Coin Rotation Paradox Calculator

Calculate the number of rotations when a coin rolls around another coin

Enter the radii of two coins to see how many full rotations the moving coin makes as it rolls around the fixed coin. This demonstrates the famous coin rotation paradox in geometry.

Examples

  • If both coins have radius 2: (2 + 2) / 2 = 2 full rotations.
  • If moving coin is radius 1, fixed coin is radius 3: (1 + 3) / 3 = 1.333... rotations.
  • If moving coin is radius 5, fixed coin is radius 2: (5 + 2) / 2 = 3.5 rotations.
  • If both coins have radius 10: (10 + 10) / 10 = 2 full rotations.

Did You Know?

When two coins of equal size are used, the moving coin makes two full rotations: one from traveling the circumference, and one from spinning around its own center!

Other Titles
Understanding Coin Rotation Paradox: A Comprehensive Guide
Explore the surprising geometry behind the coin rotation paradox, its applications, and how to use the calculator effectively.

Understanding Coin Rotation Paradox: A Comprehensive Guide

  • The paradox arises when a coin rolls around another of equal size.
  • The moving coin appears to make two full rotations, not one.
  • This result surprises many and is a classic geometry paradox.
The coin rotation paradox is a fascinating result in geometry. When a coin rolls without slipping around another coin of the same size, it completes two full rotations. This seems counterintuitive, as one might expect only a single rotation.
The paradox is resolved by considering both the distance traveled around the fixed coin and the rotation due to the coin's own movement. The total number of rotations is given by (R₁ + R₂) / R₂, where R₁ is the radius of the moving coin and R₂ is the radius of the fixed coin.
This phenomenon is not limited to coins of equal size. The formula applies to any two circles, and the result is always greater than one full rotation.

Basic Examples

  • Two coins of radius 2: (2 + 2) / 2 = 2 rotations.
  • Moving coin radius 3, fixed coin radius 1: (3 + 1) / 1 = 4 rotations.
  • Moving coin radius 1, fixed coin radius 3: (1 + 3) / 3 = 1.333... rotations.

Step-by-Step Guide to Using the Coin Rotation Paradox Calculator

  • Input the radii of both coins.
  • Click Calculate to see the number of full rotations.
  • Interpret the result and visualize the paradox.
To use the calculator, simply enter the radius of the moving coin and the radius of the fixed coin. Both values must be positive numbers.
Input Guidelines:
  • Both radii must be greater than zero. The calculator will show an error for zero or negative values.
  • The result shows the total number of full rotations the moving coin makes as it rolls around the fixed coin.
Understanding Results:
  • If both coins are the same size, the result is always 2 full rotations.
  • For different sizes, the formula (R₁ + R₂) / R₂ gives the answer, which may be a non-integer value.
  • The paradox is most striking when the coins are the same size, but the formula works for any positive radii.

Usage Examples

  • Moving coin radius 2, fixed coin radius 2: (2 + 2) / 2 = 2 rotations.
  • Moving coin radius 1, fixed coin radius 3: (1 + 3) / 3 = 1.333... rotations.
  • Moving coin radius 5, fixed coin radius 2: (5 + 2) / 2 = 3.5 rotations.

Real-World Applications of Coin Rotation Paradox Calculations

  • Understanding gear ratios and mechanical linkages.
  • Explaining phenomena in physics and engineering.
  • Educational demonstrations in mathematics classes.
The coin rotation paradox is not just a mathematical curiosity. It has practical applications in engineering, physics, and education.
Gear Ratios and Mechanisms:
  • The principle is used to explain gear ratios, where the rotation of one gear around another is analogous to the coin paradox.
Physics and Rolling Motion:
  • The paradox helps in understanding rolling motion, angular velocity, and the behavior of wheels and pulleys.
Education and Visualization:
  • Teachers use the paradox to challenge students' intuition and to demonstrate the importance of careful reasoning in geometry.

Real-World Examples

  • Explaining why a bicycle wheel makes more than one rotation when rolling around another wheel.
  • Understanding the movement of planetary gears in machinery.
  • Demonstrating angular displacement in physics labs.

Common Misconceptions and Correct Methods in Coin Rotation Paradox

  • Many believe the moving coin should make only one rotation.
  • The paradox is resolved by considering both translation and rotation.
  • Careful geometric reasoning is required to understand the result.
A common misconception is that the moving coin should make only one full rotation, since it travels a distance equal to the circumference of the fixed coin. However, this ignores the additional rotation caused by the coin's own movement around the center.
Misconception 1: Only One Rotation
  • The moving coin actually rotates once due to traveling the circumference, and once more due to spinning around its own center.
Misconception 2: Formula Only for Equal Radii
  • The formula (R₁ + R₂) / R₂ applies for any positive radii, not just equal coins.
Correct Method: Use the Full Formula
  • Always use (R₁ + R₂) / R₂ to find the total number of rotations.

Misconceptions & Corrections

  • If both coins are radius 2: (2 + 2) / 2 = 2 rotations (not 1).
  • If moving coin is radius 1, fixed coin is radius 3: (1 + 3) / 3 = 1.333... rotations.
  • If moving coin is radius 5, fixed coin is radius 2: (5 + 2) / 2 = 3.5 rotations.

Mathematical Derivation and Examples

  • Derivation of the formula for the number of rotations.
  • Worked examples for different coin sizes.
  • Visualizing the paradox with diagrams.
Let the radius of the moving coin be R₁ and the fixed coin be R₂. The center of the moving coin traces a circle of radius (R₁ + R₂), so the path length is 2π(R₁ + R₂). The moving coin rotates by the path length divided by its own circumference, 2πR₁. Thus, the number of rotations is (R₁ + R₂) / R₁. However, since the coin also spins around the fixed coin, the total number of full rotations is (R₁ + R₂) / R₂.
Example: If R₁ = 2 and R₂ = 2, then (2 + 2) / 2 = 2 rotations.
Example: If R₁ = 1 and R₂ = 3, then (1 + 3) / 3 = 1.333... rotations.
Example: If R₁ = 5 and R₂ = 2, then (5 + 2) / 2 = 3.5 rotations.

Mathematical Examples

  • Derivation: Number of rotations = (R₁ + R₂) / R₂
  • If R₁ = 2, R₂ = 2: (2 + 2) / 2 = 2 rotations.
  • If R₁ = 1, R₂ = 3: (1 + 3) / 3 = 1.333... rotations.
  • If R₁ = 5, R₂ = 2: (5 + 2) / 2 = 3.5 rotations.