Using a gyroscope to find true north—A lecture ...

Using a gyroscope to find true north—A lecture demonstration Wolfgang Rueckner

Citation: American Journal of Physics 85, 228 (2017); doi: 10.1119/1.4973118 View online: http://dx.doi.org/10.1119/1.4973118 View Table of Contents: http://aapt.scitation.org/toc/ajp/85/3 Published by the American Association of Physics Teachers

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NOTES AND DISCUSSIONS

Using a gyroscope to find true north—A lecture demonstration Wolfgang Ruecknera) Harvard University Science Center, Cambridge, Massachusetts 02138

(Received 26 August 2016; accepted 9 December 2016) [http://dx.doi.org/10.1119/1.4973118]

I. INTRODUCTION The curious behavior of a gyroscope never ceases to fascinate. It is the quintessential lecture demonstration whenever examples of angular momentum are discussed. The gyrocompass is but one example of its application. With conservation of angular momentum in mind, most students probably imagine a gyrocompass as simply a “directional” gyroscope in the sense that a gyroscope (spinning freely in a gimbal mount) maintains its axis orientation regardless of how it is moved around. They would probably be surprised to learn that if you constrain the rotational axis of a gyroscope to move in a horizontal plane, the axis will align itself with Earth’s meridian in a north-south direction—it seeks out and indicates true geographic north and in no way depends upon Earth’s magnetic field! A true gyrocompass is designed to sense Earth’s rotation and that coerces the gyro to orient its spin axis to be in the plane containing Earth’s rotation axis. Shortly after his pendulum experiment in the Pantheon (1851), Foucault tried to show the rotation of Earth by means of a gyroscope. He suspended a rapidly rotating disk, mounted in gimbals, from a nearly torsionless filament, the axle being horizontal. The apparatus failed to give the expected results, principally because he could not keep up the rotation for a long enough time.1 The first practical gyrocompass was invented around 1906 by Dr. Hermann Ansch€utzKaempfe in Germany. Soon afterward, in 1911, E. A. Sperry put the first gyrocompass on the market in America. Its subsequent success is a tribute to the resourcefulness, ingenuity, and engineering abilities of the many pioneer designers. Readers interested in its historical development, patent battles, etc., will find the book by Rawlings rich in details.2 A book edited by P. H. Savet3 provides a comprehensive treatment of the art and science of gyroscopes in modern applications. Gyroscopes have been the subject of investigation in undergraduate labs for many decades. For example, the MIT freshman physics laboratories developed an air suspension gyroscope for quantitative studies of precession.4,5 Modifying and refining the MIT design, de Lange and Pierrus6 were able to accurately measure the difference in the periods of clockwise and counter-clockwise precessions, due to the effect of Earth’s rotation. It should be noted that the precession behavior of a gyroscope is markedly different from that of a gyrocompass, which executes azimuthal oscillations about the north-south direction. A gyrocompass is a much more complicated instrument in its design and fabrication (images found on the web are testimony to that) and is probably the reason why it is not used as a physics demonstration. 228

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However, its behavior can be explained using physics concepts that are quite accessible to first year physics students. A description and mathematical analysis of a gyrocompass has been presented in a few papers appearing in this journal6–9 and will not be repeated here. For example, Knudsen8 designed a gyrocompass for use in undergraduate instructional physics labs, but it is quite complicated and not at all appropriate for classroom use. The purpose of this lecture demonstration is to show, in a direct and simple way, the remarkable behavior of this device. II. HOW A SIMPLE GYROCOMPASS WORKS It is well known10 that the rate of precession of a gyro is directly proportional to the applied torque and inversely proportional to its angular momentum: xp ¼ s=L, where xp is the angular precession rate, s is the torque perpendicular to the axle, and L is the angular momentum of the gyro. Note that xp represents a steady motion at right angles to the applied torque. What follows is a qualitative explanation. The simplest gyrocompass is a spinning disk (gyro) with one degree of freedom—its axis of spin is constrained to lie in a horizontal plane but is free to turn in that plane. From the vantage point of an inertial observer in space, the horizontal plane rotates around with Earth. Imagine that the gyro is at the equator with its spin axis aligned in the east-west direction, and suppose it is spinning in the clockwise (CW) direction as viewed from the east side so that its angular momentum vector is pointing due west. As Earth rotates toward the east, the west side of the horizontal plane will rise (from the vantage point of an inertial observer) while the east side dips down. This change in the horizontal plane puts a torque on the gyro. The vector representing this torque is parallel with Earth’s axis of rotation and points north. Thus, the torque will impart some angular momentum in the northern direction, and the gyro’s angular momentum vector will consequently move a little north of west (it will rotate CW as viewed from above). This action persists as Earth continues to rotate until the direction of the gyro’s axis ends up aligned with the meridian in the north-south direction. At that point, there will no longer be a torque on the axis (the perpendicular component of the torque varies as sin /, where / is the angle between the gyro axis and meridian, see Fig. 1). If the gyro overshoots, it will experience a counter-torque that brings it back in alignment. Instead of being at the equator, suppose the gyro is located at some latitude h. The horizontal plane of the gyro will now be tilted at an angle h with respect to Earth’s rotation axis (see Fig. 1). This has the effect of varying the torque by cos h and becomes zero at the north pole.11 Thus, assuming C 2017 American Association of Physics Teachers V

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the gyro’s angular momentum remains constant, the overall decrease in torque on the gyro (and xp ) will be due to a combination of the two orientations and will depend on the product sin / cos h. III. DESCRIPTION OF DEMONSTRATION GYROCOMPASS As mentioned, the simplest gyrocompass is a spinning disk (gyro) whose axis of spin is constrained to lie in a horizontal plane but is free to turn in that plane. To that end, a gyro (sans gimbal mount) is fixed to the middle of a round “boat.” Floating the boat in a pan of water forces the spin axis to remain horizontal while allowing it to rotate freely in that plane (see Fig. 2). If started with the gyro spin axis oriented in an east-west direction, the boat will turn until the axis is oriented north-south. The demonstration is simple, but as is often the case the devil is in the details. To work reliably, a fine balance between gyro angular momentum, overall rotational inertia, and viscous damping is required. A. Gyro Since the magnitude of the directive torque (the torque that turns the gyro) is proportional to the horizontal component of the angular momentum of the gyro, the first requirement is to use a gyro capable of providing significant angular momentum—“toy” gyros will not do. Second, because it takes one or two minutes for the gyro to orient itself, it is important to use a high-quality gyro, one that will spin at a high speed for a sufficiently long time. To that end, the Super Precision Gyroscope12 was chosen. It comes with a battery-powered starter motor that spins it up to 12,000 rpm. The rotational inertia of the gyro disk is I ¼ ð1=2ÞmR2

¼ 390 g cm2 and its angular momentum is L ¼ Ix ¼ 490 103 g cm2/s at 12,000 rpm. B. Gyro support components We want to minimize the time it takes to respond to Earth’s rotation so that the gyro settles down to its final orientation long before its angular momentum has dropped to marginal levels. Since the gyro is not continuously driven by a motor and slows down in a matter of minutes, it is imperative that the entire support mechanism be light and possesses very little rotational inertia. The rotational inertia of the aluminum casing of the gyroscope is quite small (I ¼ 176 g cm2). The casing is supported by a 9-cm diameter thin wooden disk with similar inertia. The gyro is placed in the middle of a light (20 g) saucer13 that serves as a flat-bottom boat that floats on water. With a diameter of 25.4 cm, this saucer provides a stable horizontal platform for the gyro. However, the saucer material is only 0.28 mm thick and transmits small vibrations from the gyro to the water (one can see tiny ripples in the water emanating from the edge of the saucer). So that vibrations from the gyro are not communicated to the boat and water, a piece of terrycloth separates the gyro from the saucer and damps out these oscillations. The saucer, together with the terrycloth, makes up the greater portion of the rotational inertia, which in total is 4040 g cm2. C. Damping In addition to supporting the gyro, the floatation fluid plays an important role as a damping medium. It turns out that plain water offers much too little friction and the gyro significantly overshoots the north-south alignment. Ordinarily, it should experience a counter-torque when it overshoots. However, the gyro has typically slowed down to the extent that there is not enough directive torque to bring it back. Adding glycerol to the water solves that problem. One can adjust the viscosity of the aqueous glycerol solution to provide the desired amount of friction (close to critical damping).14 D. Drift To prevent the boat from drifting to the walls of the container, the gyro can be persuaded to stay in the middle. The boat can be moved around by gently nudging the horizontal ring of the gyro’s casing with a small soft paintbrush.

Fig. 1. The geometry that defines the spin axis and local horizontal directions. 229

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Fig. 2. Gyrocompass with one degree of freedom. Notes and Discussions

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Pushing radially inward toward the center of mass results in translational motion without rotation. One can convince the students that this is so by navigating the boat around in this manner (without the gyro spinning) and observing that it does not rotate. IV. PERFORMING THE DEMONSTRATION Spin up the gyro up to 12,000 rpm and place it gently in the boat with its spin axis aligned in the east-west direction. Try to avoid imparting any rotation to the boat when releasing the gyro. A fine artist’s brush can be used to steady the boat. If oriented so that the gyro is spinning CW as viewed from the east, the boat will slowly turn CW (as viewed from above) until the spin axis is aligned in the north-south direction. This process takes 1–2 min (sometimes longer). You may have to gently coax the gyro to stay in the middle of the container while this is happening. If one starts with the gyro spinning CW as viewed from the west, the boat will turn CCW to line up in the north-south direction. Showing both cases makes for a convincing demonstration. In 2 min time Earth rotates about one half of a degree; it is truly remarkable that this simple gyro senses such a small movement. V. DISCUSSION The fact that the gyro always settles down in a particular orientation is quite satisfying, but is it truly lined up with Earth’s axis of rotation? A real gyrocompass is capable of indicating true north to within a small fraction of one degree.3 This experiment is far too crude to come even close to that kind of accuracy. If the direction of true north is not known, then a comparison with a magnetic compass could be made. Knowing your latitude and longitude, one can look up the declination of magnetic north for your locality.15 For example, a magnetic compass in Boston points about 14 west of true north (and changes by 4 min of arc toward the east each year). Unfortunately, steel beams, reinforced concrete, and metal furnishings in today’s modern buildings thwart efforts to determine magnetic north with any accuracy. To circumvent these difficulties, one might recruit a GPS device. Most cell phone compasses rely on measuring the magnetic field using Hall effect sensors, so those will not help in this situation. The author performed the experiment in the attic of a wood house and noted the gyro’s orientation using an ordinary compass. The gyrocompass indeed seeks out and settles down in the true north-south orientation, but is very sensitive to initial conditions. When released properly, the gyro will orient itself with the meridian with a reproducibility of 65 ; that is to say, sometimes it stops a little short of north and sometimes it overshoots a little. It takes some practice to release the gyro without imparting any rotation. The slightest rotation strongly influences the amount of time it takes to settle down. For example, it may take up to 8 min to settle if the gyro is initially biased (rotating) in the wrong direction. This extra time is detrimental to the experiment because the gyro’s rotation rate drops by about half in the first minute of spinning. Thereafter, it drops by approximately half every 2 min. After 5 min, its speed is only about 1/6 of its starting rpm. This reduction greatly impacts the directive torque on the gyro, which is already diminishing naturally (from the geometry) as the gyro gets closer to the meridian alignment. 230

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Consequently, the gyro’s final alignment may be off by as much as 615 . a)

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1

Richard F. Deimel, Mechanics of the Gyroscope (Dover, New York, 1950), p. 111. 2 A. L. Rawlings, The Theory of the Gyroscopic Compass and its Deviations, 2nd ed. (Macmillan, New York, 1944). 3 Gyroscopes: Theory and Design with Applications to Instrumentation, Guidance, and Control, edited by P. H. Savet (McGraw-Hill, New York, 1961). The eleven authors of this book participated for a considerable number of years in the birth and development of modern high-precision gyro instruments. Particularly recommended is Chapter 5, “The Gyrocompass,” written by C. T. Davenport. 4 Bernard H. Duane, “Air suspension gyroscope,” Am. J. Phys. 23, 147–150 (1955). A group of freshman physics students designed an air suspension gyroscope (magnetized steel sphere) for the quantitative study of gyroscopic precession. They reported that the rotation of Earth was discernible, but the effect almost lost in the noise. 5 Robert G. Marcley, “Air suspension gyroscope,” Am. J. Phys. 28, 150–155 (1960). An update on technique details and refinements of apparatus reported in Ref. 4, achieving 0.5% experimental error. 6 O. L. de Lange and J. Pierrus, “Measurement of inertial and noninertial properties of an air suspension gyroscope,” Am. J. Phys. 61, 974–981 (1993). The authors present a thorough and careful experimental study of the air suspension gyroscope. In agreement with their mathematical analysis, they find that the average period of precession is proportional to the gyro’s spin frequency. Furthermore, the periods of clockwise and counterclockwise precession differ slightly. The difference is due to Earth’s rotation (Coriolis force) and is proportional to the square of the spin frequency. 7 G. David Scott, “Precession of a gyro and model of a gyro-compass,” Am. J. Phys. 25, 80–82 (1957). Scott qualitatively explains the behavior of the gyroscope in terms of linear dynamics and discusses the north-seeking gyrocompass as an application. A small gyroscope combined with a model of Earth on a rotating platform is used to simulate the north-seeking behavior. Although not a real gyrocompass, it is a clever model that illustrates the basic principles as well as the gyro’s azimuthal oscillations about the north-south direction. 8 A. W. Knudsen, “A student’s gyrocompass,” Am. J. Phys. 41, 531–539 (1973). Knudsen designed a gyrocompass for use in an instructional physics lab. Delicate suspension bands support the gyro and a manual “followsteering” servo mechanism (which he describes as a bit tedious) is utilized to reduce the torsional stress and overall friction. An optical lever is employed to monitor the gyro’s motion and students measure the gyro’s 10-min long azimuthal oscillations over a period of 1 hour to determine gyroscopic (true) north from the mean position of these oscillations. 9 Geoffrey I. Opat, “Coriolis and magnetic forces: The gyrocompass and magnetic compass as analogs,” Am. J. Phys. 58, 1173–1176 (1990). Opat shows that the action of the Coriolis force on a rotating mass, and the action of the magnetic force on a rotating charge are formally identical. Quoting from his abstract, “Just as the action of a magnetic field is to align the axis of the rotating charge distribution (magnetic dipole) with itself, so the Coriolis force aligns the axis of a rotating mass distribution (angular momentum) with the angular velocity of the rotating frame.” Opat proposes that this analogy might enable the student to more easily understand the gyrocompass. He models the behavior of the gyrocompass by using the ubiquitous “spinning bicycle wheel on a rotating platform” demonstration. 10 € F. Klein and A. Sommerfeld, Uber die Theorie des Kreisels (B.G. Teubner/Johnson Reprint Corp., Stuttgart/New York, 1910/1965), p. 763. Much of the theory of the gyroscope has been covered in this monumental four-volume treatise. 11 Although a gyrocompass completely loses its north-seeking action at the poles, Davenport (Ref. 3, p. 79) states that it performs well even above 80 latitude, where a magnetic compass would be completely unusable because of the erratic nature of the variation error in that part of the world. 12 Available from for approximately $110 (without gimbals); details available online. The motor can be fastened to the gyroscope for continuous spinning at 12,000 rpm, but it was found that the motor and battery pack increased the rotational inertia of the supporting platform/boat far too much for this experiment. Notes and Discussions

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13

Ten-inch diameter clear vinyl flowerpot saucer (cost about $1 in most hardware stores). 14 Glycerol is about 1400 times more viscous than water and its viscosity goes down almost exponentially with the addition of water. A solution of 60% glycerol (by weight) yields a viscosity that is only about 11 times that of water alone; a 50% solution about 6 times. With the present set-up, a 55% solution seems to provide just the right amount of friction for critical damping. The “right amount” is something that needs to be determined empirically for the particular apparatus and might change with time. It is well known that solutions above approximately 85% glycerol are hygro-

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scopic whereas those below this concentration give up water when exposed to average humidity conditions. The solution’s stability with time should not be a problem if precautions are taken to obviate changes in concentration—keep the solution in a well-sealed container when not in use. In addition, temperature can be a factor; the viscosity of glycerol depends strongly on temperature. At the concentration of this experiment, a modest rise in temperature from 68 to 73 F (20–22.8  C) reduces the viscosity by 11%. 15 National Centers for Environmental Information, “A magnetic field declination calculator,” .

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