Video 1, chapter 2

The Coin Rotation Paradox

Why a coin rolling without slipping around an equal coin turns twice in the coin rotation paradox, explained by the circumference of its center's path, the turning line of centers and the frame of reference.

  • Video 1, chapter 2
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Video 1, chapter 2

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Two identical coins lie flat on a table, touching. Hold one still and roll the other once around it, keeping the edges in contact without slipping, until the moving coin is back where it started. The two rims have the same length, so it seems the moving coin should turn exactly once, yet an arrow drawn on its face shows two full turns. This page finds where the second turn comes from and gives the count for coins of any two sizes.

Key terms

Coin rotation paradox

Definition The coin rotation paradox concerns two coins of equal radius rr that lie flat and touch. One coin is held fixed. The other rolls around the outside of the fixed coin without slipping, always touching it, until it returns to its starting position after one trip around, called an orbit. Counted against a direction that stays fixed in the room, the moving coin turns twice during the orbit, in the same direction as the orbit, although its rim has rolled exactly once along the fixed coin’s rim, which has the same length as its own. The resolution is that rolling along the fixed rim accounts for one turn relative to the line joining the two centers, and that line itself turns once as the coin goes around, which adds the second turn. For a fixed coin of radius RR and a moving coin of radius rr rolling around its outside, the number of turns in the room per orbit is

N=R+rr=Rr+1.N = \frac{R + r}{r} = \frac{R}{r} + 1.

In plain words The moving coin does roll once along the fixed coin, but the path it rolls along curves all the way around. Going around once turns it one extra time, just as walking once around a table while facing forward turns you around once.

Example Start with the moving coin to the right of the fixed coin and an arrow on its face pointing up. After a quarter of the orbit the moving coin sits directly above the fixed coin and the arrow points down, so the coin has made half a turn. Halfway around, on the left, the arrow points up again after one full turn, and back at the start it points up after two full turns. The first and last pictures look identical, so the count has to follow the arrow all the way around instead of comparing two snapshots. For a fixed coin with three times the radius, R=3rR = 3r, the moving coin makes 3+1=43 + 1 = 4 turns, not the 33 that comparing the two rims suggests.

Rolling without slipping

Definition A rigid wheel rolls without slipping on a stationary track if, at every instant, the point of the wheel touching the track is momentarily at rest. Then the length of rim that has touched the track equals the length of track that has been touched. Measure angles in radians, where an angle of θ\theta radians at the center of a circle of radius rr cuts off an arc of length rθr\theta, so that a full turn is 2π2\pi radians, and let the angular speed ω\omega be the angle the wheel turns per unit of time. At each instant the wheel pivots about its point of contact, so its center, at distance rr from that point, moves at speed rωr\omega. Hence, as long as the wheel keeps turning the same way, the distance traveled by its center equals rr times the total angle it turns, measured in the room.

In plain words No slipping means the rim meets the track point by point, like a tape measure being unrolled along it. On a flat road the center moves forward exactly one circumference for each turn.

Example A coin of radius rr rolls along a straight track of length 2πr2\pi r, its own circumference. It turns exactly once, and its center, which stays at height rr above the track, also moves 2πr2\pi r. Around the fixed coin of the paradox the center does not follow the fixed rim. It follows a circle of radius 2r2r and length 4πr4\pi r, so the coin turns through 4πr/r=4π4\pi r / r = 4\pi radians, which is two full turns.

Circumference

Definition The circumference of a circle is its length, the distance once around it. A circle of radius rr has circumference 2πr2\pi r, where π≈3.14159\pi \approx 3.14159 is the same number for every circle. The circumference is proportional to the radius, so doubling the radius doubles it.

In plain words Wrap a string once around the circle, then straighten the string and measure it.

Example In the coin paradox two paths have to be kept apart. The contact point moves along the rim of the fixed coin, a path of length 2πR2\pi R. The center of the moving coin stays R+rR + r from the fixed center, so it moves along a circle of length 2π(R+r)2\pi(R + r). By the rolling rule, dividing the center path by the moving coin’s circumference gives the turns in the room,

N=2π(R+r)2πr=R+rr.\begin{aligned} N &= \frac{2\pi(R + r)}{2\pi r} \\[0.4em] &= \frac{R + r}{r}. \end{aligned}

For equal coins the contact path has length 2πr2\pi r, one circumference, while the center path has length 4πr4\pi r, two circumferences.

Line of centers

Definition The line of centers of two circles is the line through their centers. When two circles touch from outside, the point of contact lies on the line of centers between the two centers, at distance RR from one and rr from the other, so the centers are R+rR + r apart. As the moving coin goes once around the fixed coin, the segment from the fixed center to the moving center turns once, through 2π2\pi radians, in the same direction as the orbit.

In plain words The segment between the two centers sweeps around like the hand of a clock, once per orbit, and the contact point always lies on it.

Example For equal coins the segment from the fixed center points right at the start, up after a quarter of the orbit, left halfway around, down at three quarters and right again at the end. At the same moments the moving coin’s arrow points up, down, up, down and up. So the arrow turns twice while the line of centers turns once.

Frame of reference

Definition A frame of reference is a choice of reference directions against which positions and rotations are measured. The room frame uses directions that stay fixed, such as an arrow that always points up. A rotating frame uses directions that turn, for example the direction of the line of centers. Measure angles counterclockwise as positive. If a body turns through an angle α\alpha relative to a frame whose directions themselves turn through an angle β\beta relative to the room, then the body turns through α+β\alpha + \beta relative to the room.

In plain words How much something turns depends on what it is compared with. A turn measured against a turning reference has to be added to the turn of the reference itself.

Example Ride along with the line of centers. In that frame both centers stay still and the two coins turn like gears in mesh, with equal lengths of rim passing the contact point. The fixed rim has length 2πR2\pi R, so the moving coin turns R/rR/r times per orbit relative to the line, once for equal coins. The line itself turns once relative to the room, so the room frame sees α+β=1+1=2\alpha + \beta = 1 + 1 = 2 turns. The rolling is what fixes α\alpha. An arrow carried once around the fixed coin while it is kept pointing up turns 00 times in the room. Relative to the line of centers it turns once clockwise, so α=−1\alpha = -1, and indeed −1+1=0-1 + 1 = 0. Going around a circle does not by itself make an object turn.

Check yourself

  1. A coin rolls without slipping around the outside of a fixed coin with twice its radius. How many turns does it make in the room during one orbit, and how many relative to the line of centers?
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Here R=2rR = 2r. The center of the moving coin stays 2r+r=3r2r + r = 3r from the fixed center, so it travels 2π⋅3r=6πr2\pi \cdot 3r = 6\pi r, which is three circumferences of the moving coin. The coin therefore turns 33 times in the room, in agreement with R/r+1=2+1R/r + 1 = 2 + 1. The line of centers turns once, so relative to it the coin turns 3−1=23 - 1 = 2 times. This matches the contact path, because the fixed rim has length 4πr4\pi r, two circumferences of the moving coin.

  1. Change the rule. A coin of radius rr rolls without slipping around the inside of a fixed ring of radius 3r3r. How many times does it turn in the room per orbit, and in which direction?
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The coin now touches the ring on its outer side, the side away from the ring’s center. Its center stays 3r−r=2r3r - r = 2r from the ring’s center and travels 2π⋅2r=4πr2\pi \cdot 2r = 4\pi r, two circumferences, so the coin turns twice in the room.

The direction is now opposite to the orbit. Suppose the center moves counterclockwise around the ring. The rim at the contact point has to move backward relative to the center so that the contact point is at rest, and on the outer side of the coin that requires clockwise turning. The frames give the same answer. Relative to the line of centers, the contact path of length 6πr6\pi r gives 33 clockwise turns, so α=−3\alpha = -3, while the line turns once counterclockwise, so β=1\beta = 1. The room sees −3+1=−2-3 + 1 = -2, two clockwise turns. In general, rolling inside a ring of radius RR gives (R−r)/r(R - r)/r turns opposite to the orbit.

  1. Extend the pattern. A coin of radius rr rolls without slipping around the outside of a square whose sides each have length 2πr2\pi r, staying in contact all the way. How many times does it turn in the room during one trip around?
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Along each side the coin rolls on a straight track as long as its circumference, so it turns once per side, four times in all. At a corner it cannot roll along the track. Instead it pivots about the corner, with the point touching the corner at rest, while its center swings through a quarter circle of radius rr. That gives a quarter turn at each of the four corners, one more turn in all, so the total is 4+1=54 + 1 = 5.

The center path confirms the count. It consists of four straight pieces of length 2πr2\pi r and four quarter circles of radius rr, which together make one full circle of length 2πr2\pi r. So the path has length 10πr10\pi r, and the coin turns 10πr/(2πr)=510\pi r / (2\pi r) = 5 times. For the square and for the circle alike, the count is the length of the track divided by the coin’s circumference, plus one.