# Circle Revolutions and the Coin Rotation Paradox

The concept of a circle‘s revolution involves understanding its circumference, which is the distance around the circle. To calculate the circumference of a circle with radius r, use the formula:

Circumference = 2πr

Here, π is a constant approximately equal to 3.14159.

### Calculating the Number of Circle Revolutions to Cover a Distance

To determine the number of revolutions a circle with radius r makes to cover a certain distance s, divide the total distance by the circumference of the circle. The formula is:

Number of Revolutions = s / 2πr

### Example Problem

Find the number of revolutions a circle with a radius of 2 meters makes to cover a distance of 12.56 meters. Using the formula:

Circumference = 2 × π × 12.56​

Number of Revolutions = 12.56 / 12.56​ = 1

So, the circle makes exactly 1 revolution to cover 12.56 meters.

### Circle Revolutions Around a Larger Circle

Now, consider a circle of radius r rolling around a larger circle of radius 3r. The number of revolutions it makes is not directly proportional to the radii of the circles.

In this case, the smaller circle makes one complete revolution when it rolls around the larger circle. However, it also rotates on its own axis due to the motion. This additional rotation results in the smaller circle making one extra revolution for each revolution of the larger circle.

Since the circumference of the larger circle is 2π×3r, which is three times the circumference of the smaller circle, the smaller circle completes three revolutions due to the rolling motion, plus one additional revolution on its own axis.

Therefore, the total number of revolutions is 3 + 1 = 4.

In general, circle of radius r rolling around one of radius R makes R/r + 1 rotations.

### The Common Misperception

The common misperception that leads people to answer “3” instead of “4” is the oversight of the additional revolution the smaller circle completes on its own axis. Many people calculate the ratio of the circumferences (which is 3:1) and conclude that the smaller circle revolves three times. However, this calculation neglects the fact that as the circle rolls, it also spins on its axis, resulting in one extra revolution.

One of the math questions on the 1982 SAT omitted the correct answer, so don’t feel bad if you thought the answer was “3”.

The circle revolution question illustrates the coin rotation paradox, also known as the coin rolling paradox. The coin rotation paradox is the counter-intuitive phenomenon that rolling one coin around another of equal size results in the moving coin completing two full rotations, rather than one. You can easily prove this to yourself using a pair of coins.

The coin rotation paradox relates to sidereal time. A sidereal day is the time Earth needs to rotates so that a distant star appears in the same position in the sky. In contrast, a solar day is the time for the Sun to return to the same position in the sky. The year takes around 365.25 solar days, but 366.25 sidereal days because of the revolution around the Sun.

### Practical Applications of Circle Revolutions

The geometry problem of calculating the revolutions of a circle has many practical applications. Here are some notable examples:

1. Mechanical Engineering and Design: Understanding circle revolutions is crucial in designing gears, pulleys, and wheels. The principles dictate how gears of different sizes interact with each other, influencing the speed and torque in machinery. This knowledge applies to automotive transmissions, clock mechanisms, and industrial machinery.
2. Robotics and Automation: In robotics, wheel revolution calculations are essential for movement and navigation. Robots and automated vehicles use these principles to calculate the distance traveled and to navigate accurately.
3. Civil Engineering and Construction: Large-scale construction equipment, such as cranes and excavators, rely on circular motion principles. Understanding how wheels and joints rotate helps predict the range of motion and the efficiency of the equipment.
4. Astronomy and Space Exploration: Circular revolution concepts apply to orbital mechanics calculations. This is essential in satellite deployment, where understanding how celestial bodies move and orbit each other is critical for successful missions.
5. Athletics and Sports Equipment Design: The design of sports equipment like bicycles and rollerblades involves understanding wheel revolutions. This helps in optimizing performance, such as maximizing speed or efficiency.
6. Manufacturing and Conveyor Systems: In manufacturing, conveyor belts and rollers apply the principles of circular motion. This helps in calculating the speed at which items move and the efficiency of the production line.
7. Art and Animation: Animators use these principles for simulating natural movements in characters or objects.
8. Education and Demonstration Tools: This geometry problem teaches fundamental concepts in physics and mathematics in a tangible and relatable manner.
9. Medical Devices: Some medical devices, like CT scanners, use principles of circular motion for accurate imaging.
10. Wheelchair Design: For mobility aids like wheelchairs, understanding wheel revolutions optimizes the design in terms of effort required to propel the wheelchair and the ease of maneuverability.

### References

• Bartlett, A. K. (1904). “Solar and Sidereal Time“. Popular Astronomy. 12: 649–651.
• Bennett, Jeffrey; Briggs, William (2005). Using and Understanding Mathematics / A Quantitative Reasoning Approach (3rd ed.). Addison-Wesley. ISBN 978-0-321-22773-7.
• Bunch, Bryan H. (1982). Mathematical Fallacies and Paradoxes. Van Nostrand Reinhold. ISBN 0-442-24905-5.
• United Press International (May 25, 1982). “Error found in S.A.T. question“. The New York Times.