Tops And Gyroscopes

Type of physical science: Classical physics

Field of study: Mechanics

The unusual motions of tops and gyroscopes are explained through the concepts of rigid body motion. These simple devices have diverse applications in modern technology.

Overview

Gyroscopes and spinning tops are specific instances of objects whose behavior falls under the category of mechanics known as "rigid body motion," in which it is assumed that each object can be treated as perfectly rigid, that is, having no flexure. The underlying theory applies to all rigid objects during periods of rotation.

A critical quantity in the characterization of rotational motion is an object's moment of inertia, which does not depend on the object's mass directly, but rather on how the mass is distributed. When a mass rotates, it moves around an axis of rotation, which may be a physical object (as with a wheel rolling about an axle) or an imaginary line (as with a spinning ball tossed into the air). The object's moment of inertia depends on its axis of rotation; the more distant the mass is from the axis of rotation, the greater the moment of inertia relative to that axis. For example, in the case of a solid sphere, the moment of inertia about any axis that passes through the sphere's center is 2mr² (where m is the sphere's mass and r is the radius), while for a hollow sphere with the same mass (the mass is concentrated in a spherical shell), the moment of inertia is 2mr^2/3.

A second quantity that affects rotational motion is torque, a force acting either to cause rotation or to change some property of the rotation (that is, a turning effect). Torque has both magnitude and direction; it is a vector quantity. The magnitude is defined as the product of an applied force times the distance from the object's axis of rotation to the imaginary line along which the force acts. Torque also has a direction, along the line that a screw would progress if this same torque were applied, through a screwdriver, to drive the screw into a surface. For example, the torque required to start a large wheel turning is given by the radius of the wheel times the force acting tangent to the wheel's edge.

An equation analogous to Sir Isaac Newton's second law of motion (F = ma) relates the torque T to the object's moment of inertia and the rotational acceleration that results: T = I α, where I α is the product of the moment of inertia and the angular (rotational) acceleration around a given axis of rotation. It is crucial to note that this equation applies only when the axis of rotation has a fixed orientation relative to both the object and to the environment surrounding the object. Returning to the example of the two spheres, the solid sphere will have 3/5 times the moment of inertia of the hollow sphere, so that if the same torque is applied to both, the solid sphere will undergo 5/3 times the acceleration of the hollow one.

A further analogy exists between rotational motion and particle motion (as described by Newton's laws of motion): In the absence of torques, an object undergoing rotational motion will continue indefinitely without decelerating. Therefore, the object's energy of rotation will remain constant unless a torque acts to remove some or all of this energy.

As with particles in motion, a rotating body also has momentum, but it is referred to as "angular momentum," and its magnitude is given by the product of the body's moment of inertia around an axis of rotation and the body's rate of rotation around that same axis. The angular momentum is assigned a direction as well, along the axis of rotation in a direction away from an observer who sees the object rotating clockwise. For example, each wheel of an automobile (moving forward) has angular momentum directed outward to the left of the car.

In the late eighteenth century, the Swiss mathematician Leonhard Euler systematized the analysis of rotational motion for rigid bodies. A full understanding of the theory requires knowledge of differential equations, but its essence can be stated as the following: The time rate of change of a body's angular momentum (in magnitude and/or direction) is equal to the applied torque. The corresponding mathematical formulation is known as Euler's equations of rigid body motion. The equations not only explain increases or decreases in rotational speed when a torque is applied but also imply that with no torque present, the object maintains constant angular momentum (both in magnitude and in direction). A further implication is that when a torque is applied to a rotating object, after a small interval of time, the object's angular momentum will have changed direction so as to be more nearly aligned with that of the torque.

Euler's equations explain the behavior of gyroscopes. A gyroscope, in its simplest form, consists of a solid disk that is allowed to spin about an axle perpendicular to the flat face of the disk. This axle is supported at each end by a rigid frame and sets of bearings that allow the axle and disk to turn freely within the frame.

When the disk is spinning, it is evident to one holding the frame and attempting to turn it about a direction that is not aligned with the axle that the turning motion is strongly resisted.

This phenomenon, which is simply the reaction of the gyroscope to the applied torque needed to change the angular momentum's direction, is called "gyroscopic stiffness." Higher angular momentum (created by spinning the disk faster) will give greater stiffness, but any applied torque will cause some rotation of the frame. One of the great fascinations with gyroscopes is that this resulting rotation of the frame is not in the direction that one would expect. For example, if the spin axis is horizontal and the frame is held so that one views the disk rotating clockwise, then trying to turn the frame so that the far end would move downward and the near end upward (applied torque directed to the left) results instead in the entire frame rotating about the vertical axis. (Viewed from above, the direction of this rotation will be counterclockwise.) This phenomenon is explained by Euler's equations: The angular momentum changes in such a way as to align itself with the applied torque. As long as one continues to apply the torque, the rotation of the frame continues; removing the torque stops the frame's rotation.

A similar phenomenon occurs when one end of the frame is suspended (for example, by a string) and the other is released. With the disk spinning, gravity creates a torque in the horizontal direction (because it exerts a force that tries to rotate the gyro about a horizontal axis), but the gyro rotates around a vertical axis for the same reason. In both situations, the rotation about the vertical axis is called "precession." The precession rate (rate at which the frame rotates) is given approximately by the applied torque divided by the angular momentum of the disk.

A spinning top is supported by its point of contact with a horizontal surface. If perfectly balanced and set on a frictionless surface, the top would spin upright indefinitely; however, friction and imperfections in the surface cause it to tilt over. Gravity then exerts a torque in the horizontal direction and the top reorients itself as its angular momentum changes to align with the torque. Interesting behaviors of a top are caused by the additional torque created by the actual friction between the top's point and the surface. As the top loses angular momentum and begins to tilt further, the motion of its point creates frictional torque that changes the top's orientation in somewhat unexpected ways.

Applications

Even before Euler's elegant treatment of rigid body motion, Newton had analyzed the motion of simple gyroscopes and applied the same basic principles to the gyroscopic motion of Earth. Because Earth bulges slightly at the equator and because its spin axis is tilted relative to its plane of motion in the solar system, the sun's gravity pulls slightly harder on the part of the bulge closer to it, exerting a small torque on Earth (the torque direction is approximately in the earth's plane of motion). As a result, Newton predicted that Earth's spin axis must precess at a rate of about one circle every twenty-six thousand years.

In modern times, one of the principal uses of gyroscopes has been for navigation. To generate a reference direction for navigational purposes, the frame of the simple gyroscope is attached to two or more outer frames via a series of pivoting joints called "gimbals," which allow the inner frames to rotate inside the outer frames. This system provides a means of suspending the rotating disk, while allowing it to orient itself in any direction. The less friction there is in the gimbals, the more closely this approaches a torque-free system. By driving the rotor (the disk) with a small electric motor attached to the inner frame, the rotor speed can be held constant.

Placing this apparatus in a spacecraft provides a reference direction for the vehicle, since the spacecraft's motion will not affect the orientation of the rotor's spin axis. Electrical sensors at the gimbal joints provide input to the spacecraft's onboard computer, indicating how much the spacecraft has rotated relative to the reference direction. For the computer to determine completely the vehicle's orientation, there must be three such reference directions, mutually perpendicular orientations. Therefore, a minimum of three gyroscopes is required; in many spacecraft, a fourth gyroscope is included for redundancy in case of a failure. Such a system of gyroscopes constitutes part of a spacecraft's inertial reference assembly, from which ideally the onboard computer can determine the vehicle's position and orientation without having to use optical readings of star positions or other information. Realistically, though, even the small amount of friction in the gimbal bearings exerts torques on the rotor as the vehicle changes orientation, and the reference direction drifts away from its desired position. Periodically, the onboard computer must employ optical sightings of stars or other external information to determine the new reference directions of the rotors in the assembly. Nevertheless, this approach provides useful intervals of completely independent navigational capability between star sightings.

While the constant direction of a rotor axis makes the simple gimballed gyroscope ideal for navigation in space, this property is undesirable for Earth-bound navigation. As an aircraft, surface ship, or submarine crosses lines of longitude, the position of the geographic North Pole changes relative to the vehicle. For a gyroscope to indicate the direction to the North Pole, it is then necessary to make the rotor axis precess at exactly the rate that accounts for the apparent motion of the North Pole. This is accomplished by means of a spring mechanism attached to the gimbal frames and adjusted to provide the necessary torque for the current speed of the vehicle.

The rotor axis of this device, called a gyrocompass, then always points toward geographic North.

The property of gyroscopic stiffness is also applied to controlling the orientation of various objects. Historically, one of the first uses of this principle was in stabilizing the motion of a bullet. The cylindrical shape of a bullet makes it susceptible to tumbling as it experiences torques because of uneven atmospheric forces in flight. This tumbling not only causes a greater atmospheric resistance (and consequent reduction in range) but also can skew the bullet's path.

By cutting spiral grooves in a gun barrel ("rifling"), the bullet is made to spin about its long axis, giving it a high value of angular momentum and stiffening it against the torques created by the air.

It is for precisely the same reason that one deliberately imparts a spin to a football as it is thrown. Also, satellites are frequently spin-stabilized to resist the torques exerted by Earth's gravitational and magnetic fields or even by the extremely rarefied atmospheric gases at altitudes of 500 kilometers.

Context

The principles of rigid body motion have been applied widely since Newton's time.

From toy tops to modern high-speed rotating machinery, these concepts are crucial to a full understanding of how such systems behave, and in the case of engineering design, how to make the systems work in some desired fashion.

One of the most vexing problems for navigational gyroscopes has always been the elimination of friction in the gimbal joints. Improved lubricants in mechanical joints gave way to air bearings (in which the bearing elements are actually separated by a cushion of compressed air that prevents mechanical contact and thus reduces friction). That approach has, in some cases, been superseded by the use of magnetic bearings, which use electromagnets in one frame to keep it from coming into contact with the gimbal axle of the adjacent frame.

A technological innovation that will eventually replace all mechanical gyroscopes for use in inertial reference assemblies as well as gyrocompasses is the ring laser gyroscope.

Although misnamed (it has no rotating masses), this device operates by sending the two halves of a split laser beam in opposite directions around a ring (using mirrors). If the vehicle that houses this apparatus is undergoing a rotation about an axis perpendicular to the ring-plane, then the two beams will create an optical interference pattern on the screen where they are recombined. This pattern shifts on the screen at a rate proportional to the rotation rate of the vehicle, and an onboard computer (using an optical counter) can then determine in a given time interval how much rotation has occurred. As with a mechanical reference assembly, at least three ring laser gyroscopes are needed for navigation, but their great advantages are an absence of moving parts and consequent lack of friction.

Principal terms

ANGULAR MOMENTUM: the product of an object's rate of rotation and its moment of inertia around the axis of rotation

MOMENT OF INERTIA: a property of an object's mass distribution around a given axis of rotation

PRECESSION: the motion of a rotating rigid object in which the spin axis changes directions because of the presence of an applied torque

TORQUE: the effect of a force applied so as to cause a rotation

Bibliography

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Feynman, Richard P., Robert B. Leighton, and Matthew Sands. THE FEYNMAN LECTURES ON PHYSICS. Vol. 1. Reading, Mass.: Addison-Wesley, 1963. These lectures were intended to challenge some of the brightest physics students at the California Institute of Technology, yet they have exceptional clarity and appeal for general audiences. One of the few books that effectively treats the approximate nature of physical laws.

Frautschi, S. C., R. P. Olenick, T. M. Apostol, and D. L. Goodstein. THE MECHANICAL UNIVERSE: MECHANICS AND HEAT. New York: Cambridge University Press, 1986. This is the companion text to THE MECHANICAL UNIVERSE, a series of videotaped lectures produced by the Public Broadcasting System and based on Goodstein's lectures at the California Institute of Technology. The historical development of these subjects, carefully integrated with the superb graphic analysis, is one of the best treatments available.

Gamow, George. MATTER, EARTH, AND SKY. Englewood Cliffs, N.J.: Prentice-Hall, 1965. Although written as a textbook for freshman physics courses, this work is based on the author's long experience in lecturing and writing for general audiences interested in science. Successfully combines physics, chemistry, geology, and astronomy, including numerous helpful diagrams and some of Gamow's own cartoons.

Perry, John. SPINNING TOPS AND GYROSCOPIC MOTIONS. New York: Dover, 1957. This fine treatment of the dynamics of rotation presents the concepts of rigid body dynamics without any equations. There are numerous examples and detailed illustrations, many of them taken from everyday experience, to assist the nontechnical reader.

Rothman, Milton A. THE LAWS OF PHYSICS. New York: Basic Books, 1963. Using a minimum of mathematics, this book introduces the basic principles of classical and modern physics, with many excellent examples. It gives considerable attention to the philosophical aspects of human knowledge of physical reality.

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