The coin rotation paradox is the counter-intuitive math problem that, when one coin is rolled around the rim of another coin of equal size, the moving coin completes not one but two full rotations after going all the way around the stationary coin, when viewed from an external reference frame. [1] The problem can be generalized to coins of different radii; in another form, it appeared in an SAT test but none of the multiple choice answers allowed were correct.
Start with two identical coins 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. [2] : 220 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".
One way to visualize the effect is to imagine the circumference of coin A "flattened out" into a straight line, by which means it can be observed 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 curve.
From start to end, the center of the moving coin travels a circular path. The circumference of the stationary coin and the path of the centre form two concentric circles. The radius of the outer circle is the sum of the coins' radii; hence, the circumference of the path of the moving centre is twice either coin's circumference. [3] The center of the moving coin travels twice the coin's circumference without slipping; therefore, the moving coin makes two complete revolutions. [4]
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.
A coin of radius r rolling around one of radius R makes R/r + 1 rotations. [5] 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 1, 1982, SAT 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. [6] [7]
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 or closed curve which does not intersect itself. If the shape is complex, the number of rotations added (or subtracted, if the coin rolls inside the curve) is the absolute value of its turning number.
The paradox is related to sidereal time: a sidereal day is the time Earth takes to rotate for a distant star to return to the same position in the sky, whereas a solar day 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. [8] As a solar day has 24 hours, a sidereal day has around 365.25/366.25 × 24 hours = 23 hours, 56 minutes and 4.1 seconds.
A version of the puzzle arises in group theory, specifically the study of the Lie group known as the split real form of G2. One construction of this group uses the fact that a ball rolling around another ball with three times its radius will make four full turns, rather than three. [9]
In physics, the Coriolis force is an inertial force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Sidereal time is a system of timekeeping used especially by astronomers. Using sidereal time and the celestial coordinate system, it is easy to locate the positions of celestial objects in the night sky. Sidereal time is a "time scale that is based on Earth's rate of rotation measured relative to the fixed stars".
The orbital period is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars. It may also refer to the time it takes a satellite orbiting a planet or moon to complete one orbit.
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, such that when the system evolves along a path in its parameter space but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state. Nonholonomic mechanics is autonomous division of Newtonian mechanics.
A synodic 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 Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités (1889) as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the domain of possibilities is infinite.
In geometry, a deltoid curve, also known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three or one-and-a-half times its radius. It is named after the capital Greek letter delta (Δ) which it resembles.
A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.
The south-pointing chariot was an ancient Chinese two-wheeled vehicle that carried a movable pointer to indicate the south, no matter how the chariot turned. Usually, the pointer took the form of a doll or figure with an outstretched arm. The chariot was supposedly used as a compass for navigation and may also have had other purposes.
The Sagnac effect, also called Sagnac interference, named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called a ring interferometer or Sagnac interferometer. A beam of light is split and the two beams are made to follow the same path but in opposite directions. On return to the point of entry the two light beams are allowed to exit the ring and undergo interference. The relative phases of the two exiting beams, and thus the position of the interference fringes, are shifted according to the angular velocity of the apparatus. In other words, when the interferometer is at rest with respect to a nonrotating frame, the light takes the same amount of time to traverse the ring in either direction. However, when the interferometer system is spun, one beam of light has a longer path to travel than the other in order to complete one circuit of the mechanical frame, and so takes longer, resulting in a phase difference between the two beams. Georges Sagnac set up this experiment in 1913 in an attempt to prove the existence of the aether that Einstein's theory of special relativity makes superfluous.
A gear train or gear set is a machine element of a mechanical system formed by mounting two or more gears on a frame such that the teeth of the gears engage.
The Faraday paradox or Faraday's paradox is any experiment in which Michael Faraday's law of electromagnetic induction appears to predict an incorrect result. The paradoxes fall into two classes:
Aristotle's wheel paradox is a paradox or problem appearing in the pseudo-Aristotelian Greek work Mechanica It states as follows: A wheel is depicted in two-dimensional space as two circles. Its larger, outer circle is tangential to a horizontal surface, while the smaller, inner one has the same center and is rigidly affixed to the larger. Assuming the larger circle rolls without slipping for one full revolution, the distances moved by both circles' circumferences are the same. The distance travelled by the larger circle is equal to its circumference, but for the smaller it is greater than its circumference, thereby creating a paradox.
In geometry, a trochoid is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle as it rolls along a straight line. If the point is on the circle, the trochoid is called common ; if the point is inside the circle, the trochoid is curtate; and if the point is outside the circle, the trochoid is prolate. The word "trochoid" was coined by Gilles de Roberval.
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.
The Eötvös effect is the change in measured Earth's gravity caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity. When moving eastbound, the object's angular velocity is increased, and thus the centrifugal force also increases, causing a perceived reduction in gravitational force.
Earth's rotation or Earth's spin is the rotation of planet Earth around its own axis, as well as changes in the orientation of the rotation axis in space. Earth rotates eastward, in prograde motion. As viewed from the northern polar star Polaris, Earth turns counterclockwise.
The Moon orbits Earth in the prograde direction and completes one revolution relative to the Vernal Equinox and the stars in about 27.32 days and one revolution relative to the Sun in about 29.53 days. Earth and the Moon orbit about their barycentre, which lies about 4,670 km (2,900 mi) from Earth's centre, forming a satellite system called the Earth–Moon system. On average, the distance to the Moon is about 385,000 km (239,000 mi) from Earth's centre, which corresponds to about 60 Earth radii or 1.282 light-seconds.
In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.
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