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.
Highlights
This paper will use the generic term “wobble” to describe the general angular motion of the football until we can be precise about what is causing the motion
This paper addressed the question of why the symmetry-axis of a spinning football tends to align itself with its velocity vector
Previous papers correctly hypothesized that this was caused by a torque acting on the football, but did not provide mathematical evidence derived from a hypothesized physical model that this should be the case
Summary
This paper will use the generic term “wobble” to describe the general angular motion of the football until we can be precise about what is causing the motion. The results in [4] begin by assuming the incorrect characteristic frequencies from [3], and claims to derive a model for ”first-order differences between the direction of the velocity, the orientation of the long axis, and the direction of the total angular momentum.” If this is understood to mean linear dynamics near an equilibrium point, this paper presents the correct solution to that problem. In a frame of reference defined by the velocity vector, the equations of motion are approximately linear and can be solved analytically This solution provides a stability condition in terms of a relationship between the spin momentum and the rate of change of aerodynamic torque with respect to angle relative to the velocity vector. If the initial condition of the ball is such that the nose of the ball is below the velocity vector or the initial angular velocity is large enough for the angle of attack to become negative, the argument provided in [3] will cause the sideslip to have the opposite sign, the torque will have the opposite sign, and the explanation predicts the angular rotation of the ball will be in the incorrect direction
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