Partnership for Integration of Computation into Undergraduate Physics

The Intermediate Axis Theorem, or Dzhanibekov Effect, [(YouTube link)](https://www.youtube.com/watch?v=1VPfZ_XzisU) captivates students and experts alike, but working out the effect analytically is quite involved and not always illuminating. This activity shows students how the intermediate axis theorem can be computed using just the basic relationship between angular momentum and angular velocity using the moment of inertia tensor and knowledge of linear algebra. Since the shape spins with no external torque, the angular momentum is constant. However, the angular velocity is not. As the shape rotates, the principal axes rotate through Cartesian space, which affects the calculation of angular velocity from angular momentum. This changing angular momentum, in turn, affects the rotation of the object. These two aspects work together to cause the dramatic flipping back and forth. The key to getting the simulation to work accurately is the use of the midpoint method to step forward in time, rather than the simpler Euler method. Besides completing the code to see a rectangular box spin and tumble, the students are asked to take it a step farther in one of two ways: 1. Get more quantitative about investigating the rate of flipping, taking careful data of how it is affected by the initial conditions. 2. Change the shape to a more complex one, such as a T, or even a tennis racket. The principal moments of inertia and the center of mass need to be computed correctly for this to work.