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 10 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. 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 con- tainer, 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. Fig. 2. Gyrocompass with one degree of freedom.
229 Am. J. Phys., Vol. 85, No. 3, March 2017 Notes and Discussions 229 Pushing radially inward toward the center of mass results in Consequently, the gyro’s final alignment may be off by as translational motion without rotation. One can convince the much as 615 . students that this is so by navigating the boat around in this manner (without the gyro spinning) and observing that it a)Electronic mail: [email protected] does not rotate.
1 IV. PERFORMING THE DEMONSTRATION Richard F. Deimel, Mechanics of the Gyroscope (Dover, New York, 1950), p. 111. Spin up the gyro up to 12,000 rpm and place it gently in 2A. L. Rawlings, The Theory of the Gyroscopic Compass and its the boat with its spin axis aligned in the east-west direction. Deviations, 2nd ed. (Macmillan, New York, 1944). 3Gyroscopes: Theory and Design with Applications to Instrumentation, Try to avoid imparting any rotation to the boat when releas- Guidance, and Control, edited by P. H. Savet (McGraw-Hill, New York, ing the gyro. A fine artist’s brush can be used to steady the 1961). The eleven authors of this book participated for a considerable boat. If oriented so that the gyro is spinning CW as viewed number of years in the birth and development of modern high-precision from the east, the boat will slowly turn CW (as viewed from gyro instruments. Particularly recommended is Chapter 5, “The Gyrocompass,” written by C. T. Davenport. above) until the spin axis is aligned in the north-south direc- 4 tion. This process takes 1–2 min (sometimes longer). You Bernard H. Duane, “Air suspension gyroscope,” Am. J. Phys. 23, 147–150 (1955). A group of freshman physics students designed an air suspension may have to gently coax the gyro to stay in the middle of the gyroscope (magnetized steel sphere) for the quantitative study of gyro- container while this is happening. If one starts with the gyro scopic precession. They reported that the rotation of Earth was discernible, spinning CW as viewed from the west, the boat will turn but the effect almost lost in the noise. CCW to line up in the north-south direction. Showing both 5Robert G. Marcley, “Air suspension gyroscope,” Am. J. Phys. 28, cases makes for a convincing demonstration. In 2 min time 150–155 (1960). An update on technique details and refinements of appa- ratus reported in Ref. 4, achieving 0.5% experimental error. Earth rotates about one half of a degree; it is truly remark- 6O. L. de Lange and J. Pierrus, “Measurement of inertial and noninertial able that this simple gyro senses such a small movement. properties of an air suspension gyroscope,” Am. J. Phys. 61, 974–981 (1993). The authors present a thorough and careful experimental study of V. DISCUSSION the air suspension gyroscope. In agreement with their mathematical analy- sis, they find that the average period of precession is proportional to the The fact that the gyro always settles down in a particular gyro’s spin frequency. Furthermore, the periods of clockwise and counter- orientation is quite satisfying, but is it truly lined up with clockwise precession differ slightly. The difference is due to Earth’s rota- tion (Coriolis force) and is proportional to the square of the spin Earth’s axis of rotation? A real gyrocompass is capable of frequency. indicating true north to within a small fraction of one 7G. David Scott, “Precession of a gyro and model of a gyro-compass,” Am. 3 degree. This experiment is far too crude to come even close J. Phys. 25, 80–82 (1957). Scott qualitatively explains the behavior of the to that kind of accuracy. If the direction of true north is not gyroscope in terms of linear dynamics and discusses the north-seeking known, then a comparison with a magnetic compass could gyrocompass as an application. A small gyroscope combined with a model be made. Knowing your latitude and longitude, one can look of Earth on a rotating platform is used to simulate the north-seeking behav- 15 ior. Although not a real gyrocompass, it is a clever model that illustrates up the declination of magnetic north for your locality. For the basic principles as well as the gyro’s azimuthal oscillations about the example, a magnetic compass in Boston points about 14 north-south direction. west of true north (and changes by 4 min of arc toward the 8A. W. Knudsen, “A student’s gyrocompass,” Am. J. Phys. 41, 531–539 east each year). Unfortunately, steel beams, reinforced con- (1973). Knudsen designed a gyrocompass for use in an instructional phys- crete, and metal furnishings in today’s modern buildings ics lab. Delicate suspension bands support the gyro and a manual “follow- steering” servo mechanism (which he describes as a bit tedious) is utilized thwart efforts to determine magnetic north with any accu- to reduce the torsional stress and overall friction. An optical lever is racy. To circumvent these difficulties, one might recruit a employed to monitor the gyro’s motion and students measure the gyro’s GPS device. Most cell phone compasses rely on measuring 10-min long azimuthal oscillations over a period of 1 hour to determine gyroscopic (true) north from the mean position of these oscillations. the magnetic field using Hall effect sensors, so those will not 9 help in this situation. The author performed the experiment Geoffrey I. Opat, “Coriolis and magnetic forces: The gyrocompass and magnetic compass as analogs,” Am. J. Phys. 58, 1173–1176 (1990). Opat in the attic of a wood house and noted the gyro’s orientation shows that the action of the Coriolis force on a rotating mass, and the using an ordinary compass. action of the magnetic force on a rotating charge are formally identical. The gyrocompass indeed seeks out and settles down in the Quoting from his abstract, “Just as the action of a magnetic field is to align true north-south orientation, but is very sensitive to initial the axis of the rotating charge distribution (magnetic dipole) with itself, so conditions. When released properly, the gyro will orient the Coriolis force aligns the axis of a rotating mass distribution (angular momentum) with the angular velocity of the rotating frame.” Opat pro- itself with the meridian with a reproducibility of 65 ; that is poses that this analogy might enable the student to more easily understand to say, sometimes it stops a little short of north and some- the gyrocompass. He models the behavior of the gyrocompass by using the times it overshoots a little. It takes some practice to release ubiquitous “spinning bicycle wheel on a rotating platform” demonstration. the gyro without imparting any rotation. The slightest rota- 10F. Klein and A. Sommerfeld, Uber€ die Theorie des Kreisels (B.G. tion strongly influences the amount of time it takes to settle Teubner/Johnson Reprint Corp., Stuttgart/New York, 1910/1965), p. 763. down. For example, it may take up to 8 min to settle if the Much of the theory of the gyroscope has been covered in this monumental four-volume treatise. gyro is initially biased (rotating) in the wrong direction. This 11Although a gyrocompass completely loses its north-seeking action at the extra time is detrimental to the experiment because the poles, Davenport (Ref. 3, p. 79) states that it performs well even above 80 gyro’s rotation rate drops by about half in the first minute of latitude, where a magnetic compass would be completely unusable because of the erratic nature of the variation error in that part of the world. spinning. Thereafter, it drops by approximately half every 12 2 min. After 5 min, its speed is only about 1/6 of its starting Available from
230 Am. J. Phys., Vol. 85, No. 3, March 2017 Notes and Discussions 230 13Ten-inch diameter clear vinyl flowerpot saucer (cost about $1 in most scopic whereas those below this concentration give up water when hardware stores). exposed to average humidity conditions. The solution’s stability with time 14Glycerol is about 1400 times more viscous than water and its viscosity should not be a problem if precautions are taken to obviate changes in con- goes down almost exponentially with the addition of water. A solution of centration—keep the solution in a well-sealed container when not in use. 60% glycerol (by weight) yields a viscosity that is only about 11 times that In addition, temperature can be a factor; the viscosity of glycerol depends of water alone; a 50% solution about 6 times. With the present set-up, a strongly on temperature. At the concentration of this experiment, a modest 55% solution seems to provide just the right amount of friction for critical rise in temperature from 68 to 73 F (20–22.8 C) reduces the viscosity by damping. The “right amount” is something that needs to be determined 11%. empirically for the particular apparatus and might change with time. It is 15National Centers for Environmental Information, “A magnetic field decli- well known that solutions above approximately 85% glycerol are hygro- nation calculator,”
231 Am. J. Phys., Vol. 85, No. 3, March 2017 Notes and Discussions 231