The coin rotation paradox refers to the counter-intuitive observation that when one coin is rolled without slipping around the rim of another identical, stationary coin, the rotating coin completes not one, but two full rotations after traversing the circumference of the stationary coin . This is considered a paradox because it appears that since both coins have the same circumference, one rotation should be sufficient for the rolling coin to return to its original position.
The reason behind the extra rotation lies in the combined effect of two rotations:
- Rotation around its own axis: As the moving coin rolls along the circumference of the stationary coin, it completes one rotation for every length of its own circumference covered along the path.
- Rotation around the center of the stationary coin: The fact that the path itself is circular also contributes to the total rotation. This can be visualized by considering the center of the rolling coin, which travels along a circular path whose circumference is twice that of the individual coins.
The number of rotations (N) for a coin of radius r r r sub r𝑟𝑟 rolling around a stationary coin of radius r f r sub f𝑟𝑓 is given by the formula:
N=1+r f r r cap N equals 1 plus the fraction with numerator r sub f and denominator r sub r end-fraction𝑁=1+𝑟𝑓𝑟𝑟
In the case of two identical coins, where r r=r f r sub r equals r sub f𝑟𝑟=𝑟𝑓, the formula becomes:
N=1+r f r f=1+1=2 cap N equals 1 plus the fraction with numerator r sub f and denominator r sub f end-fraction equals 1 plus 1 equals 2𝑁=1+𝑟𝑓𝑟𝑓=1+1=2
This formula demonstrates that the rolling coin will always complete one more rotation than the ratio of the radii would suggest.
Imagine a stationary coin (A) and a rolling coin (B), both with a radius of ‘r’. As coin B rolls around coin A, its center travels along a path with a radius of r+r=2 r r plus r equals 2 r𝑟+𝑟=2𝑟. This larger path has a circumference of 2×π×2 r=4 π r 2 cross pi cross 2 r equals 4 pi r 2×𝜋×2𝑟=4𝜋𝑟, which is twice the circumference of coin B (2 π r 2 pi r 2𝜋𝑟). Since the coin rolls without slipping, for every 2 π r 2 pi r 2𝜋𝑟 distance its center travels, it completes one rotation. Therefore, traveling a distance of 4 π r 4 pi r 4𝜋𝑟 will result in 4 π r/2 π r=2 4 pi r / 2 pi r equals 2 4𝜋𝑟/2𝜋𝑟=2 rotations.
The phenomenon can also be understood by considering the movement of the moon around the Earth. The moon is tidally locked, meaning it rotates once on its axis during one complete orbit around the Earth. As a result, we always see the same side of the Moon. However, if you were to observe the moon from space, you would see it completing a full rotation on its axis during its orbit, [according to GraphicMaths] . This is analogous to the coin rotation paradox, where the observed rotation depends on the chosen frame of reference.
What is the equation for the coin rotation paradox?
A coin of radius r rolling around one of radius R makes R/r + 1 rotations. That is because the center of the rolling coin travels a circular path with a radius (or circumference) of R + r/r = R/r + 1 times its own radius (or circumference).
What is the coin rotation test?
This task is a validated bedside test that requires the participant to rotate a coin (the approximate size of a US nickel) through consecutive 180° turns, using the thumb and index, and middle fingers, as rapidly as possible for 10 seconds. …
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