Going in Circles

August 31, 2020

Circles and circular motion are ubiquitous in nature, so they are also ubiquitous in physics. One of my uncles embraced this concept when he would explain to people that his nephew did "pi-r-squared physics." The Greek philosophers considered the circle to be the most perfect of shapes, so it's no wonder that astronomy embraced Ptolemy's model of the Solar System and its circular orbits, including the circular epicycles, for many centuries.

Motion in a circle and the principle of angular momentum can be found in the gyroscope. Toy gyroscopes are easy to manufacture, so they are a part of many a child's extracurricular education, including my own. Gyroscopes are fascinating, since their spin axis will stay pointed in the direction of their initial spin-up if not perturbed by an external force. This principle was important in the early days of navigation, and my former employer, an aerospace giant, manufactured a variety of mechanical gyroscopes for inertial navigation. Spinning mass gyroscopes have been displaced by ring laser gyros and fiberoptic gyroscopes, and a variety of MEMs devices.

A gyroscope maintaining its direction when held in a frame of three gimbals.

This gyroscope is held in place by jewel bearings at each end. While these offer low friction, energy loss through friction at the bearings and with air will cause the gyroscope to slow and eventually stop.

electrostatic suspension in some gyroscopes mitigates such frictional energy losses by suspending a ball in a vacuum using electric fields created by electrodes, generally an octant of electrodes, around the sphere.[1] An orthogonal trio of motor windings is used to bring the ball up to speed and orient it in a desired direction. A circumferential air jet is sometimes used for the initial spin-up.

(Wikimedia Commons image by Lucas Vieira.)

So, why does a gyroscope maintain its direction? As I discovered as a student in high school physics class, the "why" answers that are usually accepted are not all that satisfying. An early question in the venerable PSSC physics textbook (I remember it's being question one in chapter one) asked why light bends when it enters water. The expected "answer" was Snell's law. The real answer is that the speed of light is slower in water; so, light, in taking the the path which takes the least time, makes the bend into water. This phenomenon, called Fermat's least time principle, is a special case of the more general least action principle.

The accepted answer of why a gyroscope maintains its direction is conservation of angular momentum. Angular momentum will only be conserved if the rotating gyroscope maintains its direction. However, the origin of angular momentum is a mystery.[2] As stated in the tagline to the article in ref. 2, "...we still don't know how a gyroscope stays pointed in a fixed direction."[2]

The gyroscope was given its name by Foucault, who used one in an 1852 experiment to demonstrate the Earth's rotation, much like his eponymous Foucault pendulum of 1851. Friction forces allowed gyroscopic action for just a few minutes, but the time was sufficient for a measurement. Foucault named the device a gyroscope from the Greek γῦρος (gyros, circle or rotation) and σκοπέω (skopeein, to see), as in seeing the rotation of the Earth.

An 1882 portrait of French physicist and astronomer, Léon Foucault (1819–1868).

Before his gyroscope experiment, Foucault built a massive, long period, pendulum, that had its swing fixed in direction. Just as the motion of a falling body is independent of its mass, a pendulum's period is independent of mass.

Making the pendulum massive, however, helps to overcome friction forces at its pivot. The period is a function of the square root of the length for small swing amplitude.

(Wikimedia Commons image by Zátonyi Sándor.)

Foucault would have believed that the directions of his pendulum and gyroscope were fixed with respect to absolute space. The idea of an absolute space was proposed by Isaac Newton (1642-1727) in his Principia. The concept of an absolute space was challenged by physicist and philosopher, Ernst Mach, who is better known for his Mach number.

Mach, in his book, "The Science of Mechanics," proposed that instead of absolute space, that local physical laws are determined by the large-scale structure of the universe.[3] In particular, the inertia of a body arises from its interaction with everything else in the universe. This idea is now known as Mach's principle. In Mach's scheme, the rotation of a gyroscope is fixed with respect to the distant stars and galaxies.

An interesting case of rotation occurs in superconducting loops, where electron pairs circulate to produce a persistent current without an applied voltage. One important application for this is the superconducting electromagnet, but the phenomenon is used in the superconducting quantum interference device (SQUID), a sensitive magnetometer based on the Josephson effect that involves quantum tunneling through an insulating barrier. SQUIDs can detect extremely low magnetic fields, of the order of a few attotesla (a refrigerator magnet has a field of about 1016 attotesla).

Schematic diagram of a DC SQUID.

The applied magnetic flux Φ causes an imbalance in the circulating supercurrents in the two portions of the loop to cause a voltage to appear across the entrance and exit points of the loop.

(Modified Wikimedia Commons image by Miraceti.)

Physicists from Los Alamos National Laboratory, Los Alamos, New Mexico, and Miami University, Oxford, Ohio, have created an atomtronic version of a SQUID that's sensitive to mechanical rotation.[4-5] Just like a gyroscope, this atomtronic Superconducting Quantum Interference Device has the potential for sensing ultrasensitive rotation measurements.[5] The research team describes this device in an open access paper in a recent issue of Nature Communications.[4]

Says study author, Changhyun Ryu of the Los Alamos National Laboratory,
"In a conventional SQUID, the quantum interference in electron currents can be used to make one of the most sensitive magnetic field detectors... We use neutral atoms rather than charged electrons. Instead of responding to magnetic fields, the atomtronic version of a SQUID is sensitive to mechanical rotation."[5]

The atomtronic device consists of a layer of cold atoms created by a laser to form a Bose–Einstein condensate (BEC) with a second laser imprinting a pattern of two semicircles with Josephson junction gaps (see figure).[5] The circular pattern is about ten micrometers in diameter.[4-5] When the device is rotated, the number of atoms in each semicircle changes as a consequence of the phase mismatch of atomic flow caused by quantum effects. Rotation rate is measured by counting the number of atoms in each semicircle.[4-5]

Schematic diagram of the Los Alamos National Laboratory atomtronic SQUID. Clouds of atoms are trapped within each semicircle, and rotation causes them to quantum mechanically interfere to change the number of atoms in each segment. (Los Alamos National Laboratory image.)

References:

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