Gyroscope Bicycle Wheel

Introduction

Why is a moving bicycle much more stable than one that is stopped? The answer lies in the momentum of the wheels. The following activities can be used to demonstrate the properties of a rotating bicycle wheel (or a gyroscope).

Concepts

• Gyroscopic motion
• Precession
• Conservation of angular momentum
• Torque

Materials

Safety Precautions

Care should be taken when spinning the bicycle wheel rapidly. Keep hands and other body parts clear of the spokes when the wheel is spinning. Hold the handles of the wheel firmly, with arms fully extended to prevent the wheel from rubbing against the body (and possibly injuring arms, legs and chest). Wear a lab coat or long-sleeved shirt when holding the spinning bicycle wheel. Wear safety glasses. When standing or sitting on a rotating stool, chair or table, be sure to keep your center of gravity low and over the center of the rotating apparatus in order to maintain optimal balance. Twist or tilt the bicycle wheel carefully. Do not use bare hands to slow down the spinning bicycle wheel. Stop the bicycle wheel by setting it on the floor.

Prelab Preparation

Practice operating the bicycle wheel before performing the demonstration in front of the class. For best results it is important to have the bicycle wheel spin at very high revolutions per minute (RPMs). See Teaching Tips section.

Procedure

Activity 1

1. Obtain the gyroscope bicycle wheel with axle handles.
2. Grip the handles tightly and carefully rotate the bicycle wheel rapidly in a vertical position (with the handles horizontal).
3. As the wheel spins, carefully, yet firmly, twist or tilt the handles. Observe and feel the strange gyroscopic phenomenon. [You will feel a force (torque) from the bicycle wheel that rotates it at a right angle to the applied twist.]
4. Discuss the observations with the students.

Activity 2

1. Obtain a swiveling stool, chair or a rotational turntable (such as the Rotational Turntable, Flinn Catalog No. AP4609). Make sure the stool or chair has very sturdy base and legs.
2. Grip the handles of the bicycle wheel tightly, and carefully rotate the bicycle wheel rapidly in a vertical position (with the handles horizontal) (see Figure 1). Be sure to keep your center of gravity low and over the center of the rotating apparatus.
   {10364_Procedure_Figure_1}
3. As the wheel spins, carefully, yet firmly, twist or tilt the handles. What happens? [Twisting the handles counterclockwise will cause the swiveling stool to rotate clockwise, and vice versa. When the handles are returned to their original position, the rotation stops.]
4. Discuss the observations with the students.

Activity 3

1. Repeat the steps in Activity 2, except hold the spinning wheel so that it is horizontal to the ground (with handles held vertical) (see Figure 2)
   {10364_Procedure_Figure_2}
2. Make sure your center of gravity is low and over the center of the rotating apparatus.
3. While sitting on the rotating apparatus, carefully rotate the wheel 180° upside-down (so your “bottom” hand is now the “top” hand) (see Figure 2).
4. Observe what happens. [The demonstrator will begin to rotate very rapidly on the rotating apparatus.]
5. Rotate the wheel back to its original position.
6. Observe what happens. [The rotation stops.]
7. Discuss the observations with the students.

Activity 4

1. Obtain a heavy-duty rope. Tie or loop one end of the rope to the eye screw in one of the handles (see Figure 3). (The other end of the rope can be tied to a support device, or it can be held in a hand for this demonstration.)
   {10364_Procedure_Figure_3}
2. Hold the wheel by the handle, with the rope tied to it, with one hand.
3. Rapidly spin the bicycle wheel in the vertical position (see Figure 3).
4. Once the wheel is spinning quickly, grip the rope with your free hand while still holding the handle with your other hand.
5. Pull the rope so that it is vertical and taut. Then, release the handle quickly so the wheel is held up only by the rope.
6. Observe the gyroscope. [The wheel should remain suspended vertically and begin to spin around the support rope—it will precess. See Discussion.]
7. Discuss the observations with the students.
8. Hanging weights can be added to the free handle to show the effect of added mass on the rotational precession speed. The rotational speed of the bicycle wheel (the RPMs) can be varied to show the inverse relationship between the speed of the wheel and the speed of precession (see Discussion).

Activity 5

1. A variation of Activity 4 is to tie the eye screws in both handles of the bicycle wheel to a horizontal support rod with rope and allow the wheel to hang in the vertical position (see Figure 4).
   {10364_Procedure_Figure_4}
2. Spin the wheel rapidly in the vertical position.
3. Once the wheel is spinning rapidly, ask the students to predict what will happen if one of the ropes is cut.
4. Obtain some heavy duty scissors and cut one of the support ropes.
5. Observe what happens. [The wheel will remain vertical and begin to precess around the remaining support rope.]
6. Discuss the observations with the students.

Activity 6

1. Tie a long rope to a ceiling support structure in the classroom (such as an I-beam or other rigid beam).
2. Make a gyroscopic pendulum by tying the other end of the rope to the eye screw on one of the handles of the bicycle wheel (see Figure 5). Make sure there are no objects in the path of the pendulum. It will not travel in a straight line.
   {10364_Procedure_Figure_5}
3. Rotate the wheel rapidly in the horizontal position.
4. Slowly raise the spinning wheel pendulum approximately 30–40 cm from the base of its swing.
5. Release the pendulum.
6. Observe the strange motion of the pendulum.
7. Discuss the observations with the students.

Teacher Tips

• An effective tool that can be used to rapidly and consistently rotate a large bicycle wheel is a variable speed drill. Insert a blank steel shaft into a oneholed rubber stopper and then connect this to the drill (see Figure 6). The rubber stopper can be set against the bicycle wheel and slowly accelerated as the drill speed is increased. Significant rotational speed (rpm’s) can be attained with this method. Be sure to use this technique cautiously and wear safety glasses. Start the drill at a low rpm and make sure the rubber stopper is firmly attached with no sharp metal protruding from the end of the stopper.
  {10364_Tips_Figure_6}
• If you feel comfortable, ask for student volunteers to perform the activities in front of the class. Alternatively, you can perform the demonstration first, and then allow students to operate the gyroscope bicycle wheel so that they can “feel” the gyroscopic effect. Some of the activities require that the demonstrator has good balance. Make sure to inform the students that they may be spinning rapidly before appointing a volunteer. Some students may not feel comfortable spinning. During the activities, make sure the volunteer maintains a low center of gravity and is positioned over the center of the rotating stool, chair, or turntable.
• Gyroscopic motion is generally a very complex topic. The explanation involves abstract (and not always well understood) concepts like torque, angular momentum and vectors. The activities can be performed simply as interesting demonstrations, or they can be used to illustrate and explain complex physics principles. Your own knowledge base, the information provided in your textbooks, the information provided in the Discussion, along with your goals for the class should help determine how to discuss the observations with your students.
• The handles are glued onto the metal axle. If the handles come unglued or become loose, they can be secured to the metal rod again using hot glue, white glue, or double-sided tape. Hot glue works better than white glue because it cools and sets faster. Add two loops of glue around the metal rod about ½" and 1½" from the end of the rod. Then, press and twist the rod handle onto the metal rod. Allow the glue to dry. Repeat for the other handle. Be sure to insert the metal rod through the hub of the wheel before securing the second handle. If double-sided tape is used, insert the metal rod through the hub of the wheel. Next, affix a loop of tape to each end of the metal rod about 1½" from the ends (close to the wheel hub). Then, carefully, yet firmly, slide the handles over the tape to fasten them to the rod.

Discussion

The sensation that is felt while operating a hand-held gyroscope is often difficult to explain because it is very unfamiliar. However, under close observation the strange behavior of a gyroscope can be followed. In the first activity, for example, the spinning gyroscope is held so that the handles are horizontal, and then a hard up-and-down twisting force is applied to the handles (i.e., the right handle goes down and the left handle goes up). However, the gyroscope does not twist down. Instead it twists in the horizontal plane (the handles go away from you and toward you). When you try to bring the bicycle wheel back to its original position by twisting the handles in the horizontal plane, the wheel handles move in the vertical plane. The gyroscope rotates in a plane that is at a right angle to the applied force. This is a very unusual phenomenon to most students.

In order to understand gyroscopic behavior, a few important physical properties need to be discussed. To begin with, all objects have mass and therefore they have inertia. When mass is set in motion, it has the property known as momentum (or, “inertia in motion”). As long as no external forces act on the object after it is set in motion, the momentum of the object will always be conserved—this is known as the law of conservation of momentum. The mass in an object that is rotating around an axle (such as a bicycle wheel) also has momentum, but it has a special type of momentum called angular momentum (or rotational momentum; “inertia of rotational motion”).

Angular momentum is slightly different from linear momentum because it is dependent on not only the mass of the object, but also on where the mass is distributed in relation to the rotational axis (axle). Angular momentum is also always conserved as long as no external forces act on the spinning object after it is initially rotated. A force acting on a rotating object actually acts at a distance away (through a lever arm) from the axis of rotation. For example, spinning a bicycle wheel requires a force to be applied to the outer rim, away from the axle, in order to make the wheel rotate around the axle. The combination of a force operating through a lever arm and applied to an axis of rotation is known as a torque. Torque is the “force-equivalent” for rotational motion. Therefore, angular momentum is always conserved as long as no outside torque acts on the rotating object. An example of conservation of angular momentum is seen in figure skating. When a figure skater spins, her (or his) angular momentum is fixed based on the initial position of her arms (moment of inertia) and the initial speed of rotation (see Equation 1). Since there is negligible friction between the ice skates and the ice during the spin there are no external torques applied to the vertical axis of rotation, and thus angular momentum will be conserved. If the figure skater starts her spin with her arms extended in a T-shape, her moment of inertia will be large. As the figure skater draws her arms towards her body, her moment of inertia decreases (because her mass is closer to the axis of rotation), so her rotational speed must increase in order to balance her decreased moment of inertia and maintain a constant angular momentum.

L = Angular momentum
I = Moment of inertia* of the spinning object
ω = Rotational speed of the spinning object

*Moment of inertia is determined from the mass of the object multiplied by the square of the relative distance between the mass and the axis of rotation. The general form is I = ∑mr², in which the moments of inertia of all the infinitely small masses that make up the object are determined about an axis of rotation and then added together. Please refer to the references at the end of this activity or other textbooks for further explanation.

The gyroscope bicycle wheel can be explained using the above terms—torque and angular momentum. When the wheel spins, it has angular momentum. Angular momentum will maintain its orientation in space unless it is acted on by an external torque (just as linear momentum will maintain its straight-line motion unless acted on by an outside force). This means that the axis of rotation (and therefore the rotational plane) of the spinning wheel will not rotate unless acted on by an outside torque. When a torque is applied to twist or tilt the spinning wheel’s axis of rotation (the axle of the wheel), the angular momentum of the wheel resists the change caused by the outside torque because “it” wants to preserve the same orientation and remain constant. Therefore, the gyroscope induces a torque in another direction in an attempt to retain the same angular momentum. This torque is verifiably felt when a person twists the spinning bicycle wheel. A more dramatic effect of this torque is observed when a person twists the bicycle wheel 180 degrees while sitting on a rotating turntable (Activity 3). The entire system’s angular momentum is always conserved (the system meaning the wheel, the person, and the rotating table) and the only object contributing to the angular momentum of the system, initially, is the spinning wheel. So, if the angular momentum of the spinning wheel is changed as the result of an internal torque (the person tilting the wheel is in the system—no external torques), there must be a torque induced internally in order to maintain the system’s original angular momentum. This internal torque causes the person on the turntable to spin in the direction that will maintain the initial angular momentum of the system. For example, a person holds a wheel that is spinning counterclockwise in the horizontal plane (when viewed from above), and applies a torque to flip it 180 degrees so the wheel will then be rotating clockwise in the horizontal plane. The spinning wheel will induce a torque on the person that generates a counterclockwise spin in order to counteract the change in angular momentum and keep the system’s angular momentum constant. The rotation of the individual on the turntable will stop when the spinning wheel is returned to its original orientation (see Activity 3).

When the spinning gyroscope is balanced vertically by a rope at the end of one of the handles a somewhat perplexing event occurs (Activities 4 and 5). Instead of the wheel toppling over as might be expected, it appears to defy gravity as it maintains its vertical orientation and begins to rotate around the support rope. This rotation of the wheel around the support rope is known as precession. Precession is a complicated result of the changing angular momentum and the torque applied to the spinning wheel. However, an explanation can be achieved by referring to the simplified pictorial view of the forces acting on the gyroscope during a short time interval. An approximation has been made that the gyroscope is spinning much faster than it is precessing so the time represented by the figures is very small. Also, the twisting force applied to the handles is constant for the short time interval. In the first image in Figure 7, the twisting force applied to the handles tries to rotate the wheel counterclockwise in the vertical plane (the axis of rotation is horizontal through the middle of the wheel, and perpendicular to the handles). This twisting force (a torque) is then translated to upper and lower area of the rim (mass points A and B, respectively) as shown by the black force arrows. Mass points C and D are not affected by the twisting force because they are in line with the axis of rotation of the applied force. A very short time later, mass points A, B, C and D have rotated 45 degrees. The original twisting force that affected mass points A and B when they were in the upper and lower regions of the wheel still affects these mass points during their rotation in this short time span. So, the “residual” force on these mass points still points in the same direction, as shown by the dashed arrows. As the wheel rotates, the residual force on these mass points moves away from, and remains perpendicular to, the vertical axis of rotation of the wheel (Image 2). Therefore, the residual force creates a twisting force in the horizontal plane about the vertical axis of rotation through the center of the wheel. In Image 3, mass points C and D are now acted on by the constant twisting force, while the residual force still affects mass points A and B. Image 4 shows that a residual force affects all four mass points and continues to produce a twist in the horizontal plane. Image 5 shows that mass points A and B have rotated so that A is now in B’s original location, and vice versa. The residual force that still affects these mass points is now balanced by the constant twisting force that is applied to the handles. Therefore, the net force at these points returns to zero (the dashed arrow and black arrow point in opposite directions). The sixth image shows that the force on mass points A and B is zero, but the residual force on mass points C and D is still perpendicular to the vertical axis of rotation. The next image shows the residual force and twisting force balance out on mass points C and D. The final image shows no forces acting directly on the mass points at that instant, and the cycle repeats. Consequently, the twisting force on the handles actually causes a twisting force in the horizontal plane which makes the wheel precess in the horizontal plane instead of rotate in the vertical plane. (When the spin of the wheel slows down, the behavior of the gyroscope becomes more complex as the gyroscope begins to dip and undulate. The undulation is known as nutation, which is a result of the changing angular momentum of the wheel as it slows and precesses.)

Please refer to the next section for a more formalized, mathematical explanation of precession.

Precession
Angular momentum is equal to the moment of inertia (also known as rotational inertia) multiplied by the rotational speed (Equation 1). Angular momentum is a vector quantity, meaning it has a defined value and a defined direction. A simple way to determine the direction of angular momentum is by curling the fingers from your right hand in the direction of the spin. When this is done, your thumb will point in the direction of the angular momentum vector. (This is known as the right-hand rule.) See Figure 8. When a spinning wheel’s axle is supported horizontally by a pivot at one end (the their end is free to fall), gravity pulls down on the center of mass of the spinning wheel to produce a torque about the pivot point. Since the force and the lever arm are at a right angle to each other, the torque (τ) will be equal to the distance between the pivot point and the center of mass along the lever arm (R) multiplied by the force of gravity (mg) pulling the center of mass downward, as shown in Equation 2. Torque is also a vector quantity and the direction of the torque is also governed by the right-hand rule. If you point your fingers in the direction of the lever arm, extending from the pivot point to the position of the applied force, and then curl your fingers in the direction of the force, your thumb will be pointing in the direction of the torque vector (see Figure 9). According to the law of conservation of angular momentum, the angular momentum of a rotating system will only change if an external torque is applied to the system. This result is represented mathematically according to Newton’s second law of rotational motion. An applied torque yields a change in (Δ) angular momentum in a given time period (Equation 3). An interesting result of the applied torque is that the change in angular momentum must be in the direction of the torque. Therefore, since the torque at the pivot point produced by gravity is in a horizontal direction, the change in angular momentum must also be in the horizontal direction (see Figure 10). The direction of the angular momentum changes with time due to the torque, but the magnitude stays the same. So, the pull of gravity that acts to topple the wheel over and change its rotational axis actually causes the gyroscope to precess in a horizontal circular path about the pivot point. (This is why a moving bicycle is stable and does not tip over.) It can be determined mathematically that the angular precession speed of the gyroscope about the pivot point is inversely proportional to the angular momentum (and therefore the angular speed) of the spinning wheel (Equation 4). Substituting Equation 1 into Equation 4: Equation 5 shows that the faster the wheel is spinning (ω), the slower the wheel will precess (ω_p).

References

Tipler, Paul A. Physics for Scientists and Engineers, 3rd Ed., Vol. 1; Worth Publishers: New York, 1990; pp 255–262.