If we try to rotate a spinning top by pressing down on one end of its axis of rotation and pulling up on the other end, the top will turn at a right angle to the direction we intend. Instead of pitching down, the top turns to the right. In order to understand this unexpected response we will create a top, using eight masses to form its body and two to define its long axis. The cyan velocity vectors in figure 1 show that the top rotates counterclockwise about its axis. You may watch this rotation by using the arrows at the bottom of the figure to advance through a series of screenshots. In figure 2 we give the top a sharp shove on each end by assigning the mass at the near end of the top's axis a downward velocity and the mass at the far end an upward velocity.

The first frame in figure 3 shows the top before it has responded to our pushing. 3D Mechanics models motion by initially letting masses move inertially. In the second frame we see how this inertial motion distorts the top as each mass moves along a straight line in accordance with its velocity. Connectors stretch or compress as they respond passively to the movement of the masses to which they are joined. Stretched connectors are tinted red and compressed ones green. We would expect that pushing down on the near end of the top would stretch the connectors on its upper half and compress those on its lower half, with the reverse holding true for the result of pulling up on the far end. We see exactly this pattern in the second frame. These distentions are at a maximum for the mass at the top of the figure and decrease in severity as we move away from this mass in either direction along the perimeter of the top.

After the inertial motion has progressed a bit the program takes the connector forces into account. In the next three frames of the animation the connectors gradually return to their original lengths, dragging their attached masses with them. In particular the uppermost mass is shoved toward us as the green connectors lengthen and the red connectors contract. This motion adds a velocity component which causes the net velocity of the mass to pivot towards us. The velocities of the remaining masses are similarly affected, with those on the upper half of the top swinging towards us while those on the lower half pivot away from us.

Figure 4. The final alignment of velocities causes rotation about the purple vector.

Figure 5. The top wobbles

In Figure 4 we have a slightly different view of the top after the connectors have returned to their original lengths. Note that the velocities of the masses suggest that the top will rotate about the axis indicated by the purple vector. Since this axis is not coincident with the axle, the top wobbles as is shown in figure 5. The lavender dots in the figures mark the previous positions of the near end of the axis.

After letting the top complete half a wobble we shall again push down on its near end and pull up its far end as shown in figure 11. Once more we expect that, following their initial distortion, as the connectors regain their original lengths they will cause the velocities of the masses on the upper half of the top to pivot towards us while those on the lower half will swing away from us.

But as figure 6 shows, this time these motions will cause the velocities of the masses to align parallel to one another so that rotation will once more be about the axis of the top.

The first time we shove on the top we cause it to wobble, but after half a wobble the velocities become so aligned that a subsequent shove undoes all that the first did. We leave the top once more spinning about its axis, the only effect of our efforts being a rotation of the top at right angles to the direction in which we shoved on it.

Figure 7. The top spins about its axis after completing the first half wobble.

Figure 8. The top wobbling.

Figure 7 provides another view of the top's motion as it responds first to one shove and then the other. Initially the top wobbles, tracing out the curved path shown. The second shove then arrests this looping motion and returns the top to a rotation about its own axis. By advancing the animation you can watch this rotation.

Had we applied continuous forces to the top instead of our two sharp blows the results would have been similar. Initially wobbling would have increased to a maximum and then fallen away to zero. This pattern would repeat over and over again, causing the top to make a series of half loops as it turns. In figure 8 the top was subjected to continuous forces and shows exactly this behavior. If the top spins very rapidly, the bobbing will be quite small and we will only notice that the top swings at a right angle to the direction in which we push.

Figure 6. The top responds to another push

Figure 9 shows that we can get the top to trace out even more interesting patterns if we give its ends an initial velocity at the moment we begin to push on it.