Coin Rotation Paradox Calculator
Explore the fascinating paradox of coins rolling around each other with mathematical precision
Calculate Coin Rotation Paradox
Radius of the stationary coin
Radius of the coin that rotates around the fixed coin
With slippage, the rotating coin always shows the same face to the fixed coin
Paradox Results
Example Calculations
Example 1: Identical Coins
Fixed coin radius: 12.13 mm (US Quarter)
Rotating coin radius: 12.13 mm (US Quarter)
Calculation: N = 1 + (12.13/12.13) = 1 + 1 = 2
Result: Exactly 2 rotations - the classic paradox!
Example 2: Different Sized Coins
Fixed coin radius: 15 mm (Large coin)
Rotating coin radius: 10 mm (Small coin)
Calculation: N = 1 + (15/10) = 1 + 1.5 = 2.5
Result: 2.5 rotations for the smaller coin
Key Concepts
Path Curvature
Curved paths add "extra" rotation compared to straight paths
Reference Frame
The paradox depends on your reference frame
No Slippage
Rolling without slipping creates the paradox
Common Coin Sizes
Understanding the Coin Rotation Paradox
What is the Paradox?
When you roll one coin around another identical coin without slipping, the moving coin completes 2 full rotations by the time it returns to its starting position. This seems counterintuitive because the path length equals only one circumference of the coin.
Why Does This Happen?
- •Path rotation: Rolling on a straight path = 1 rotation
- •Path curvature: Circular path adds 1 extra rotation
- •Total: 1 + 1 = 2 rotations for identical coins
Mathematical Formula
N = 1 + (Rf / Rr)
- N: Number of rotations
- Rf: Radius of fixed coin
- Rr: Radius of rotating coin
- 1: Base rotation from rolling
- Rf/Rr: Extra rotation from path curvature
Special Case: With slippage (like Moon-Earth), the coin always shows the same face and completes only 1 rotation.
Reference Frames
External Observer
From outside the system, you see the rotating coin complete N = 1 + (Rf/Rr) rotations.
Fixed Coin Frame
From the fixed coin's perspective, the rotating coin appears to complete Rf/Rr rotations.
Contact Point Frame
From the contact point's view, both coins appear to rotate, each completing half a turn.
Real-World Examples
Earth and Moon
The Moon is tidally locked to Earth, showing the same face. This is equivalent to the "slippage allowed" case - one rotation per orbit.
Gears and Wheels
In mechanical systems, understanding this paradox is crucial for calculating gear ratios and planetary gear systems.