The Magnus Effect and the Curved Path of a
Spinning Ball
Suppose a spherical ball is moving in a stationary, viscous fluid with
velocity \(\vec{v}_O\) and spinning with angular velocity
\(\vec{\omega}\). Then there is a lift force \(\vec{F}_{mag}\), called the Magnus force, that acts on the
ball and affects its path. This force has the properties
where \(C_L\) is the lift coefficient,
which is 1) itself a function of the shape of the blunt object, the
Reynolds number, and the Mach number, and 2) read off standard tables,
\(\rho\) is the density of the fluid, and
\(A\) is the relevant surface area, usually \(\pi d^2\) for a sphere
(where \(d\) is
the diameter of the sphere). Note that \(\frac{1}{2}\rho||\vec{v}_O||^2
\) is the formula for the dynamic pressure of a fluid, so
\(\frac{1}{2}\rho||\vec{v}_O||^2 A\) is indeed a kind of force.
(My fascination with modeling the phenomenon of "banana kicks" of
soccer
balls off of corner kicks into the goal is actually what got me into
STEM way back in
1986-9. In 1986, I was taking Calc I as a junior in high school and
captain of the soccer team, but I just couldn't figure out for the life
of me how to model
the angular velocity of a spinning soccer ball with just my 1D Calc I
knowlege. Then, in 1989, in
my Calc III course, we went over 3D math, it showed me how to
model spin angular velocity vectors (which are actually pseudo-vectors, I found out decades later), and I was hooked on STEM. Finally, in
my Fluid Mechanics course in 2025, I learned how to model the Magnus
effect itself.)
