The forbidden coin makes a full rotation, returning to the lapp stead after alone rotating about half of the inner coin. amply rotating around the inner coin it will have completed two rotations . The path of a one point on the edge of the moving coin is a cardioid The coin rotation paradox [ citation needed ] is the counterintuitive notice that, when one coin is rolled around the rim of another mint of equal size, the moving mint completes two full rotations after going all the way around the stationary mint. [ 1 ]
Reading: Coin rotation paradox – Wikipedia
description [edit ]
Start with two identical coins touching each early 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 antonym slope, the two heads will again be parallel ; B has made one revolution. Continuing to move B brings it back to the starting military position and completes a irregular revolution. Paradoxically, mint B has rolled a distance equal to twice its circumference. As mint B rotates, each degree on its circumference describes ( moves through ) a cardioid bend .
psychoanalysis and solution [edit ]
From startle to end, the center of the moving coin travels a round path. The border of the stationary coin and said path form two concentric circles. The radius of the path is the sum of the coins ‘ spoke ; hence, the circumference of the path is twice either coin ‘s circumference. [ 2 ] 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 mint 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 way. That the coin rotates doubly a described above and focusing on the edge of the moving mint as it touches the stationary coin are distractions .
inadequate radius and other shapes [edit ]
R = 3r. In figure 1, with R straightened out, the number of rotations (number of times the arrow points upward) is R/r = 3. In figure 2, as R has been restored into a circle, the coin makes an extra rotation, giving R/r + 1 = 4. Example where= 3. In figure 1, withstraightened out, the total of rotations ( numeral of times the arrow points up ) is= 3. In figure 2, ashas been restored into a circle, the mint makes an extra rotation, giving+ 1 = 4 . rotation of a little coin around a larger one
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A coin of spoke r rolling around one of radius R makes R/r + 1 rotations. [ 3 ] That is because the center of the rolling coin travels a round path with a radius ( or circumference ) of (R + r)/r = R/r + 1 times its own radius ( or circumference ). In the limit case when R = 0, the mint with radius r makes 0/ r + 1 = 1 simple rotation around its bottom point. The May 1st, 1982 SAT had a question concerning this trouble, and, ascribable to homo error, had to be regraded after three students proved there was no decline answer among the choices. [ 4 ] The determine around which the coin is rolled need not be a lap : one extra rotation is added to the ratio of their perimeters when it is any simple polygon or closed arch which does not intersect itself. If the determine is complex, the number of rotations added ( or subtracted, if the coin rolls inside the curl ) is the absolute value of its turning issue .
References [edit ]

 This upvoted answer includes animations and intuitive explanations about the original question where r of the “outer coin” was 1/3 of the inner coin’s radius.