Suggested activity. Calculate the axial and transverse central principal moments of inertia for a homogeneous cylinder of mass m whose height (length) equals h, and whose radius equals R. What is the ratio of the axial and transverse moments of inertia? At what value of h/R these moments of inertia are equal, that is, the cylinder becomes a spherical top? Verify your answer by a simulation experiment.

The geometrical interpretation of inertial rotation of a circular cylinder considered above is also applicable to a special case of a cylinder whose longitudinal and transverse moments of inertia are equal. For such a body called a spherical top moments of inertia for all possible axes passing through the center of mass are equal. For such bodies, the angular velocity and the angular momentum always have a common direction, and an arbitrary axis is the axis of free rotation. This means that for a spherical top vector w of the angular velocity and the momentary axis of rotation keep their directions in space fixed (they do not precess), and the immovable cone degenerates into a ray directed along the angular momentum L (vertically on the screen). Rolling of the moving cone attached to the body is reduced in this case to a uniform rotation of this cone around the vertical ray. This ray (the degenerate immovable cone) lies on the lateral surface of the moving cone and is directed along ω. Any point of the body (say, the endpoint of an arrow sticked into the body) generates a circular path whose center is located on the axis of rotation. Click here to simulate the inertial rotation of a spherically symmetrical body. Click also here to put the applet on the upper part of the screen.

For a sufficiently short cylinder (disc with h < 1.732 R) the imaginary cone fixed to the cylinder is touching the immovable space cone internally, by its interior surface. In this case the geometrical interpretation of inertial rotation is less obvious and even counterintuitive in some way. We note that when the symmetry axis of the cylinder performs precession counterclockwise and the wide moving cone is rolling by its interior surface without slipping over the enveloped immovable cone, the cylinder itself is spinning (rotating around its axis) clockwise – in the opposite sense with respect to the precession. This is clearly seen from the decomposition of the angular velocity vector (yellow arrow on the screen) onto components (red arrows) corresponding to precession and axial rotation: the component that corresponds to spin makes an obtuse angle with L (is directed downward on the screen). Click here in order to simulate the inertial rotation of a thin circular disc. Click also here to put the applet on the upper part of the screen.

Another example of a symmetrical top is given by a rectangular prism whose base is a regular polygon. Below you can see an applet with the simulation of a torque-free rotation of a prism whose base is a square.

If the height h (length) of the prism equals the side a of the square, the prism becomes a cube. For a cube all three principal moments of inertia are equal. This means that a cube is dynamically equivalent to the spherical top. At arbitrary direction of the angular velocity the angular momentum vector has the same direction. Hence the torque-free rotation of a cube is very simple: the cube always rotates uniformly about an axis whose direction in space is constant. Try to verify this by the simulation experiment at h = a for different directions of the angular velocity.

A detailed theoretical description of an inertial rotation of an axially symmetrical body can be found in the paper "Inertial Rotation of a Rigid Body".

Controlling the program.
The simulation program allows you to vary parameters of the system and conditions of the simulation. You can rotate the cube with the image of the cylinder around the vertical and horizontal axes (for a more convenient point of view) simply by dragging the pointer of the mouse inside the applet window (moving the pointer with the left button pressed). If in addition you will hold the "Control" key on the keyboard, the image will move in the desired direction. If you will hold the "Shift" key, dragging the mouse pointer will change the scale – the image will move closer or farther.

Rotation of the cylinder can be represented in a convenient time scale. To change the time scale, you can vary the delay value (in milliseconds) by dragging a slider in the lower part of the control panel. The functions of other controls are rather obvious. The upper button starts the simulation and makes a pause. The second button allows you to execute the simulation by steps. The third button ("Initialize") retrieves the initial conditions. The fourth ("Reset") restores the default values.

At first acquaintance with the program, instead of entering the values of parameters, you can open the list of predefined examples and choose from it a suitable example. Performing experiments on your own, you can vary the values of parameters either by dragging the sliders, of by typing the desired values with the keyboard. Before changing parameters, you should pause the simulation by using the "Start/Pause" button. When you type new values into the window, the colour of its background changes to bright yellow. You should finish the input by pressing the "Enter" key. If the value is admissible, the window assumes its usual colour.

Inertial properties that characterize rotation of a symmetrical body are given by two central moments of inertia: one about the axis of symmetry, the other about any transverse axis passing through the center of mass. Only the ratio of these moments of inertia is essential for the simulation. For the cylinder, this ratio depends on the h/R ratio. In the program, you can arbitrarily change this ratio.

Another parameter that you can vary in the simulation program is the angle between the axis of symmetry of the cylinder and direction of the angular velocity vector (direction of the momentary axis of rotation). On the control panel of the program, this angle is labelled "Tilt." The value of this angle should be entered in degrees. Admissible values lie in the interval from 0 up to 90 degrees. The magnitude of the angular velocity can be varied in the interval from 0.5 up to 10 arbitrary units. Variation of this parameter influences the speed of axial rotation, but doesn't change characteristic features of the body motion.

We have already mentioned earlier how you can switch on drawing of the trajectory of an arbitrary point of the body by marking the check box "Show point trace" and indicating the angular position of the desired point ("TraceAngle" parameter). The background color of the applet window depends on the state of the check box "Dark background".

The applet presented here was created with the help of Easy Java Simulations tool developed by Francisco Esquembre, professor at University of Murcia, Spain.