Figure 4. The calculated trajectories of the contact point of the rattle back. The blue represents a torque in one direction due to tipping and the red represents torque in the opposite direction.
Hermann Bondi, "The rigid body dynamics of unidirectional spin," Proc. R. Soc. A 405, 265-274, (1986).
Robert Walgate, "Tops that like to spin one way," Nature 323, 204 (1986).
G. T. Walker, Q. Jl. pure appl. Math. 28, 175 (1986).
Fig. 3 When the Rattleback shown above rocks from side to side, its skew symmetry makes it rock along the diagonal line. When it rocks to the left, the contact point is above the horizontal line and the object center of mass tends to fall downward relative to the contact point. Friction thus acts downward to prevent slipping at the contact point. When the Rattleback rocks to the right, the contact point is below the horizontal line and the object center of mass tends to fall upward relative to the contact point. Friction thus acts upward to prevent slipping at the contact point. The forces acting on either side of the center of mass tend to produce a counterclockwise rotation.
Fig. 2 The forces acting on a falling ladder in contact with the ground are the weight W, the normal force N, and the frictional force f. Because there is no force balancing the frictional force, the center of mass accelerates to the right.
If you look carefully at the Rattleback, it has an unusual symmetry. The elliptical shape seen from above rotates slightly about a vertical axis as one progresses down towards the bottom contact point (say clockwise as represented by in figure 3). It has what is called a skew symmetry. As the Rattleback rocks back and forth approximately along its long axis, the center of mass alternately rolls about that axis above and below the contact point with the ground (see Figure 3.) for the five Rattlebacks positioned as shown above. When the center of mass lies above the contact point, the actual contact point is also below the center of mass line and the frictional force must push upward to prevent the Rattleback from slipping. However, like a falling ladder, the center of mass is off to the side of (above) the contact point and the frictional force acting pushes in the direction of the center of mass. As a result, the center of mass moves (accelerates) away from the contact point. This is obviously true for the falling ladder. For the Rattleback, where the center of mass is above the contact point on one side and below on the other, the frictional force pushes to produce the same rotation about the center on both sides. So we can understand the preferential rotation. This is true only if acceleration does not compensate for this force. Thus we must solve the problem completely to be sure our intuition is correct.
The toy shown above goes by many names: space pets, the Celt, or the Rattleback. When it is spun in one direction, it spins freely without stopping. When spun in the other direction, it slows down rapidly, begins to rock back and forth wildly (rattle), and then spins in the opposite direction. This behavior is counter-intuitive to most spinning behavior we observe. Is it contrary or does it violate the laws of classical physics that we know so well? What properties are required for this behavior?
The strange behavior of this "toy" is discussed by us (unpublished) intuitively and analysed following the work of Bondi (1986), however, the body coordinates are expressed differently and lead to more transparent results. Rather than investigate the stability for initial motion, the equations of motion are integrated numerically and demonstrate the strange behavior of the rattleback. While Bondi produced mathematics supporting the rattleback behavior, he claimed no intuition as to the behavior. We supply such intuition.
To understand the rattleback intuitively, consider first a ladder that is allowed to fall without slipping. The vertical components of forces on the ladder are the weight located at the center of mass of the ladder and the normal force from the ground acting upward. The normal force does not necessarily balance the weight and the difference determines the downward acceleration of the center of mass. There is also a frictional force that keeps the ladder from slipping as long as the ladder is contact with the ground. This force is unbalanced and leads to a horizontal acceleration of the center of mass. The ladder is pushed (to the right) by the frictional force.