# The Mystical, Mythical, Magical Gyroscope Demonstration

This page discusses the March 15, 2011 Fun Physics Facts for the Family Demonstration...

• In the first demonstration (Feb 15, 2011) we examined the principle of Conservation of Angular Momentum, and showed how a body will spin in the same orientation unless some external force is applied. Here is a link to the wiki page for this demonstration: FPF-01 Astronaut: Turn Around? .
• In the second demonstration (Mar 15, 2011) we studied the gyroscope, the embodiment of angular momentum, and discussed a simple Newton's Law explanation (here is a link to the wiki page discussing Newton's Laws: FPF Newton's Laws ) for its intriguing behaviors, such as precessional motion. Here is a link to the wiki page for this demonstration: FPF-02 Mystical, Mythical, Magical Gyroscope .
• In the third demonstration (Apr 12, 2011) we continued the examination of the properties of gyroscopes and show how a very clever trick can be employed to produce a compass that always points to the true North direction and always works at all locations and in all vessels. Here is a link to the wiki page for this demonstration: FPF-03 The Most Marvelous Mechanical Machine .
• In the fourth demonstration (May 10, 2011) we yet again continue the study of the magical behaviors of the gyroscope, only this time since we now understand the mechanism underlying these mystical behaviors we endeavor to dispel some of the mythical behaviors attributed to gyroscopes. In particular, a famous British engineer by the name of Eric Laithwaite claimed antigravity capabilities for gyroscopes leading to a plethora of similar claims, and we will debunk these claims. Here is a link to the wiki page for this demonstration: FPF-04 Anti-Antigravity: Eric Laithwaite .

• ## The Mystical, Mythical, Magical Gyroscope: March 15, 2011

Doesn't it appear that gyroscopes violate the laws of physics?!? Here is a fun, intuitive model of the gyroscope that predicts and explains all of its qualitative behaviors...
 Click for full-size image

The Mystical, Mythical, Magical Gyroscope

 Click for full-size image

While the gyroscopic mechanism was first seen in 1817 and then again in 1833, called the rotascope, Leon Foucault built and named the gyroscope in 1852 in order to demonstrate that the Earth rotated. This photograph shows what I believe is an 1867 reproduction of Foucault's original (Musee des artes et metiers, Paris).

Now gyroscope's display amazing behaviors, behaviors that at first appear to violate the principles of physics, such as illustrated in the following photograph. How is it possible that the gyroscope does not fall?

The following explanation is a simple application of Newton's First Law of Motion to the gyroscope. It is not found in very many places. In fact, all of the following physics texts do not talk about this mechanism, and I even tried Wikipedia (word of caution, though, in my experience I have noticed a number of mistakes on Wikipedia's articles on mathematics and physics topics). One of these books has a simple explanation based upon the Coriolis Force, but in my opinion the Coriolis Force is not all that well understood either and the following Newton's First Law explanation also works to explain the genesis of the Coriolis Force.

 Click for full-size image
All of these physics books discuss gyroscopes or spinning tops, but none of them give the following simple Newton's Law explanation.

Thus I rather like the following, simple, Newton's Law explanation. It works to explain the qualitative behaviors of a gyroscope, such as the direction of precession. I know of only a single case where this explanation fails, and that is the case when the angular momentum is zero. Recall angular momentum from last month's lecture, it is the momentum generated by a rotating object. When the angular momentum is zero, the following argument does not work, which makes a certain amount of sense. In any case, let's get started.

If we hang a non-spinning gyroscope from a string around one of its ends, the force of gravity (cyan arrow) on the massive rotor will cause the gyroscope to rotate around in a vertical plane and fall from the string loop, just as we expect. But when the rotor is spinning, the gyroscope does not rotate around its supporting string loop, as we also would expect! Rather the gyroscope ends up rotating slowly around its support in the horizontal plane, not the vertical plane. Thus the spinning gyroscope does not fall.

Doesn't this appear to violate physics and our everyday experiences? Yes it does. But, indeed, the gyroscope is following Newton's First Law of Motion, the one that states that a body in motion will stay in motion unless acted upon by an unbalanced force. Let's see how this works...

In Part A of the above diagram, we see that two small voxels, or mass particles, on the rim of the spinning rotor (the red and blue dots) would tend to move in the directions shown by the red and blue arrows as the gyroscope fell from the force of gravity (cyan arrow). But a fraction of a second later the rotor spins around a quarter of a revolution as shown in Part B. Notice that, by Newton's First Law, the red voxel continues to move to the right (red arrow) and the blue voxel continues to move to the left (blue arrow). These two motions, red right, blue left, would cause the gyroscope to twist around in the horizontal plane, in the direction shown by the green arrow.

Next we consider what happens after another fraction of a second when the rotor has spun a second quarter of a turn, as shown in Part C. Now the red voxel is at the bottom and its initial Newton's Law motion to the right (red arrow) is being countered by the blue motion resulting from the tendency to fall caused by the gravity force (cyan arrow). The motion to the right (Newton's Law) thus cancels the motion to the left (gravity). In this diagram C, the red and blue arrows cancel one another, leaving no motion in the vertical plane. In other words, the gyroscope does NOT fall! Rather the "falling" motion is converted by the spin into the twisting around in the horizontal plane (Part B), known as precession, and shown in Part D.

Therefore this Newton's First Law argument not only explains why the gyroscope does not fall, but it also explains why the gyroscope precesses and predicts the correct direction of precession.
The following video (Gyroscope-Newtons_Law_Explanation_Part_I-1080p.mov) explains and demonstrates the Newton's Law Explanation for the precessional motion of a gyroscope. This explanation predicts the directions of the precession when a torque is applied to the gyroscope's axle as well as explains why the rotor does not simply "fall over" under the influence of gravity.
(YouTube encodes the video at a number of different resolutions.)
Description of the video's experiments: This video discusses the Newton's Law Explanation, based upon Newton's First Law of Motion, for the precessional motion of a gyroscope. It then demonstrates that this explanation predicts the correct direction of the precession as well as the fact that the gyroscope does not "fall over".

In fact, there is an alternative view of the precession of a gyroscope that is sometimes useful. Consider that the torque applied to the gyroscope's axle is along some direction to the axle, like the vertical force of gravity. The gyroscope does not respond in this direction, however, rather it precesses in a direction perpendicular to the original torque, like the horizontal direction of the precession. In other words, it appears as if the gyroscope "reflects" the input force resulting in an output force at 90 degrees to the input.

We can even push this Newton's Law argument a little further to say that if the rotor is spinning very fast, then the red voxel at the top of the rotor has very little time to accelerate to the right and thus its velocity to the right will not be very large. In fact, the faster the spin rate of the rotor, the shorter the time interval that the red voxel "falls" and consequently the slower the red voxel at the top moves to the right. A slower right velocity for the red voxel means that after a quarter revolution the horizontal twisting motion illustrated in Part B above will also be slower. Thus we predict that the faster the rotors spins, the slower the gyroscope precesses, and vice versa.

Also from Part A in the above Figure we notice that the "falling" of the red voxel is caused by the torque generated by the force of gravity (the mg cyan arrow). By Newton's Second Law, if the force is greater then the acceleration would also be greater. A greater acceleration would lead to a greater right-directed velocity (red arrow) for the red voxel at the top of the rotor. And a greater velocity, after a quarter revolution, would yield a faster precession. Thus we also predict that the greater the torque on the gyroscope's axle, the faster the precessional motion.

Let's see if this is all true experimentally.

(At this point I demonstrate with two gyroscopes that indeed a faster spin causes a slower precession, and a greater torque produces a faster precession.) Here is a video (Gyroscope-Precession-1080p.mov) demonstrating that a greater torque produces a faster precession and that a faster rotor spin rate yields a slower precession.
Click on the following link to view the video on YouTube: http://youtu.be/dwU3veKRH88
(YouTube provides several alternative resolutions for the video.)
Description of the video's experiment: This experiment characterizes the functional dependency of the precession rate on both the torque and the rotor spin rate. In words, a greater torque produces a faster precession, and a slower rotor spin rate yields a faster precession rate. The precession rate is found to be directly proportional to the torque and inversely proportional to the rotor spin rate (see the equation below).

Now the theoretical equation for the precession rate is

Here r_CMmg is the radius arm r_CM times the gravity force mg, lambda_k is a constant [lambda_k is the moment of inertia about the rotor's spin axis], omega_k is the spin rate of the rotor, and overline{dot(alpha)_-} is the average precessional rate. Notice that the spin rate, omega_k, is in the denominator. Thus, the larger omega_k the smaller the fraction becomes and thus the slower the precessional rate, just as we predicted from our simplistic Newton's Law explanation for the gyroscope. Also notice that r_CMmg is in the numerator of this fraction, thus as the torque gets larger the precessional rate also grows larger. Actually, the r_CMmg numerator does not include the sin(beta) factor that would yield the actual torque, rather the numerator is proportional to the torque and at a constant tilt angle beta the numerator is a constant proportional to the torque at that angle. But if the tilt angle beta changes, the numerator still stays the same and the the average precession rate is independent of the tilt angle beta. Detailed calculations of this is treated on the wiki page covering nutation and its PDF file: Gyroscope Precession and Nutation <-- click to go to the Nutation derivation.

The average precessional rate, overline{dot(alpha)_-}, is directly proportional to the torque at a given tilt angle, r_CMmg, and inversely proportional to the rotor spin rate, omega_k. Thus if the radius arm time the gravity force doubles, the precessional rate doubles, and if the rotor spin rate doubles, the precessional rate is cut in half. This quantifies the qualitative behavior predicted by our Newton's Law explanation: at a given tilt angle, the greater the torque the faster the precession, and the faster the rotor spin rate the slower the precession.

In fact, we might have even predicted these functional relationships. For instance, since the force is proportional to the torque, and from Newton's F=ma, the acceleration of the red voxel to the right is also proportional to the force which is proportional to the torque. Since the voxel's right-directed velocity is proportional to the acceleration, v=a\Delta(t), and after a quarter revolution this velocity becomes the tangential velocity component of the precessional rotation, we could then conclude that the precessional rate would be proportional to the torque, r_CMmg. At the same time, since v=a\Delta(t) where \Delta(t) represents the time interval that the red pixel is at the top during the rotors spin, then as the rotor spin rate increases, the time the red voxel has to accelerate decreases, that is, \Delta(t) is smaller and thus v is proportionally smaller. Hence we would conclude that the precessional rate is inversely proportional to the rotor spin rate, omega_k. Thus we could have even guessed the potential functional form of the precessional rate from our simple Newton's Law argument. It turns out that a rigorous derivation indeed yields precisely these dependencies on the torque and rotor spin rate.
This video (Gyroscope-Newtons_Law_Explanation_Part_II-1080p.mov) provides the Second Law extension to the Newton's Law Explanation for the behaviors of a gyroscope. It shows how Newton's Second Law may be used to find the functional dependencies of the precession rate on the applied torque and the rotor spin rate.
(YouTube encodes multiple alternative resolutions for the video.)
Description of the video's experiments: This video discusses the extension of the Newton's Law Explanation, based upon Newton's Second Law of Motion, for the precessional motion of a gyroscope. It describes how the Second Law predicts the functional dependencies of the precessional rate on the torque and the rotor spin rate. In particular, the precessional rate is directly proportional to the torque and inversely proportional to the rotor spin rate. The video then shows demonstrations of these torque and rotor spin rate effects on the precessional motion.

In summary, our Newton's Law explanation for the gyroscope explains why the gyroscope does not simply fall over when suspended from only one side of is axle, why it precesses instead. It predicts the direction of precession, and it predicts the qualitative nature of that precession, that is, that a greater torque produces a faster precession while a faster rotor spin yields a slower precessional rate. And if we combine the First Law with the Second Law, we can even predict the functional dependency for the precessional rate on the torque and rotor spin rate.

The following PDF gives this Newton's Law explanation for the gyroscope in slightly different language:

## A "little" South Dakota History:

Did you know that South Dakota was a major naval power? That without her we might have even lost the war in the Pacific theater during WWII? And that a memorial park in Sioux Falls is dedicated to the USS South Dakota and has several relics of the battleship's "big" guns? (There is a sign on Interstate I29 in Sioux Falls that directs you to this park.)
Draper Labs, in Cambridge, MA, built the first gyroscope stabilized and aimed gun sight. Draper convinced the Navy to try their new sight on the antiaircraft guns of the USS South Dakota. On October 24, 1942 during the Battle of Santa Cruz the USS South Dakota shot down 32 attacking Japanese fighters using the Draper Mark 14 gun sights. It was assessed that the guns with gyro-stabilized sights were effective 65% of the time against attacking Japanese dive bombers while the other, non-gyro-stabilized sighted guns, were effective only 5% - 30% of the time. This antiaircraft result was unprecedented, and it convinced the US Navy of the amazing utility of the Draper gyro-stabilized sights, which were then installed on all of the Navy's battleships. Subsequently, these sights saved many of our navy ships in the Pacific. (This Mark 14 history is little known with few historical accounts written about it. What I have summarized above comes from the writings of the Draper Labs itself.)

## An Interesting Application: Precession of the Equinoxes

The expression for the precessional rate,

has an interesting application.

Because of the centrifugal force generated by the daily rotation of the Earth around its polar axis, the Earth is not a sphere but rather is an oblate spheroid (a squashed sphere, in other words). The polar axis of the Earth also points at an angle of 23.4 degrees from the perpendicular to the ecliptic plane (the plane of the orbit of the Earth around the Sun). This angle, in conjunction with the "bulge" around the Earth's equator, means that the Sun's gravity (and also the Moon's) tugs on the Earth's bulge off-axis and therefore generates a slight torque being applied to the Earth's spin axis (F_1>F_2). This torque, through the above equation, generates a precession of the polar axis, and this precession rate is calculated (measured) to be roughly 25772 years, give or take a few years.

Today the polar axis points nearly directly at Polaris (the North Star), but after 12886 years it will be pointing in a direction about 47 degrees away from Polaris as the polar axis precesses around at a rate of one revolution every 25772 years. This precession of the spin axis causes the equinoxes to also precess around at the same rate. Unimaginatively, this effect is called the Precession of the Equinoxes, although in 2006 it was suggested that the name be changed to Precession of the Equator. The equinoxes (the two dates during the year when daytime equals nighttime) very slowly (25772 year rate) move westward along the ecliptic plane and thus also along the calendar. Today the equinoxes are near March 21 and September 22, but in a few thousand years the equinoxes will occur at other dates, and after 25772 years they will be back to roughly their current dates.

Although Aristarchus of Samos (c.310-c.230 BCE) writings have been lost, save possibly one, other authors, including Archimedes in his The Sand Reckoner and Plutarch, have referred to Aristarchus's astronomical results. In particular, Aristarchus is often cited as the progenitor of heliocentrism: the concept that the Sun and not the Earth is the center of the Universe. In addition, it is thought that Aristarchus may also have been the first to observe the precession of the equinoxes. But it was up to Hipparchus of Rhodes (c.190-c.120 BCE), however, to accurately measure the precession of the equinoxes to be roughly 1 degree every century, remarkably close to the 1.397 degrees that we know it to be today. Thus Hipparchus is most often cited as the discoverer of the precession of the equinoxes. Now I ask, weren't those ancient Greek astronomers simply amazing?!?

And who knows what a Great Year is? No, it is not the year you were born. But it is 25772 Julian years long, the torque-induced precessional period of the Earth!

## Homework:

(Working through these Homework Problems will cement your understanding of the qualitative behaviors of gyroscopes.)

You have now seen how to use Newton's Laws of Motion to predict the behaviors of a gyroscope whose axle is horizontal, as shown in the photograph below.
 Click for full-size image

• Problem 0: Notice that in the Newton's Law argument above, we have described what happens to the red voxel as it rotates a quarter revolution (90 degrees) and a half revolution (180 degrees). What happens at the three-quarter revolution (270 degrees) case?

• Problem 1: Please give this Newton's Law argument for a gyroscope whose axle is initially pointing vertically, as shown in the following photograph.

The following PDF file gives the Newton's Law explanation for the spinning top (vertically oriented spin axis):

Here is a video (VerticalGyroscope-Precession-1080p.mov) demonstrating what happens in for this vertical gyroscope when it's rotor is spun in both directions.
(YouTube provides a number of different resolutions for the video.)
Description of the video's experiments: This video shows the precessional motions of a vertically oriented gyroscope when its rotor is spun in both directions. When the rotor spins clockwise, the precessional motion is also clockwise, and vice versa.

• Problem 2: And how about the counterweight balanced spinning gyroscope, please predict what happens in this case.

This PDF file describes the operation of the counterweight balanced gyroscope:
The following video (Gyroscope-Precession_Counterweight-1080p.mov) demonstrates what happens when the gyroscope is spun in both directions and the spindle is twisted around.
Click on the following link to view the video on YouTube: http://youtu.be/VTTZ6JwgD94
(YouTube provides several alternative resolutions for the video.)
Description of the video's experiments: This video shows a counterweighted gyroscope. When the direction of the gyroscope's rotor matches the direction of the twist around the spindle, the gyroscope "lifts". When these directions are opposite, the gyroscope "drops".

• Problem 3: The counterweighted gyroscope is a "clean" experiment, in that the counterweight eliminates the torque on the gyroscope produced by gravity and thus there is no initial precession of the spinning gyroscope. Predict what happens in the case where the gyroscope is not counterweighted, as shown in the following photograph.

The following video (HorizontalGyroscope-Precession-1080p.mov) shows the complications that arise when the gyroscope is not counterweighted. Our Newton's Law Explanation is still sufficient to predict this behavior.
Click on the following link to view the video on YouTube: http://youtu.be/vOBv0UzuReo
(YouTube encodes multiple different resolutions of the video.)

Description of the video's experiments: This video shows a gyroscope whose axle is horizontal. The gyroscope is not counterweighted, so the gravity force on the rotor applies a torque to the gyroscope's axis of rotor spin. This torque produces an initial precessional motion. Additional applied torques to the spindle axis then speed up the initial precessional motion and "lift" the gyroscope, or slow down the initial precession motion and "drop" the gyroscope.

• Problem 4: Oh no! Predict what happens in the case of two gyroscopes on the same axle. What happens when they are both spinning in the same direction, what happens when they are spinning in opposite directions? What happens when they aren't spinning?
 Click for full-size image

Here is the first discussion of the double gyroscope:

And here is a second discussion of the properties of the double gyroscope:

The following video (DoubleGyroscope-Precession-1080p.mov) demonstrates the solutions for the double gyroscope.
(YouTube encodes the video at a number of alternative resolutions.)
Description of the video's experiments: In this video two gyroscopes mounted on a common axis are spun in both the same direction and in opposite directions. The video shows what happens in both cases when the gimbal mount is twisted around.

• Problem 5: And here is another test of your gyroscopic acumen... If the following gyroscope is spun up to speed and the string is raised until it lifts the gyroscope, what would happen?
Don't be fooled! Look carefully at the picture and use the Newton's Law explanation.

The following video (Gyroscope-String_Suspensions-1080p.mov) demonstrates what happens when the gyroscope is suspended from a string at two alternative locations.
Click on the following link to view the video on YouTube: http://youtu.be/tCjjaavBfE4
(YouTube provides a variety of different resolutions.)
Description of the video's experiments: This video shows a gyroscope suspended from a string at two alternative locations. When the gyroscope is suspended along its axis, precession occurs and the gyroscope may be lifted by the string. When the gyroscope is suspended at a point perpendicular to the axis, no precession occurs and the gyroscope simply falls off the string.

• Problem 6: If you are worried about the approximation sign in the above expression for the precession rate ... then this problem is for you. [This is the way physicists really work...they end up with equations that either cannot be solved exactly and/or their solutions are too complicated to gain much insight into what they mean physically, therefore physicists end up approximating the solutions in order to better understand what they predict and gain some insight into their physical mechanisms.]
[This is an Advanced Problem, requiring a thorough understanding of the Algebra of quadratic equations and, especially, Newton's Binomial Theorem. A PDF file providing the details of Newton's Binomial Theorem is included at the end of this problem.]
In particular, for the precession rate of a gyroscope problem, the rigorously exact expression for this precession rate is the negative (-) root of the following two solutions:

 Click for full-size image

but this complicated expression gives us little insight into the nature of the precession rate. Notice that for MgR = 0, that is, when there is no torque applied to the gyroscope, then the negative root is zero: dot(alpha)_- = 0 . We might thus be tempted to approximate this negative root by zero whenever omega_3 >> MgR and thus the first term under the square root sign is far greater in magnitude than the second term and thus we could neglect the second term (4 lambda_1 cos(beta) MgR = 0), but if we performed this approximation then our torque induced precession of the gyroscope would also be zero. No, it turns out that a more careful approximation of the negative root yields the expression for dot(alpha)_- :

In fact, I like to call these approximations the Quadratic Mystery Solutions since they initially don't appear to derive from the standard roots of the quadratic equation.

For instance, following elementary algebra, the quadratic equation,
has the exact solutions,
which, when b^2 >> 4ac, are approximated by

Your problem is to show why this is true!

Note: If b^2 >> 4ac , then we might estimate b^2-4ac by b^2 to arrive at
 Click for full-size image
and
 Click for full-size image

Now x_- is the same as that given above for the - Approximation, but x_+ is zero and thus not equal to the + Approximation which is -c/b . Not only is -c/b not equal to 0, but how did the b get in the denominator in the first place? So, you can now see why this is a mystery.

Hint: Use Newton's Binomial Theorem to derive the approximation to the square root term.

Here is the PDF file deriving the Quadratic Mystery Solutions:
And here is a PDF deriving Newton's Real Exponent Binomial Theorem that is employed in the Quadratic Mystery Solutions:

P.S. Don't worry about the +/- labeling of the solutions. I have labeled the torque induced precession dot(alpha)_- with a minus because it stems from the term with a minus sign. The confusion arises because the b coefficient is negative in the precession rate equation.

This functional form for the precession rate, that it is directly proportional to the torque and inversely proportional to the rotor spin rate as found by the Quadratic Mystery Solutions, can be justified by a simple Newton's Second Law Explanation. The following video demonstrates this: (Gyroscope-Newtons_Law_Explanation_Part_II-1080p.mov).
(YouTube provides a number of different resolutions.)
Description of the video's experiments: This video discusses the extension of the Newton's Law Explanation, based upon Newton's Second Law of Motion, for the precessional motion of a gyroscope. It describes how the Second Law predicts the functional dependencies of the precessional rate on the torque and the rotor spin rate. In particular, the precessional rate is directly proportional to the torque and inversely proportional to the rotor spin rate. The video then shows demonstrations of these torque and rotor spin rate effects on the precessional motion.

• Problem 6B: We have seen that one approximation to the quadratic roots, x \approx -c/b, gives us the readily interpreted torque induced precession of a gyroscope (precession rate is directly proportional to the torque and inversely proportional to the rotor spin rate):
The other approximation, x \approx -b/a, yields:
This dot(alpha)_+ is also a real physical precession known as Free Precession. Free precession occurs without any applied torque. It happens because the direction of the angular momentum vector is slightly misaligned with the direction of the angular velocity vector (rotor spin axis). Notice that this precession is directly proportional to omega_3, the spin rate of the rotor. So, if the spin rate doubles, the rate of the free precession also doubles. Depending upon the ratio lambda_3/lambda_1 (the ratio of the moments of inertia about the principal axes), the free precession rate is much faster than the torque induced precession rate which is inversely proportional to the spin rate. Besides the torque induced precession, the Earth also undergoes a free precession. It is called the Chandler Wobble, named after the American amateur astronomer who discovered it in 1891. The Chandler Wobble completes one revolution every 433 days, in contrast to the torque induced Precession of the Equinoxes that completes one revolution every 25772 years. So indeed, the Chandler Wobble is much faster than the Precession of the Equinoxes, as our above approximations predict.

You may have also experienced free precession if you happened to be riding in a car when one of the wheel balancing weights fell off (these are the heavy lead weights attached to the wheel rim to balance the wheel). Without the balance weight, the angular momentum of the wheel is slightly misaligned with the wheels axle, producing a free precession of the wheel as you drive. This you feel as a fast shaking, or shimmy, of the car. If the rim weight fell off while you were driving at highway speeds, you may even think your car is trying to shake itself apart.
Here is a PDF file describing the mismatch between the angular momentum vector and the angular velocity vector, and its consequences for wheel balance:

## Summary Concepts:

• We examined Newton's First Law Explanation of the qualitative behaviors of a gyroscope (the gyroscope supported on only one end of its axle does not fall, rather it precesses). The Newton's First Law Explanation also predicts the direction of precession.
• We also discussed Newton's Second Law Explanation of the functional dependency of precession rate (the precession rate is directly proportional to the torque and inversely proportional to the rotor spin rate).
• A gyroscope acts to "reflect" a force through 90 degrees, that is, an input force along some direction perpendicular to the rotor's axle results in an output force perpendicular to the input.
• Unusual and somewhat counterintuitive approximations to the roots of a quadratic equation yield the functional forms for the torque-induced precession as well as the free precession (non-torque induced) of a rotating rigid body. Applications of these approximate solutions to the Earth as a whole explain the Precession of the Equinoxes (25,772 years) and the Chandler Wobble (433 days).

This month's Puzzlers deal in wooden blocks and marbles, the Means and the Pigeonhole Principle...

# Mean Means

Allow me to digress for a moment to ask,

"Who here plays with wooden blocks?"
"What? Do you mean that I'm the only one who still plays with blocks?"

Well then, allow me to show you something that we can learn from blocks!

Okay, who can tell me how to take the average of two numbers, say 8 and 10? Yup, that's correct, the average of 8 and 10 is (8+10)/2 = 18/2 = 9 . Good job! (The average is two numbers is the midpoint between those two numbers, it is computed by summing the numbers and dividing by two.)

To a mathematician, the average is known as the arithmetic mean, to distinguish it from two other types of means, the geometric and harmonic means. Instead of summing two numbers and dividing by two, as is done to compute the arithmetic mean, the geometric mean is computed by multiplying the two numbers and taking the square root. Thus the geometric mean of 8 and 10 is calculated as sqrt(8*10) = sqrt(80) = 8.944271909999... < 9 . Notice that the geometric mean of 8 and 10 is less than the arithmetic mean of 8 and 10 . Is this true for all numbers and not just 8 and 10 ?

So, with this quick introduction to means, let's see what we can do with those blocks, shall we?

## Wow! Isn't that great?!? We just proved an important theorem, that the geometric mean is less than the arithmetic mean.

Let's see how we accomplished this.

Consider the black square in the figure below. It has sides of lengths x+y and is dissected into four light red rectangles whose lengths are y and whose widths are x. Notice that these four xy rectangles do not cover the x+y square fully, rather they leave a small light blue square open having sides of z where z=y-x. Notice that the area of the black outer square is (x+y)^2 while the sum of the areas of the four pink rectangles and small light blue square is 4xy + z^2 . Since these two areas are the same, we see that (x+y)^2 = 4xy + z^2 .

Now notice that if z=0, then x=y (or if x=y, then z=0) and the four light red rectangles (now squares) fully cover the black x+y square, as shown in the figure below.

Now from the upper figure, we see that when x is not equal to y, then z=y-x>0 and the sum of the areas of the four rectangles is less than the area of the x+y square:
4xy < (x+y)^2
and even when z=0, then x=y and we have 4xy = (x+y)^2. We thus see that for any x and y,
4xy <= (x+y)^2 .
Dividing both sides by 4 we find
xy <= (x+y)^2/4 = [(x+y)/2]^2
and taking the square root of both sides leaves
sqrt(xy) <= (x+y)/2 .
The right-hand side of this inequality is simply the Arithmetic Mean, or average, of the two numbers x and y. The left-hand side, on the other hand, is known as the Geometric Mean. This inequality thus says that the Geometric Mean of two numbers is less than or equal to their Arithmetic Mean. For two distinct numbers, the Geometric Mean is strictly less than the Arithmetic Mean, and only if the numbers are the same will the Geometric Mean equal the Arithmetic Mean. While we have demonstrated that the Geometric Mean is less than or equal to the Arithmetic Mean for just two numbers, this fact is true in general, for any count of numbers. Thus it is true for three numbers, four numbers, and so forth.

This Theorem, that the Geometric Mean is less than or equal to the Arithmetic Mean, is employed in several of the Puzzlers. For instance, this theorem may be used to answer the following query, "Of all possible rectangles having a given perimeter length, what rectangle encloses the greatest area?" Another application is to compute bounds for the factorial function, n! = (1)(2)(3)(4)...(n), used extensively in statistics. And the Geometric Mean is less than or equal to the Arithmetic Mean Theorem can be employed to prove that the square of the average is less than or equal to the average of the squares:
((x+y)/2)^2 <= (x^2+y^2)/2
another quite useful fact. Can you derive this?

Here is a little more discussion on the Geometric and Arithmetic Means:

With that introduction to the Geometric and Arithmetic Means, let's get to the Butcher puzzler.

## Honest Fritz, the Boston Butcher ... or was he?

There once was an old Boston butcher, named Fritz, who just loved his tripe and wanted to share it. So each week he ran a special on this delicacy, two pounds for \$2.00! I know, "Wow!" you say.

Priding himself on his honesty, Fritz was worried that his balance beam scale on which he meted out the meat was a mite unbalanced. You see, his balance was as antiquarian as he.

Now Fritz had a one pound balance weight in which he had every confidence, but with a potentially unbalanced balance beam, what was he to do?

So he decided upon the following satisfying strategy for weighing the tripe: he would place his pound weight in which he had every confidence on the right-hand pan of his scale and thus weigh 1 lb of tripe in the left pan. He would then swap pans for the second pound, weighing the tripe in the right-hand pan using his pound weight for the left. Beaming, Fritz would thus measure two pounds of tripe in this fashion to fill his customers's orders.

The question is, was Fritz shortcharging, shortweighing, or being fair to his tripe patrons?

Given a set of calibrated balance weights, can you devise a way to check if one beam of Fritz' balance was, scandal of scandals, shorter than the other beam?

At times, Fritz also has the occasion to weigh a quantity of gold on his befuddled balance beam. Surely he would want an accurate determination of the gold's weight. Can you help Fritz out by designing a scheme by which he can weigh the gold without error?

Hint 1: Remember the Principle of Moments? In order for the beam to balance, the opposing moments must be numerically equal. In other words, in the following diagram where f denotes forces and l denotes lengths, the beam is balanced around its pivot point P when (f_1)(l_1)=(f_2)(l_2). (See the first part of the Thinking Solution if you do not understand what this means.)

Hint 2: Now you may be thinking that taking the average, or arithmetic mean, is the proper thing for Fritz to do, but it turns out that if you pursue the physics behind this problem you will find that the arithmetic mean is incorrect and that Fritz should rather employ the geometric mean in order to accurately weight the tripe and the gold.

Honest Fritz, the Boston Butcher Puzzler therefore employs the Geometric Mean is less than or equal to the Arithmetic Mean Theorem. Here is a link to the Puzzlers: 2 page where this Puzzler and its solution are located: Puzzlers: 2

# Water and Wine, Wine and Water

So, we can learn something from wooden blocks...let's next see what we can learn from pigeons...

Last month we discussed the Water and Wine, Wine and Water Thinking Problem. Here is a link to the Wiki page for this Puzzler: Puzzlers. Below the Puzzler is repeated...

1. Two glasses, one containing precisely 1 cup of water, the other with precisely 1 cup of wine.
2. Take exactly one teaspoon of water from the water glass and add it to the wine glass. Stir well.
3. Take exactly one teaspoon from the water in wine mixture glass and return it to the water glass.

## Puzzler: Is there more wine in the water glass or water in the wine glass?

[Note: While wine is roughly 90% water, for this puzzler we consider wine to be make up of "wine" molecules. Thus wine is 100% wine molecules, and water is 100% water molecules.]
This problem has spawned many a fight at dinner parties, why even I have come dangerously close to suffering a punch from a dinner guest who refused to believe the Pigeonhole Principle.

## The Pigeonhole Principle:

Say there are 10 pigeon holes and 11 pigeons, what must happen?

Yup, that's right, this means that at least one hole must have two or more pigeons in it! Now there can be more than one hole with two or more pigeons (then a hole will also be empty), but at least one hole must have at least two pigeons. Thinking about this another way, it means that if there are only ten pigeonholes, only ten pigeons can be accommodated (assuming only one pigeon per hole).

Let's see how we can put this concept to use in solving the Water and Wine, Wine and Water Puzzler...

Let's model the glass of wine with 12 red marbles and the glass of water with 12 green marbles, as shown below...

...placed in the glass jars, thus here is our model of the two glasses, one of wine, one of water...

...we take one teaspoon of water from the water glass (say one teaspoon is modeled by 3 marbles), then we take 3 green marbles from the water glass...

...and add them to the wine glass, and mix well...

...next we take one teaspoon (3 marbles) from the water in wine mixture glass...

...which just happens to be two red marbles and one green marble (2/3 wine, 1/3 water) in this case, and add it back to the water glass...
...with the results that each glass once again contains precisely 12 marbles (one cup in each glass, in other words). Notice that the wine glass has 10 red marbles and 2 green marbles while the water glass has 10 green marbles and 2 red marbles. Indeed, since each glass has precisely 12 marbles, then for every green marble in the wine glass there must be a corresponding red marble in the water glass! Or, for every red marble in the water glass there must be a corresponding green marble in the wine glass. This follows immediately from the Pigeonhole Principle. Hence, there is the same amount of wine in the water glass as there is water in the wine glass!
Notice that we didn't even have to stir the mixture before taking the second teaspoon to replace in the first glass. Even if we didn't stir the mixture before taking the second teaspoon, we would still end up with equal amounts of water in the wine as wine in the water. The point is that for every molecule of water in the wine there must be a molecule of wine in the water. In other words, there is the same amount of water in the wine glass as there is wine in the water glass!
Here is a link to the Puzzlers page where the Water and Wine, Wine and Water Puzzler and its solution is located: Puzzlers . The solution begins with a couple of mathemagical card tricks employing the same Pigeonhole Principle in order to illustrate its use in alternative circumstances.