Rattleback Top

The rattleback is a top that defies convention.

It spins freely in one direction, but if spun in the other direction, its rotational motion becomes unstable; it starts to wobble and then will begin to spin in the opposite direction. In this regard, the rattleback seems to defy the physical law of conservation of angular momentum.

The rattleback is shaped as an elongated asymmetric semi-ellipsoid with a flat top. The unintuitive reversal of the rotational motion occurs because of its asymmetry. The asymmetry of the rattleback and frictional force from the table surface cause the rotational motion of the rattleback to become unstable. The friction transfers energy from the spinning (rotational energy) to the wobble (vibrational energy). The vibrational energy is then transferred back to rotational energy but in the preferred direction. Rotation in the preferred direction can also be initiated by pushing down on one end of the rattleback. This vibration is then transferred to a rotational energy.

The rattleback provides an opportunity to demonstrate many physics concepts in the classroom.

What are the axes of rotation?

The rattleback has three primary axes of rotation. It can spin about its center vertical axis. It can roll about its horizontal long axis, and it can rock or wobble about its horizontal short axis. In boating terms, these are called yaw, roll and pitch.

Have your students identify these axes and sketch the shape of the rattleback in each case. The asymmetry in the shape can be seen when you look at the rattleback along the line of the long horizontal axis.

Why isn’t angular momentum conserved?

Conservation of angular momentum is really just a special case of Newton’s second law of rotation when there is no net external torque (force applied at some distance from a rotation axis). In the case of the rattleback, however, there is a torque caused by the friction from the table top.

Try spinning the rattleback on various surfaces to illustrate the importance of friction.

What is Newton’s second law of rotation? dtLdext=Στ [1]

Angular momentum (L) is only conserved when there is no net external torque (∑Ŧ_ext=). In the rattleback case, the torque comes from the frictional force, which acts in a direction opposite to the direction of the spinning. Angular momentum depends on the angular velocity (ω) and the rotational inertia (I):

ωIL= [2] where ∑=iidMr_ⁱ² [3]

The rotational inertia is defined by the distribution of mass (M) at a distance r from the axis of rotation. In rotation, objects behave differently based on the distribution of mass about the axis of rotation.

You can demonstrate this with a simpler example than the rattleback:
Take two PVC pipes of equal length. Add a high density material (like clay or steel washers) to the middle section of one. Divide the same mass in half and add equal parts to the end sections of the second pipe. The pipes will weigh the same, but the distribution of the mass
and, therefore, the rotational inertia will vary.
Have students quickly rotate the pipes back and forth (by gripping them at the middle) and describe the differences in their movement. Which is harder to rotate?

How does stability come into play?

Objects in rotation preferentially spin around the axis with the smallest rotational inertia (eqn. [3]). Try spinning a book-sized block of wood in the air to demonstrate this. Spinning the wood around the longest or shortest axis of rotation is stable and the wood continues to spin. However, it wobbles when spun around the intermediate length axis.

The rattleback is more complicated in its instability due to the asymmetry around the axis with the smallest rotational inertia. However the instabilities are demonstrated when you tap the end of the rattleback. In this case, you are rotating the rattleback around the intermediate axis (short horizontal axis), which is the most unstable, similar to the book example. The wobble (rotation about the intermediate axis) is unstable and turns into rotation about the axis that is most stable.

Physics at Work

The rattleback is a great example of physics at work and the complexities of describing what appears to be a simple toy. Researchers have worked over decades to describe the motion of the rattleback in both physical and mathematical terms. Complicated mathematical equations and numerical models have been developed to simulate this behavior. But physicists who try to describe the physical behavior still struggle to intuitively understand it! The world’s most pressing problems in science, medicine, engineering and our natural environment are complex in much the same way.

References:

Physics

http://www.4physics.com:8080/phy_demo/rattleback.htm
Walker, J. The Flying Circus of Physics, 2nd ed., Wiley, 2007
(www.flyingcircusofphysics.com)

Crane, H.R. “How things work: The rattleback revisited.”
The Physics Teacher, 29(5):278-9, 1991.

Simulations

http://www.tu-chemnitz.de/ifm/english/e_ifm_ala4bsp_rattle.htm
http://www.autolev.com/WebSite/SampleProblemRattleback/Rattleback.html

History

http://128.174.130.156/LectDemo/descript/1148/more%20info.html
http://en.wikipedia.org/wiki/Rattleback