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Gyroscope

flywheel angular momentum torque

A gyroscope is heavy disk placed on a spindle that is mounted within a system of circles such that it can turn freely. When the disk, called a flywheel, is made to spin, the gyroscope becomes extremely resistant to any change in its orientation in space. If it is mounted in gimbals, a set of pivot and frame mountings that allow it freedom of rotation about all three axes, a fast-spinning gyroscope will maintain the same position in space, no matter how the frame is moved. Once the flywheel is set spinning, the spindle of a gyro in a gimbal mount can be aimed toward true north or toward a star, and it will continue to point that direction as long as the flywheel continues spinning, no matter what kind of turning or tilting the surface bearing it experiences. This stability has allowed the gyroscope to replace the magnetic compass on ships and in airplanes.

An interesting aspect of gyroscope motion occurs when a the flywheel is set rotating and one end of the spindle is set on a post using a frictionless mount. Intuitively, it would appear that the gyroscope should fall over, but it instead describes a horizontal circle about the post, flywheel still spinning. In apparent defiance of natural laws, it is simply obeying one of the simplest laws of physics, that of conservation of angular momentum. To understand the motion of a gyroscope, you must first understand angular momentum and torque. Angular momentum can be thought of as a rigid body's tendency to turn. Specifically, it tells us how much a given bit of the gyroscope flywheel contributes toward turning the flywheel about the spindle at any instant in time. For a small rotating object, angular momentum L is defined as

where m is the mass of the object, r is the distance of between the mass and the origin of rotation, and v is the instantaneous velocity of the mass. For the gyroscope, the angular momentum is obtained by considering it as composed of tiny bits, and adding up the contributions of each piece. Vectors have magnitude and a direction.

Torque, on the other hand, can be thought of as the rotational analog of force. Its effect depends upon the distance it is applied from the pivot point. Torque (τ) is defined as τ = r × f, where f is the applied force and r is the distance between the pivot and the point at which the force is applied. In this case Newton's second law permits us to state that force equals mass times acceleration (f = ma), so and we can define torque as τ = r × ma = r × mg, where g is acceleration due to gravity. When you hold a weight out horizontally with your arm fully extended, you feel the torque that the weight mg is applying at r, the length of your arm from your shoulder, the pivot point.

How does this apply to the gyroscope? When the flywheel spins at a high rate, the angular momentum vector is pointing straight along the spindle. The vector r points along the spindle also, until it reaches the flywheel, the center of mass m. On Earth's surface, gravity g is acting on the flywheel, pulling it downward. According to the righthand rule, the fingers of your righthand point in the direction of r, then bend to point in the direction of g, and your thumb will point in the direction of the torque vector t. Notice that t is in a horizontal direction, the same direction that the gyroscope describes as it turns. To conserve angular momentum, the gyroscope will pivot about the support post, or precess, in an effort to align the angular momentum vector with the torque vector.

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