The Precessing Top

When a symmetric top is set spinning with angular velocity ω c about its own axis, it traces out a circle about a vertical direction with angular velocity ω z , as shown in Fig. 1. In most textbooks [1, 2, 3] the precessional angular velocity ω z is calculated under the assumption that it is much smaller than the spin angular velocity ω c . This makes for a simplification of the calculation by considering the total angular momentum as due to the spin motion only, and thus directed along the symmetry axis. In this case, the angular momentum along the symmetry axis is L = Iω c , where I is the moment of inertia about this same axis. The tip of this vector describes a circle of radius L sin θ around the vertical direction. The torque of gravity is mgl sin θ, where m is the mass of the top and l is the distance of its center of mass C to O. In a time interval ∆t the torque changes the angular momentum by Lsin θω z ∆t in a direction tangent to the circle. Equating torque to time change in angular momentum yields the usual formula ω z = mgl/Iω c for slow precession [2]. In this note I drop the simplifying assumption that the total angular momentum is solely along the symmetry axis in order to obtain the following general expression for ω z ,