Here, we explore the behavior of a book tossed in the air by developing a model that governs the book’s motion. Through simulation and animation, we wish to confirm the well-known result that rotation about the book’s intermediate principal axis is unstable in the face of rotational perturbations; in contrast, the spin behavior about the book’s other two principal axes is stable. This topic is discussed in various textbooks, such as [1] and [2]. A related scenario involves tossing a tennis racket into the air while giving it an initial angular velocity about an axis parallel to the racket’s face, which causes the racket to flip through 180. This phenomenon is explained in [3] and can be observed in the video in Figure 1.

## Equations of motion

Referring to Figure 2, we represent the book by a rigid and uniform rectangular prism of mass , length , width , and thickness . The book’s reference configuration is defined such that the book’s mass center is located at the origin and the space-fixed basis is aligned with the book’s principal axes. We locate the mass center relative to the origin using a set of Cartesian coordinates, . The book’s corotational basis is related to the fixed basis by a rotation such that .

We can obtain the equations of motion governing the translation of the mass center and the book’s orientation by applying balances of linear and angular momenta, respectively. Neglecting the effect of air drag, the only force exerted on the book after being tossed into the air is the weight acting at the center of mass, and so a balance of linear momentum yields the system of equations

while a balance of angular momentum with respect to the mass center gives

where are the corotational components of the book’s angular velocity , and are the principal moments of inertia:

Suppose we use a set of 3-2-1 Euler angles , , and to parameterize the rotation tensor . In this case, the tensor’s components , when arranged in a matrix , have the representation

Additionally, the corotational angular velocity components are related to the Euler angles , , and and their rates of change by

Lastly, by introducing the state vector

we can express the the governing equations in (1), (2), and (5) for the translation of the book’s mass center and the book’s orientation in the first-order matrix-vector form for numerical integration in MATLAB, where

## Simulation and animation

In simulating the book’s behavior when tossed in the air, we take the book to initially be in an unrotated state with its mass center at the origin: and . The book is launched vertically, in which case and , and an initial spin is imparted to examine the stability of rotation about each principal axis. Suppose we first toss the book with an initial spin rate about its longitudinal axis (i.e., its first principal axis) and with small perturbations about the other two principal axes: and . As shown in the animation in Figure 3, the resulting motion, using values of the physical parameters representative of a small textbook, indicates that the tossed book spins stably about this axis. One corner of the book is highlighted with a blue marker to help better visualize the rotation.

Next, launch the book with a spin largely about its third principal axis, which corresponds to the axis perpendicular to the book’s front and rear covers: and . The animated response in Figure 4 demonstrates that rotation about this axis is also stable.

Lastly, impart a spin mostly about the book’s intermediate (i.e., second) principal axis such that , , and . As expected, and similar to the tennis racket behavior shown in Figure 1, we see from the animation in Figure 5 that the book tumbles in the air instead of stably spinning about this axis.

For convenience, an interactive, web-based JavaScript simulation that demonstrates the same behavior animated in Figures 3–5 is accessible through Figure 6.

## Downloads

The MATLAB code used to generate the animations in Figures 3, 4, and 5 is available here.

## References

- Marsden, J. E., and Ratiu, T. S.,
*Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems*, 2nd ed., Springer-Verlag, New York (1999). - O’Reilly, O. M.,
*Intermediate Dynamics for Engineers: A Unified Treatment of Newton-Euler and Lagrangian Mechanics*, Cambridge University Press, Cambridge (2008). - Ashbaugh, M. S., Chicone, C. C., and Cushman, R. H., The twisting tennis racket,
*Journal of Dynamics and Differential Equations***3**(1) 67-85 (1991).