🪙 Coin Rotation Paradox Calculator
Discover why a coin rotates more than you expect!
The Paradox: If a coin rolls around another coin of the same size without slipping, how many rotations does it make? The surprising answer is 2 rotations, not 1!
The radius of the coin/circle that stays fixed
The radius of the coin/circle that rolls around
How to Use This Calculator
Enter the Fixed Circle Radius
Input the radius of the circle that stays fixed (the one being rolled around).
Enter the Rolling Circle Radius
Input the radius of the circle that rolls around the fixed circle.
Calculate
Click "Calculate Rotations" to see how many rotations the rolling circle makes.
Try the Classic Case
Enter radius = 1 for both circles to see the classic paradox: 2 rotations!
Formula and Explanation
Number of Rotations = (R + r) / r
Where R = fixed circle radius, r = rolling circle radius
Why the Paradox Occurs:
- The rolling coin rotates due to two components: its own rotation and the rotation from traveling the curved path
- When R = r (same size), rotations = (r + r) / r = 2r / r = 2
- The extra rotation comes from the fact that the coin's center travels in a circle, adding one full rotation
- This is different from rolling in a straight line, where rotations = distance / circumference
Example 1: Classic Case (Same Size)
R = 1, r = 1
Rotations = (1 + 1) / 1 = 2 rotations
This is the famous paradox: a coin rotating around an identical coin rotates twice!
Example 2: Different Sizes
R = 2, r = 1
Rotations = (2 + 1) / 1 = 3 rotations
The smaller coin rotates 3 times when rolling around a coin twice its size.
Example 3: Rolling Around a Point
If R = 0 (rolling around a point), then r > 0
Rotations = (0 + r) / r = 1 rotation
Rolling around a point gives exactly 1 rotation, as expected.
About the Coin Rotation Paradox
The Coin Rotation Paradox, also known as the "rolling coin paradox" or "satellite paradox," is a counterintuitive result in geometry. When a coin rolls around another coin of the same size, it completes two full rotations, not one as intuition suggests.
The History
This problem was popularized by Martin Gardner in his "Mathematical Games" column. It has fascinated mathematicians and puzzle enthusiasts because it challenges our intuitive understanding of rotation.
Why Use This Calculator?
- ✅ Understand the Paradox: See why the result is counterintuitive
- ✅ Explore Different Cases: Test with various radius combinations
- ✅ Educational: Learn about rotation and curved paths
- ✅ Verification: Check your manual calculations
- ✅ 100% Free: No registration required
Understanding the Physics
The key insight is that the rolling coin's motion has two components:
- Orbital Rotation: The coin's center travels in a circle around the fixed coin's center
- Spin Rotation: The coin rotates about its own center
- When R = r, the orbital path adds exactly one extra rotation
- This is why we get 2 rotations instead of 1
Real-World Applications
Planetary Motion: This principle applies to satellite orbits and planetary rotations, explaining why the Moon always shows the same face to Earth despite orbiting around it.
Engineering: Understanding this rotation is important for designing gears, pulleys, and rolling mechanisms.
Education: Demonstrates how intuition can be misleading in mathematics and physics.
Frequently Asked Questions
Why does a coin rotate twice when rolling around an identical coin?
The rolling coin rotates once from its own spinning motion and once more because its center travels in a circular path. The curved path adds an extra rotation, making it 2 total rotations.
What if the coins are different sizes?
The formula is (R + r) / r, where R is the fixed coin's radius and r is the rolling coin's radius. If R > r, you'll get more than 2 rotations. If R < r, you'll get between 1 and 2 rotations.
Does this apply only to coins?
No, this applies to any circle rolling around another circle without slipping. It works for wheels, gears, planets, or any circular objects.
What happens if the coin rolls in a straight line?
In a straight line, rotations = distance / circumference. There's no extra rotation from orbital motion, so you get exactly what you'd expect.
Is this related to why the Moon always faces Earth?
Yes! The Moon rotates once on its axis while orbiting Earth once, which is why we always see the same side. This is similar to the coin rotation paradox.
Can I use this for elliptical paths?
The formula (R + r) / r applies specifically to circular paths. For ellipses or other curves, the calculation becomes more complex and depends on the specific path geometry.
What if there's slipping?
The paradox assumes no slipping (pure rolling). If there's slipping, the number of rotations would be different, and you'd need additional information about the friction and motion.