Author's home page A small simulation program (Java-applet) presented here visualizes torque-free rotation (or "inertial rotation") of a circular cylinder (or a circular disc). This is a special case of a symmetrical top, that is, a body whose two of three principal moments of inertia are equal. Kinematically, such inertial rotation is called precession. General properties of the torque-free rotation are discussed on the web page "Torque-free rotation of a symmetrical top."
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. (Click here to put the applet on the upper part of the screen.) We can associate also this obvious geometrical interpretation of the free precession (as rolling of the moving cone without slipping over the surface of the immovable cone) with the decomposition of vector ω of momentary angular velocity onto the vector sum of two components shown by red arrows in the applet. One of these components corresponds to the axial rotation (spinning) of the body around its axis of symmetry. This component of the angular velocity is directed along the axis of symmetry, that is, its direction inside the body does not change. In space, this component generates a circular cone together with the axis of the body. The other component does not change its direction in space: this component corresponds to precession of the axis of symmetry about angular momentum L, whose direction in space (vertical on the screen) is preserved in the absence of external torques. Points of the cylinder located on its axis of symmetry trace circular paths whose centers lie on the axis of the immovable cone, that is, on the axis of precession (vertical on the screen). We can consider complicated motions of points that doesn't lie on the axis of the cylinder as a superposition of two relatively simple motions, namely of a rotation around the axis of the cylinder with simultaneous motion of this axis in space (precession) along the surface of a vertical cone. The simulation program can show trajectories of such points. To make the program do this, mark the check box on the control panel labelled "Trace of a point", and indicate position of the desired point by entering the angle between the axis of the cylinder and the direction to the point in question. For more convenient observation, the program shows the trajectory traced by the end point of an imaginary long arrow fixed to the cylinder. The end point of this arrow lies beyond the surface bounding the cylinder. All points of this arrow (that starts at the center of mass) trace similar trajectories, but the path of its end point shows their peculiarities in a larger scale. Click here in order to observe the trajectory of a point located on the surface of a moving cone. (Initially this point lies on the momentary axis). Click also here to put the applet on the upper part of the screen.
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. 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". Simulation of inertial rotation of an arbitrary symmetrical top can be found on the web page "Torque-free rotation of a symmetrical body." To the top ... The applet presented here was created with the help of Easy Java Simulations tool developed by Francisco Esquembre, professor at University of Murcia, Spain.
Last revision – 20 June 2012. |
Simulations in Physics |