Precession - Torque-induced - Classical (Newtonian)

Classical (Newtonian)

Precession is the result of the angular velocity of rotation and the angular velocity produced by the torque. It is an angular velocity about a line that makes an angle with the permanent rotation axis, and this angle lies in a plane at right angles to the plane of the couple producing the torque. The permanent axis must turn towards this line, since the body cannot continue to rotate about any line that is not a principal axis of maximum moment of inertia; that is, the permanent axis turns in a direction at right angles to that in which the torque might be expected to turn it. If the rotating body is symmetrical and its motion unconstrained, and, if the torque on the spin axis is at right angles to that axis, the axis of precession will be perpendicular to both the spin axis and torque axis.

Under these circumstances the angular velocity of precession is given by:

In which I_s is the moment of inertia, is the angular velocity of spin about the spin axis, and m*g and r are the force responsible for the torque and the perpendicular distance of the spin axis about the axis of precession. The torque vector originates at the center of mass. Using =, we find that the period of precession is given by:

In which I_s is the moment of inertia, T_s is the period of spin about the spin axis, and is the torque. In general, the problem is more complicated than this, however.

There is a non-mathematical method of visualizing the cause of gyroscopic precession. The behavior of spinning objects simply obeys the law of inertia by resisting any change in direction. A solid object can be thought of as an assembly of individual molecules. If an object is spinning, each molecule's direction of travel constantly changes as that molecule revolves around the object's spin axis. If a force is applied to the object to induce a change in the orientation of the spin axis, the object behaves as if that force was applied 90 degrees ahead, in the direction of rotation. Here is why: imagine the object to be a spinning bicycle wheel, held at the axle in the hands of a subject. The wheel is spinning clock-wise as seen from a viewer to the subject’s right. Clock positions on the wheel are given relative to this viewer. As the wheel spins, the molecules comprising it are travelling vertically downward the instant they pass the 3 o'clock position, horizontally to the left the instant they pass 6 o'clock, vertically upward at 9 o'clock, and horizontally right at 12 o'clock. Between these positions, each molecule travels a combination of these directions, which should be kept in mind as you read ahead. If the viewer applies a force to the wheel at the 3 o'clock position, the molecules at that location are not being forced to change direction; they still travel vertically downward, unaffected by the force. The same goes for the molecules at 9 o'clock; they are still travelling vertically upward, unaffected by the force that was applied. But, molecules at 6 and 12 o'clock ARE being "told" to change direction. At 6 o'clock, molecules are forced to veer toward the viewer. Then, as molecules pass 12 o'clock, they are forced to veer away from the viewer. The inertia of those molecules resists this change in direction. The result is that they apply an equal and opposite force in response. At 6 o'clock, molecules exert a push directly away from the viewer. Molecules at 12 o'clock push directly toward the viewer. This makes the wheel as a whole tilt toward the viewer. Thus, when the force was applied at 3 o'clock, the wheel behaved as if the force was applied at 6 o'clock--90 degrees ahead in the direction of rotation.

Precession causes another peculiar behavior for spinning objects such as the wheel in the scenario described. If the subject holding the wheel removes one hand from the axle, the wheel will remain upright, supported from only one side. However, it will immediately take on an additional motion; it will begin to rotate about a vertical axis, pivoting at the point of support as it continues its axial spin. If the wheel was not spinning, it would topple over and fall if one hand was removed. The initial motion of the wheel beginning to topple over is equivalent to applying a force to it at 12 o'clock in the direction of the unsupported side. When the wheel is spinning, the sudden lack of support at one end of the axle is again equivalent to this force. So instead of toppling over, the wheel behaves as if the force was applied at 3 or 9 o’clock, depending on the direction of spin and which hand was removed. This causes the wheel to begin pivoting at the point of support while remaining upright.

The special and general theories of relativity give three types of corrections to the Newtonian precession, of a gyroscope near a large mass such as the earth, described above. They are:

• Thomas precession a special relativistic correction accounting for the observer's being in a rotating non-inertial frame.
• de Sitter precession a general relativistic correction accounting for the Schwarzschild metric of curved space near a large non-rotating mass.
• Lense-Thirring precession a general relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.