with radius 3r. How many revolutions does Circle A make? Official choices: (A) 3/2, (B) 3, (C) 6, (D) 9/2, (E) 9.
Analytical Strategy: ETS mistakenly assumed revolutions = circumference ratio 3r/r = 3 (Option B). High schooler Matt Vander-Schoot proved all options were incorrect: the center travels a radius of (3r+r)=4r, covering 8πr, so A rotates 8πr/(2πr) = 4 times. ETS awarded credit to all 300,000 students.
Step-by-Step Derivation:
- Step 1: Circumference of fixed circle = 6πr; rolling coin = 2πr.
- Step 2: Center of moving coin travels at radius R + r = 3r + r = 4r.
- Step 3: Distance travelled by center = 2π × (4r) = 8πr.
- Step 4: Total revolutions N = 8πr / (2πr) = 4.
- Step 5: Formula check: N = 1 + R/r = 1 + 3 = 4 (1 from orbit + 3 from rolling).
⚠️ Common Misconception Pitfall: Confusing straight line rolling with orbital rolling: straight paths have zero orbital rotation; circular paths inherently impart 1 extra full revolution.
Student Inquiry Worksheet & Practice Problems
Q1. A 5 cm radius coin rolls on a straight table for 31.4 cm. How many turns does it rotate? (π=3.14)
Answer Key: 1 turn (straight rolling has no orbit: 31.4 / (2 × 3.14 × 5) = 1)
Q2. If a 10 mm coin rolls around a 20 mm fixed coin, how many total rotations does it make?
Answer Key: 3 turns (N = 1 + 20/10 = 3)
Frequently Asked Questions & Misconceptions
What is the rotation formula if the coin rolls on the INSIDE of the fixed circle?
For interior rolling, orbit opposes spin: N = R/r - 1. When R=2r, N = 2 - 1 = 1 turn, and every point on the rolling coin traces a straight diameter line (Tusi Couple)!