NOTES and DISCUSSIONS Using a Gyroscope to Find True North—A
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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 However, its behavior can be explained using physics con- cepts that are quite accessible to first year physics students. The curious behavior of a gyroscope never ceases to fasci- A description and mathematical analysis of a gyrocompass nate. It is the quintessential lecture demonstration whenever has been presented in a few papers appearing in this jour- examples of angular momentum are discussed. The gyro- nal6–9 and will not be repeated here. For example, Knudsen8 compass is but one example of its application. With conser- designed a gyrocompass for use in undergraduate instruc- vation of angular momentum in mind, most students tional physics labs, but it is quite complicated and not at all probably imagine a gyrocompass as simply a “directional” appropriate for classroom use. The purpose of this lecture gyroscope in the sense that a gyroscope (spinning freely in a demonstration is to show, in a direct and simple way, the gimbal mount) maintains its axis orientation regardless of remarkable behavior of this device. how it is moved around. They would probably be surprised to learn that if you constrain the rotational axis of a gyro- II. HOW A SIMPLE GYROCOMPASS WORKS scope to move in a horizontal plane, the axis will align itself with Earth’s meridian in a north-south direction—it seeks It is well known10 that the rate of precession of a gyro is out and indicates true geographic north and in no way directly proportional to the applied torque and inversely pro- depends upon Earth’s magnetic field! A true gyrocompass is portional to its angular momentum: xp ¼ s=L, where xp is designed to sense Earth’s rotation and that coerces the gyro the angular precession rate, s is the torque perpendicular to to orient its spin axis to be in the plane containing Earth’s the axle, and L is the angular momentum of the gyro. Note rotation axis. that xp represents a steady motion at right angles to the Shortly after his pendulum experiment in the Pantheon applied torque. What follows is a qualitative explanation. (1851), Foucault tried to show the rotation of Earth by The simplest gyrocompass is a spinning disk (gyro) with means of a gyroscope. He suspended a rapidly rotating disk, one degree of freedom—its axis of spin is constrained to lie mounted in gimbals, from a nearly torsionless filament, in a horizontal plane but is free to turn in that plane. From the axle being horizontal. The apparatus failed to give the the vantage point of an inertial observer in space, the hori- expected results, principally because he could not keep up the zontal plane rotates around with Earth. Imagine that the gyro rotation for a long enough time.1 The first practical gyrocom- is at the equator with its spin axis aligned in the east-west pass was invented around 1906 by Dr. Hermann Anschutz-€ direction, and suppose it is spinning in the clockwise (CW) Kaempfe in Germany. Soon afterward, in 1911, E. A. Sperry direction as viewed from the east side so that its angular put the first gyrocompass on the market in America. Its subse- momentum vector is pointing due west. As Earth rotates quent success is a tribute to the resourcefulness, ingenuity, toward the east, the west side of the horizontal plane will rise and engineering abilities of the many pioneer designers. (from the vantage point of an inertial observer) while the Readers interested in its historical development, patent bat- east side dips down. This change in the horizontal plane puts tles, etc., will find the book by Rawlings rich in details.2 a torque on the gyro. The vector representing this torque is A book edited by P. H. Savet3 provides a comprehensive parallel with Earth’s axis of rotation and points north. Thus, treatment of the art and science of gyroscopes in modern the torque will impart some angular momentum in the north- applications. ern direction, and the gyro’s angular momentum vector will Gyroscopes have been the subject of investigation in consequently move a little north of west (it will rotate CW as undergraduate labs for many decades. For example, the MIT viewed from above). This action persists as Earth continues freshman physics laboratories developed an air suspension to rotate until the direction of the gyro’s axis ends up aligned gyroscope for quantitative studies of precession.4,5 Modifying with the meridian in the north-south direction. At that point, and refining the MIT design, de Lange and Pierrus6 were able there will no longer be a torque on the axis (the perpendicu- to accurately measure the difference in the periods of clock- lar component of the torque varies as sin /, where / is the wise and counter-clockwise precessions, due to the effect angle between the gyro axis and meridian, see Fig. 1). If the of Earth’s rotation. It should be noted that the precession gyro overshoots, it will experience a counter-torque that behavior of a gyroscope is markedly different from that of a brings it back in alignment. gyrocompass, which executes azimuthal oscillations about Instead of being at the equator, suppose the gyro is located the north-south direction. A gyrocompass is a much more at some latitude h. The horizontal plane of the gyro will now complicated instrument in its design and fabrication (images be tilted at an angle h with respect to Earth’s rotation axis found on the web are testimony to that) and is probably the (see Fig. 1). This has the effect of varying the torque by reason why it is not used as a physics demonstration. cos h and becomes zero at the north pole.11 Thus, assuming 228 Am. J. Phys. 85 (3), March 2017 http://aapt.org/ajp VC 2017 American Association of Physics Teachers 228 the gyro’s angular momentum remains constant, the overall ¼ 390 g cm2 and its angular momentum is L ¼ Ix ¼ 490 3 2 decrease in torque on the gyro (and xp) will be due to a Â10 gcm/s at 12,000 rpm. combination of the two orientations and will depend on the product sin / cos h. B. Gyro support components We want to minimize the time it takes to respond to III. DESCRIPTION OF DEMONSTRATION Earth’s rotation so that the gyro settles down to its final ori- GYROCOMPASS entation long before its angular momentum has dropped to As mentioned, the simplest gyrocompass is a spinning marginal levels. Since the gyro is not continuously driven by a disk (gyro) whose axis of spin is constrained to lie in a hori- motor and slows down in a matter of minutes, it is imperative zontal plane but is free to turn in that plane. To that end, a that the entire support mechanism be light and possesses very little rotational inertia. The rotational inertia of the aluminum gyro (sans gimbal mount) is fixed to the middle of a round 2 “boat.” Floating the boat in a pan of water forces the spin casing of the gyroscope is quite small (I ¼ 176 g cm ). The axis to remain horizontal while allowing it to rotate freely in casing is supported by a 9-cm diameter thin wooden disk with similar inertia. The gyro is placed in the middle of a light that plane (see Fig. 2). If started with the gyro spin axis ori- (20 g) saucer13 that serves as a flat-bottom boat that floats on ented in an east-west direction, the boat will turn until the water. With a diameter of 25.4 cm, this saucer provides a stable axis is oriented north-south. The demonstration is simple, horizontal platform for the gyro. However, the saucer material but as is often the case the devil is in the details. To work is only 0.28 mm thick and transmits small vibrations from the reliably, a fine balance between gyro angular momentum, gyro to the water (one can see tiny ripples in the water emanat- overall rotational inertia, and viscous damping is required. ing from the edge of the saucer). So that vibrations from the gyro are not communicated to the boat and water, a piece of A. Gyro terrycloth separates the gyro from the saucer and damps out Since the magnitude of the directive torque (the torque these oscillations. The saucer, together with the terrycloth, makes up the greater portion of the rotational inertia, which in that turns the gyro) is proportional to the horizontal compo- 2 nent of the angular momentum of the gyro, the first require- total is 4040 g cm . ment is to use a gyro capable of providing significant angular momentum—“toy” gyros will not do. Second, because it C. Damping takes one or two minutes for the gyro to orient itself, it is In addition to supporting the gyro, the floatation fluid plays important to use a high-quality gyro, one that will spin at a an important role as a damping medium. It turns out that plain high speed for a sufficiently long time. To that end, the water offers much too little friction and the gyro significantly 12 Super Precision Gyroscope was chosen. It comes with a overshoots the north-south alignment. Ordinarily, it should battery-powered starter motor that spins it up to 12,000 rpm. experience a counter-torque when it overshoots. However, The rotational inertia of the gyro disk is I ¼ð1=2ÞmR2 the gyro has typically slowed down to the extent that there is not enough directive torque to bring it back.