The coin rotation paradox is the counter-intuitive observation that, when one coin is rolled around the rim of another coin of equal size, the moving coin completes two full rotations after going all the way around the stationary coin.

Description

Start with two identical

coin A coin is a small, flat (usually depending on the country or value), round piece of metal or plastic used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order t ...

s touching each other on a table, with their "head" sides displayed and parallel. Keeping coin A stationary, rotate coin B around A, keeping a point of contact with no slippage. As coin B reaches the opposite side, the two heads will again be parallel; B has made one revolution. Continuing to move B brings it back to the starting position and completes a second revolution. Paradoxically, coin B appears to have rolled a distance equal to twice its circumference. In reality, as the circumferences of both coins are equal, by definition coin B has only rolled a distance equal to its own circumference. The second rotation arises from the fact that the path along which it has rolled is a circle. This is analogous to simply rotating coin B “in situ”. The best way to visualise the effect is to imagine the circumference of coin A “flattened out” into a straight line, by which means it will be readily observed, and indeed self-evident, that coin B has rotated only once as it travels along its, now flat, path. This is the “first rotation”. Equally, sliding coin B around the circumference of coin A, instead of rolling it, whilst maintaining its current specific point of contact, will impart a rotation representative of the “second rotation” in the original scenario. As coin B rotates, each point on its perimeter describes (moves through) a

cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal ...

curve.

Analysis and solution

From start to end, the center of the moving coin travels a circular path. The edge of the stationary coin and said path form two concentric circles. The radius of the path is the sum of the coins' radii; hence, the circumference of the path is twice either coin's circumference. The center of the moving coin travels twice the coin's circumference without slipping; therefore, the moving coin makes two complete revolutions. How much the moving coin rotates around its own center en route, if any, or in what direction – clockwise, counterclockwise, or some of both – has no effect on the length of the path. That the coin rotates twice as described above and focusing on the edge of the moving coin as it touches the stationary coin are distractions.

Unequal radii and other shapes

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). In the limiting case when ''R'' = 0, the coin with radius ''r'' makes 0/''r'' + 1 = 1 simple rotation around its bottom point. The May 1st, 1982

SAT The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and scoring have changed several times; originally called the Scholastic Aptitude Test, it was later called the Schol ...

had a question concerning this problem, and, due to human error, had to be regraded after three students proved there was no correct answer among the choices. The shape around which the coin is rolled need not be a circle: one extra rotation is added to the ratio of their perimeters when it is any

simple polygon In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise ...

or closed curve which does not intersect itself. If the shape is

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

, the number of rotations added (or subtracted, if the coin rolls inside the curve) is the absolute value of its

turning number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...

Application

The paradox is related to

sidereal time Sidereal time (as a unit also sidereal day or sidereal rotation period) (sidereal ) is a timekeeping system that astronomers use to locate celestial objects. Using sidereal time, it is possible to easily point a telescope to the proper coord ...

: a

sidereal day Sidereal time (as a unit also sidereal day or sidereal rotation period) (sidereal ) is a timekeeping system that astronomers use to locate celestial objects. Using sidereal time, it is possible to easily point a telescope to the proper coord ...

is the time Earth takes to rotate for a distant star to return to the same position in the sky, whereas a

solar day A synodic day (or synodic rotation period or solar day) is the period for a celestial object to rotate once in relation to the star it is orbiting, and is the basis of solar time. The synodic day is distinguished from the sidereal day, which is ...

is the time for the sun to return to the same position. A year has around 365.25 solar days, but 366.25 sidereal days to account for one revolution around the sun.Bartlett, A. K.
''Solar and Sidereal Time''
Popular Astronomy, vol. 12, pp.649-651 A solar day is 24h so a sidereal day is around

\frac\times 24\text

= 23h 56min 4.1s.

References

External links

* {{Cite web , last=Nguyen , first=Huyen , date=June 4, 2020 , title=answer to the question What proven math fact surprised you the most... , url=https://www.quora.com/What-proven-math-fact-surprised-you-the-most-when-you-learned-it/answer/Huyen-Nguyen-111 , website=

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