Spinning tops need to be visualised in reference to 2 frames - the space frame ( x , y , z ) that is effectively the
'real life' frame that we have been using up until now when describing the position of the pendulums; and the
body frame ( x' , y' , z' ) that is basically another set of ( x , y , z ) axes taken with the z'-axis pointing up through
the middle of the spinning top. This body frame obviously changes its orientation with respect to the space
frame as the spinning top moves.
There are 3 degrees of freedom needed to describe the position of the spinning top - , which is the angle between the z-axis of the space frame and the z'-axis of the body frame; , which is the rotation angle of the z'-axis of the body frame around the z-axis of the space frame; and , which is the rotation angle of the spinning top around its own z'-axis (that of the body frame).
A diagram of the symmetric spinning top is shown below:
Lagrange's equations are needed again to derive the equations of motion used to model the
spinning top. The equations of motion for the symmetric top are shown below:
(Click Here to see the proof)
The applet below simulates the motion of the symmetric spinning top:
It is possible to adjust the initial values for , , , , and . It is also possible to alter the values for the mass m of the top, and l, which is the length of the top along the z'-axis from the origin (point of the top) to the centre of the circle at the other end of the top.
In addition to this, you can activate/deactivate gravity; adjust the framerate of the animation (if you want to watch the applet in slow motion, for example); and turn a reference dot on/off. This reference dot highlights a fixed point on the top, and is designed to help visualise how fast the top is spinning.
© Michael Hart 2004