FREE ROTATION OF AN AXIALLY SYMMETRICAL BODY (Eugene Butikov)

[Fu-Kwun Hwang]

http://faculty.ifmo.ru/butikov/Applets/Precession.html


You can make a pause in the simulation and resume it by clicking on the button "Start/Pause" on the control panel located on the left-hand side of the applet window. This control panel allows you also to vary parameters of the system and conditions of the simulation. By dragging the pointer of the mouse inside the applet window (moving the pointer with the left button pressed) you can rotate the image around the vertical and horizontal axes for a more convenient point of view.

The vector of angular velocity (the yellow arrow on the black background) shows in space the direction of momentary axis of rotation. The set of these momentary axes at different moments of time forms in space a circular cone whose vertex is located at the center of mass and whose axis is directed along vector L of the angular momentum (vertically on the screen). This space cone is also called the immovable axoid (see the right-hand side of the applet window).

Next we imagine one more circular cone, this time attached firmly to the body. The vertex of this cone is also located at the center of mass, and its axis is directed along the axis of symmetry of the body. Let the generator of this cone be the vector of angular velocity ω (the yellow arrow on the screen), that is, the momentary axis of rotation. In other words, the lateral surface of this cone associated with the body is formed by the set of momentary axes of rotation at different moments of time, and shows how these axes are located inside the body (relative to the body). For this reason, this imaginary cone, associated with the rotating body, is called the moving axoid (or the body cone).

The moving and immovable cones touch one another (outwardly for a prolate body, whose transverse moment of inertia is greater than longitudinal) by their lateral surfaces along vector ω that shows the momentary axis of rotation. All the points of the body, which are located at a given moment of time on the momentary axis of rotation, have zero linear velocities. This means that the moving cone (attached to the body) is just rolling without slipping over the surface of the immovable cone. This clear geometrical interpretation of kinematics of the free precession (of the inertial rotation) is shown in the right-hand side of the applet.