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Unformatted text preview: CHAPTER GYROSCOPES To those who study the progress of exact science, the common spinning
top is a symbol of the labours and the perplexities of men who had
successfully threaded the mazes of planetary motions. The mathematicians of
the last age, searching through nature for problems worthy of their analysis,
found in this toy of their youth ample occupation for their highest mathematical
powers. No illustration of astronomical precession can be devised more perfect
than that presented by a properly balanced top, but yet the motion of rotation
has intricacies far exceeding those of the theory of precession. James Clerk Maxwell, “On a Dynamical Top" (1857) 24.1 AN ANCIENT QUESTION In ancient times, people much more familiar with the night sky than we are helped
themselves memorize its configurations by seeing heroes and creatures in clusters of stars.
The constellations were patterns formed by stars fastened to a great sphere that surrounded
the earth and formed the boundary of the universe. This celestial globe rotated on an axis
through the earth, causing the stars to move along cirCular paths across the sky. 457 458 GYROSCOPES Figure 24.1 Time exposure photograph of the night sky when a
camera is pointed at the North Star. (Lick Observatory Photograph.) Likewise the lifegiving sun was fixed to its own sphere whose rotation made the
sun seem to travel across the sky each day, rising in the east and setting in the west. But
unlike the stars, the sun gradually changed its path each day, rising and setting more
northerly in the summer and more southerly in the winter. Ancient astronomers accounted
for the yearly motion of the sun by an extra annual rotation of the sun’s sphere about an
axis tilted by 235°. They named the plane of the sun’s orbit the ecliptic, and the zodiac
was the circular 200 of constellations through which the sun moved on its yearly journey
around the stationary earth. Figure 24.2 illustrates the ecliptic and zodiac. Twice a year, once in the spring and once in the autumn, the orbit of the sun passes
through the equatorial plane of the earth. On these dates, called the vernal and autumnal
equinoxes, the sun rises due east and sets due west, and the lengths of day and night are
equal. The points were marked by the position of the sun in the zodiac. Through amazingly careful observations in the second century B.C. the Greek as
tronomer Hipparchus discovered that the position of the equinoxes in the zodiac slowly
drifts westward. For this phenomenon, called the precession of the equinoxes, he reported
a value of 36 seconds of are per year. But to him the precession of the equinoxes merely
represented an empirical fact, like so many astronomical facts, which had to be considered
in the compilation of calendars for planting and harvesting. Not until 1543 was an explanation proposed for the precession of the equinoxes. In
his book De Revolutionibus, Copernicus conjectured that the earth orbits the sun and that
the earth spins on an axis tilted by 23%o to the ecliptic. In his model of the universe, the
precession of the equinoxes was due to the fact that the axis of the earth slowly traces
out a circle, as shown in Fig. 24.3. As the axis shifts, the star it points at, the pole star,
also seems to drift until it’s replaced by another. Copernicus concluded that the preces
sional rate is about 52 seconds of are per year, which corresponds to a precessional period
of 26,000 years. By the time of Newton, the descriptive account of the precession of the equinoxes
was well documented. However, a physical cause remained a mystery until Newton
himself solved this great astronomical problem, the key to which lies in the gyroscope. 24.2 THE GYROSCOPE 459 Celestial
equator Figure 24.3 Precession of the equinoxes is caused by the rotation of
the earth’s axis. 24.2 THE GYROSCOPE Before we can explain the precession of the equinoxes, we first need to understand the
underlying physics of a type of motion called gyroscopic precession. The simplest gy
roscope is a wheel which is free to spin on an axle, which we’ll call the spin axis. A spinning top is another example of a gyroscope.
Figure 24.4a shows a nonspinning bicycle wheel of mass M at the center of a horizontal axle OP of negligible mass along the x axis. Initially the axle is supported at
both endpoints O and P. If the support at P is removed, the torque due to the weight Mg
causes the wheel to fall and, in so doing, to rotate counterclockwise about the y axis as
shown in Fig. 24.4b, where the wheel is viewed from along the positive y axis. 460 GYROSCOPES (b) Figure 24.4 (a) Axle supported at both ends. (b) Support at P
removed. If the same wheel is set spinning rapidly on its axle, as indicated in Fig. 24.5, an
extraordinary phenomenon occurs when the support is removed at P. The wheel does not
fall as before but instead the axle remains almost horizontal and begins to revolve, or
precess, about the x axis, as shown in Fig. 24.5. This apparently paradoxical motion,
called gyroscopic precession, can be explained by changes in angular momentum. precession Figure 24.5 A rapidly spinning wheel does not fall but exhibits
precession. As the nonspinning wheel in Fig. 24.4 begins to topple we can calculate its torque
about 0 as follows. Let ai denote the vector from O to the center of the wheel. When
the support at P is removed the torque has initial value 1 = (ai) x (—Mgli) = aMgaE x i) = ang. When the axle has dropped through an angle 9 from the x axis, as in Fig. 24.6, where
0 S 6 S 11/2, the corresponding torque about 0 is T = [a(cos int] x (—Mgﬁ) = aMg(cos (9)3. This torque is the rate of change of some angular momentum vector L0, 242 THE GYROSCOPE 461 Figure 24.6 Torque about 0 when axle drops through angle 6. where L0 has the same direction as 1' (parallel to the plane of the wheel) since L0 = 0
when t = 0. Now suppose you start the wheel spinning with a large constant angular speed in the
counterclockwise direction, as shown in Fig. 24.5, before the support at P is removed.
This gives the wheel an amount of angular momentum about the center which we denote
by L, a vector of constant length. We call this the spin angular momentum of the wheel.
The faster the wheel spins the longer the vector L. Let’s analyze what happens now if
you remove the support at P. The spin angular momentum L plus the angular momentum L0 due to the weight
Mg have vector sum L + L0. If the wheel spins rapidly enough, L0 will be very small
compared to L. For the moment we neglect L0 and assume all the angular momentum
is represented by the spin vector L. Gravity still exerts a torque 1' about the point 0 and
this torque is parallel to the plane of the wheel. The torque is related to L by the equation dL
= — 23.7
T dt ( ) where now L is perpendicular to 1' because a vector of constant length is always per—
pendicular to its derivative. On the other hand, L is always directed along the axle and,
since the torque dL/dt is not zero, L must change its direction, so L moves in a circle.
As the axle moves about this circle in response to the torque, the direction of L changes
and because of Eq. (23.7) the direction of 1 changes as well. This is described by saying
that L tries to follow 1', which leads it around in a circle, as suggested in Fig. 24.7. The
resulting motion is uniform gyroscopic precession. 462 GYROSCOPES / X Figure 24.7 Angular momentum vector L tries to follow torque
vector 1. Example 1
A gyroscope with angular momentum L is spinning in space, where it is acted on by two
equal but opposite forces F, shown in the diagram, equidistant from the center of mass at O. (a) What is the torque acting on the gyroscope about point 0?
(b) For the instant shown in the diagram, in what direction is endpoint A moving as a
result of precession? (a) From 1' = r x F, we can find the direction of the torque created by each force
using the “righthand rule. Gravity creates no torque about 0. Each force F creates a
torque rF k in the z direction. Therefore the total torque is 1' = 2rFl:L (b) Since the angular momentum vector L tries to follow the torque, endpoint A
(along with L) actually moves upward as a result of the horizontal forces! This precessional
motion is such that L describes a circle in the yz plane; so A moves counterclockwise
when viewed from the positive x axis. 242 THE GYROSCOPE 463 Note that although the net force acting on the gyroscope is zero, the net torque is
nonzero. A pair of forces equal in magnitude and oppositely directed are known as a
couple; they produce rotation without acceleration of the center of mass. Now that we have some idea of the cause of gyroscopic motion, let’s consider the
angular momentum once more as well as energy associated with this motion. In the foregoing description of gyroscopic precession we neglected L0 and assumed
that all the angular momentum of the gyroscope is purely spin momentum lying in the
horizontal xy plane. But when you remove the support at P of a real gyroscope as in Fig.
24.4a you observe that the axle actually tilts down a bit, so L now has a vertical component
—L2 as well as horizontal components. Since there is no torque to produce angular
momentum in the z direction, any angular momentum in that direction must be conserved.
How does this take place? It is the gyroscopic precession itself that creates the angular momentum that balances
the downward vertical component ~LZ. The resulting rotation of the axle about the z
axis is directed so that the angular momentum associated with it is in the positive 2
direction. The energy of the gyroscope must also be conserved. The precession of the gyro has
a kinetic energy associated with it. That kinetic energy must come from somewhere, and
the source is a change in gravitational potential energy of the center of mass. When the
center of mass drops slightly as the gyro initially falls, the gravitational potential energy
of the center of mass decreases. That decrease in potential energy appears as kinetic
energy of the precession of the gyroscope. The simple gyroscope displays even more intricacies in its motion. Consider what
happens to the center of mass. Originally, the center of mass lies in the xy plane of Fig.
24.5. Once the gyro is released, the center of mass falls a bit as precession begins. The
new (lower) height of the center of mass is one of stable equilibrium. But the gyroscope
didn’t start off in a stable equilibrium position. We recall from Chapter 14 that a system
slightly disturbed from its stable equilibrium position oscillates. That’s what happens to
the center of mass of the gyroscope: the center of mass oscillates about the stable equi
librium position. Figure 24.8 illustrates the potentialenergy curve as a function of the
angle of precession. This oscillatory motion which takes place in addition to the precession
is called nutation (after the Latin word for nodding). Figure 24.9 illustrates the nutation
we’ve just described, where the curved, cusplike path represents the motion of the center
of mass. Eventually these oscillations are damped out by friction at the pivot point and
air resistance and the gyro’s motion turns into uniform precession. ————————————————— starting point equilibrium 0 (angle of precession)
Figure 24.8 Potentialenergy curve for a nutating gyroscope. 464 GYROSCOPES Figure 249 In nutation, the center of mass of a gyroscope traces a
path like the one shown. Let’s brieﬂy recapitulate the motion of a gyroscope. There are three different kinds
of motion simultaneously occurring. First, there is rotation about the gyro’s spin axis;
this spin is at some angular velocity which we’ll call 0). Second, there is precession
about the point of suspension. The precession has a different angular velocity which we’ll
call 0 (capital omega), the angular velocity of precession. The third kind of motion is
the oscillation about the direction of precession, and that’s called nutation. Example 2 Suppose that instead of releasing one end of a spinning horizontally oriented gyroscope,
you impart a slight horizontal velocity to the end as you release it. Describe the nutation
you would expect. By imparting a slight velocity to the gyroscope when releasing it, you are adding
kinetic energy to it. At first, you might think that this added energy will become the
needed kinetic energy of precession. However, the gyroscope still must conserve angular
momentum in the vertical direction, and the only way it can do that is to drop down a
little. Therefore the gyroscope will nutate for reasons cited above. If the initial velocity
is in the same direction as the precession, then the nutation will consist of oscillations
about the stable equilibrium position which are “stretched out” more along the path of
the center of mass, as illustrated below: Questions 1. Suppose you had two gyroscopes of identical dimensions with one twice as massive
as the other. Would you expect the rate of precession of the more massive gyro
scope to be greater than, less than, or equal to that of the lighter one? Why? 24.2 THE GYROSCOPE 465 2. Consider the two gyroscopes shown in the diagram. They have equal masses, but
the center of mass of gyroscope A is twice as far from the suspension point as that
of B. How would their angular velocities of precession compare? [email protected] 3. The angular momentum of the gyroscope shown below points along the negative y
axis. If the support at B is removed, describe the precessional motion. 4. Imagine a gyroscope spinning in outer space. At one instant, its intrinsic angular
momentum is directed along the negative x axis as shown in the diagram. Two
equal but oppositely directed forces act at ends A and B. From the resulting preces
sion, end A is moving in the xy plane in the positive y direction. (a) In what direction is the torque about 0 acting on the gyroscope?
(b) In what direction are the forces at A and B acting? 5. Shown below is a photograph of the motion of the end of a nutating gyroscope
taken with a stroboscope. As the end moves around the loops, it doubles back as From Kleppner, D., and Kolenkow, R. J. An Introduction to Mechanics. McGraw—Hill Book Co.,
New York (1973). By permission of the publisher. 466 GYROSCOPES the dots show. This nutation was achieved by impartng a small velocity to the
end of the gyroscope as it was released. Relative to the way the end is precessing,
in what direction was the initial velocity? Explain your reasoning. 6. Everyone knows that to turn a bicycle to the right, you lean in that direction. In
terms of torque and precession of the bicycle wheel explain how leaning makes the
bicycle turn. 24.3 THE GYROCOMPASS Another simple and initially surprising behavior of gyroscopes makes them useful in
navigation. Suppose you have a toy gyroscope (you should try this experiment for yourself)
and tie strings to the frame at points A and B on oppostie sides midway between the
bearings, as shown in Fig. 24.10. With the gyroscope spinning, suppose that you hold
the strings taut at arm’s length with the spin axis horizontal. Now if you slowly pivot to
your left arm so that the gyroscope moves in a circle with arm’s length radius, what
happens? Surprisingly, the gyroscope suddenly ﬂips over and tilts out of the horizontal
plane. After a few oscillations, which are damped out by friction, the spin axis of the
gyroscope comes to rest with its axis vertical, parallel to your axis of rotation. Figure 24.10 A gyroscope moved in a circle ﬂips over. The gyrocompass is based on this effect. At the heart of a gyrocompass is a gyroscope
which has an axis of rotation which is itself free to rotate about the horizontal axis. Figure
24.11 illustrates a gyrocompass; the outer gimbal allows the inner gimbal and spin axis
to rotate (taking the place of the strings in our experiment). We can understand the behavior of a gyrocompass by simple vector arguments. When the outer frame is rotated, the pivots on which the inner gimbal is mounted
provide a torque. Let’s find the direction of this torque. Rotating the outer frame in a
circle, that is, turning it about the z axis, produces horizontal forces at the gimbal mounts,
as shown in Fig. 24.11. These forces create a torque 1' in the z direction. Since torque
is equal to the change in angular momentum, dL
1' = — ,
dt
the spin angular momentum moves in the direction of the torque and swings toward the z direction. This motion is precession, with the torque now provided not by gravity but
by the gimbal pivots. (23.7) 24.3 THE GYROCOMPASS 467 Figure 24.11 A gyrocompass consists of a gyroscope mounted on a
gimbal mounted inside another gimbal which allows rotation of the
gyroscope about a horizontal axis. We can understand why the effect is so pronounced by considering angular momentum
in the y direction of Fig. 24.12. An attempt to rotate the spin axis of the gyroscope in
the horizontal plane would, by itself, introduce a component of angular momentum in
the y direction. However, the pivots at A and B allow the gyroscope to rotate freely about
the y axis, so angular momentum in that direction must be conserved. Since the angular
momentum along the y direction is initially zero, it must remain zero. Now as the
gyrocompass begins to rotate about the z axis, the spin angular momentum L begins to
point slightly in the y direction. At the same time, the gyroscope and its mount begin to
tilt and rotate about the y axis. The angular momentum arising from motion is in the
negative y direction and exactly cancels out the component of L in the y direction; this Figure 24.12 Torque creating a rotation of a gyrocompass about a
vertical axis causes the spin axis to rotate about a horizontal axis. 468 GYROSCOPES way angular momentum in the y direction remains zero. When L finally comes to rest
in the z direction, parallel to the axis of rotation, the motion of the frame no longer
changes the direction of L, so the spin axis remains stationary in the z direction. Example 3
Suppose a gyrocompass frame is slowly rotated in a horizontal plane. How does the rate
at which the spin axis ﬂips over depend on the magnitude of the spin angular momentum?
From the discussion in the text and Fig. 24.12, we see that rotating the gyrocompass
causes the spin angular momentum to acquire a y component. Since angular momentum
in the y direction is conserved and is initially zero, the ﬂipping of the spin axis about
the y axis creates the necessary angular momentum to cancel out the newly acquired y
component of the spin angular momentum L. The greater the magnitude of L, the faster
the gyroscope and its mount must rotate about the y axis. Therefore, the faster the
gyrocompass is spinning, the faster it will ﬂip over as it is rotated in a horizontal plane. The rotation of the earth causes a torque like that described above on a gyroscope
mounted and spinning in a horizontal plane. The gyroscope would, therefore, slowly ﬂip
until its axis is pointing parallel to true north as shown in Fig. 24.13. After the spin axis
is pointed northward, it will continue to act as a compass indicating that direction. A ‘
gyrocompass is a nonmagnetic compass. Earth’s
rotation .{}_ 3
Figure 24.13 Precession of a gyroscope on the rotating surface of the
earth. There is another way in which gyroscopes can be used to aid navigation. It depends
on the fact that a spinning gyroscope, carefully balanced ”and mounted to eliminate torques,
maintains its spin axis in a fixed direction in absolute space. This application of gyroscopes
is called inertial guidance. One important application of inertial guidance is in aircraft navigation. A gyroscope
can give a constant indication of a chosen direction. Another gyroscope which precesses
about the vertical direction under the torque provided by gravity can be used to identify
the true vertical and so define an artificial horizon against which the angles of banks and
climbs can be measured. Inertial guidance is also indispens...
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 Fall '07
 Goodstein

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