Period of Precession - (Torque-induced - Classical Newtonian)
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called nutation. In physics, there are two types of precession: torque-free and torque-induced.
In astronomy, “precession” refers to any of several slow changes in an astronomical body’s rotational or orbital parameters, and especially to Earth’s precession of the equinoxes. (See section Astronomy below.)
Torque-induced precession (gyroscopic precession) is the phenomenon in which the axis of a spinning object (e.g.,a gyroscope) describes a cone in space when an external torque is applied to it. The phenomenon is commonly seen in a spinning toy top, but all rotating objects can undergo precession. If the speed of the rotation and the magnitude of the external torque are constant, the spin axis will move at right angles to the direction that would intuitively result from the external torque. In the case of a toy top, its weight is acting downwards from its center of mass and the normal force (reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess.
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, because 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.
The torque vector originates at the center of mass. Using omega=(2*pi)/T we find that the period of precession is given by the formula shown here.In general, the problem is more complicated than this, however. Related formulas
|Tp||period of precession (sec)|
|Is||moment of inertia (kg*m2)|
|r||perpendicular distance of the spin axis about the axis of precession (m)|
|Ts||period of spin about the spin axis (s)|