Aerodynamic-Torque Induced Motions of a Spinning Football and Why the Ball’s Longitudinal Axis Rotates with the Linear Velocity Vector

Abstract

This paper presents an explanation of why a spinning football rotates so that the spin axis remains nearly aligned with the velocity vector, and approximately parallel to the tangent to the trajectory. The paper derives the values of the characteristic frequencies associated with the football’s precession and nutation. The paper presents a graphical way of visualizing how the motions associated with these frequencies result in the observed “wobble” of the football. A solution for the linearized dynamics shows that there is a minimum amount of spin required for the motion to be stable and for the football not to tumble. This paper notes the similarity of this problem to that of spun projectiles. The results show that the tendency of a football to align itself with and rotate with the velocity vector is associated with an equilibrium condition with a non-zero aerodynamic torque. The torque is precisely the value required for the football to rotate at the same angular rate as the velocity vector. An implication of this is that a release with the football spin axis and velocity vector aligned (zero aerodynamic torque) is not the condition that results in minimum motion after release. Minimum “wobble” occurs when the ball is released with its symmetry axis slightly to the right or left of the velocity vector, depending on the direction of the spin. There are additional forces and moments acting on the football that affect its trajectory and its stability, but it is not necessary to consider these to explain the tendency of the ball to align with the velocity vector and to ”wobble.” The results of this paper are equally applicable to the spiral pass in American football and the screw kick in rugby.