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Introduction
In this article I explain how to use Arbor Scientific’s new Gyroscope Wheel to explain precession and other rotation related phenomena associated with gyroscopes.
In many ways the gyroscope wheel is a modern piece of physics teaching equipment. It has very low friction bearings, it is light and easy to get spinning, and it has safe-grip handles that help you to demonstrate with confidence.
The Gyroscope Wheel is a product unique to Arbor Scientific and we are proud to offer a durable, safe, and elegant apparatus. You do not have to worry about getting hurt by rotating spokes because it has no spokes, your hair won’t get caught in the wheel because it is well shielded, and the smooth design will help you demonstrate these concepts without distractions and hazards. For example, the flat masses fit snugly in during rotation which makes it safer and easier to use.
The fact that this is not a bicycle wheel can also be a great benefit, because the physics of the gyroscope and its many applications exist independent of bicycle wheels (discussed later). The dark stripe also allows you to easily measure the rate of rotation, for example in a video you might make, which can help you perform quantitative experiments.
Figure 1. James Lincoln demonstrating how easy it is to handle the device.
Angular Momentum
The first thing you are going to want to do is to demonstrate the concept of angular momentum. In object in motion remains in motion (including rotation).In the case of regular momentum, an object (like a bowling ball) is difficult to stop in its forward motion because it has momentum (mass and velocity). Similarly, a spinning or rotating object has rotational momentum or angular momentum. This is the rotational inertia (different for every object, derived from the rotating masses’ positions) times its angular speed. The formula for this is L = I ω, where ω is the Greek letter omega, measured in radians per second. (One radian is 57.3 degrees.)
Spinning the gyroscope wheel will give you a feeling for angular momentum. The analogy is that it takes a force to change momentum and therefore it takes a torque (force at a radius) to change angular momentum. The resistance is always a surprise.
Figure 2. James feels the angular momentum when he tries to change the direction of rotation. The easy grip handle prevents slipping.
Angular momentum is a conserved quantity, like energy or linear momentum. For this reason, it is possible to demonstrate that this quantity will not decrease in an isolated system. Rather it will be redirected or redistributed. This is usually achieved by standing on a rotating platform.
To achieve this demonstration, get the wheel spinning as fast as you can and hold it so that it is parallel to the floor. Stand on a low-friction rotating platform and invert it. This will change its direction of rotation and the result is that you will begin to spin! You are indeed part of the system. By Newton’s third law, the action you took on the gyroscope is now having the reaction of causing you to rotate. The total angular momentum is unchanged.
Figure 3. James spins on rotating platform, subjecting himself to the conservation of angular momentum.
Precession
Of course, the unique behavior of the gyroscope is precession. When a perpendicular torque is applied to an object that is already rotating, the result is precession. This might remind you of circular motion. In that case, when a force is applied perpendicular to a velocity it will change its direction and make it move in a circle. The same effect is at play here.
Precession: the slow rotation of a spinning object that changes its rotation axis, wobbling.
First, we acknowledge that the direction of spin controls for the direction of precession. For example, if you spin the gyroscope clockwise it will precess in a different direction from when you have it spun counterclockwise. We also notice that the rate of spin controls the rate of precession, you can easily see this with the black stripe.
Figure 4. The rate and direction of spin controls for the rate and direction of precession.
Next, we also note that the location of the string affects the rate of precession. This, we conclude, is because the torque from the weight acting on the handle increases with the length of the lever arm. So, it really is torque that causes the precession.
Figure 5. Attaching the string closer reduces the torque and therefore decreases the rate of precession.
Lastly, we note that if we reduce the inertia of the gyroscope, by moving the weight, that it makes the precession rate faster. So, we can now assemble a formula based on these observations. The precession rate increases with the torque but decreases with the spin rate and decreases with the inertia of the gyroscope, generally:
Precession = torque / spin / inertia
This formula can be rewritten in a more specific form. Precession is an angular speed, so I will use the symbol Ω capital omega. Torque is force times lever arm, specifically τ = mgL (mass x gravity x the length of attachment ). The spin rate is of course little omega ω. The inertia is variable, but proportionate to disk mass m and radius of the disk r squared, so I = Cmr2, where C is some constant between .5 and 1, which depends on where you put the weights (it is .5 if you remove them). The result of all this is…
Ω = mgL / ω / Cmr2 à Ω = 2 g L / ( ω r2 )
Which you can verify experimentally. In this case, I assumed C = .5, which is true for the no masses removed case. It is nice to point out that the mass does cancel in the equation. So now we know how fast the gyroscope will precess, but we have not yet said anything about which direction it will precess. This can be quite a challenge.
Figure 6. The adjustable weights allow you to change the rotational inertia for experiments and demonstrations.
The Direction of Precession
How can we predict the direction of precession? Let’s follow an individual part of the wheel to make sense of the direction of precession. I have illustrated this in Figure 7.
- Point P moves from side A to side B during a half rotation.
- As it moves it is subject to the tipping torque from the weight of the wheel itself, and the leverage of the handle.
- This begins to accelerate point P in the y direction as it moves from A to the top, then B
- The maximum acceleration occurs at the top, but the maximum velocity occurs at point B
- Thus, when P reaches B, it is ready to move in the y direction, which is exactly what it does. We have successfully predicted the direction of precession.
Figure 7. As point P moves from A to B it is at all times accelerated in the Y-direction. Thus, when it finally arrives to point B it moves with its velocity in that direction. Point A will then move in the other direction.
But have we only looked at half of the story? What about the bottom half?
The exact reverse happens. This time, moving from B to A and accelerating in the opposite direction. Thus, when P reaches A, it is ready to move in the negative Y direction. Again, we have successfully predicted the direction of precession.
There is another way to explain all this, using rotation vectors. We imagine the wheel is spinning clockwise again. By curling the fingers of your right hand from A to B, we find that this spin is directed into the screen (+Y) But the torque vector, caused by the weight of the wheel applied L centimeters along the lever arm at the point of connection, is in the - X direction. The net result is that the face of the spin is redirected from +Y toward – X, thus B turns left, which is the same effect we predicted earlier. This approach can be a bit tricky, which is why I supplied it second. However it is the more common explanation.
Applications of Gyroscopes
Of course, there are many applications for such a gyroscope. They are used in spacecraft for directional adjustment, aircraft for flying blind, guided missiles for their attitude adjustment, and stabilizing space-based telescopes like the Hubble and James Webb.
Figure 8. Whenever you need to know what direction you are pointed, such as flying through a fog, the gyroscope gauge tells you the “attitude” of your aircraft or space craft.
But there is also the earth itself, which is of courses a very large gyroscope, undergoing rotation. However, it is s also precessing, but very slowly, as explained by Isaac Newton. The equatorial bulge allows for a gravitational torque from the sun. This pulls earth into a very slow wobble, or precession, that has a period of around 26,000 years. Thus, the north star Polaris, will not always be the north star. Eventually the Earth will wobble and tilt toward a new star in the sky, Vega. Look forward to it.
Figure 9. The earth is the largest gyroscope you have access to. The precession was predicted CORRECTLY by Isaac Newton. This astonishing prediction is lost in the unfathomable wealth of his accomplishments.
James Lincoln
Physics Instructor
James Lincoln is an experienced physics teacher with graduate degrees in education and applied physics. He has become known nationally as a physics education expert specializing in original demonstrations, the history of physics, and innovative hands-on instruction. The American Association of Physics Teachers and the Brown Foundation have funded his prior physics film series and SCAAPT's New Physics Teacher Workshops. Lincoln currently serves as the Chair of AAPT's Committee on Apparatus and has served as President of the Southern California Chapter of the AAPT, as a member of the California State Advisory for the Next Generation Science Standards, and as an AP Physics Exam Reader. He has also produced Videos Series for UCLA's Physics Demos Project, Arbor Scientific, eHow.com, About.com, and edX.org.
