All that is necessary to create lift is
to turn a flow of air. The airfoil of a wing turns a flow, but so
does a spinning baseball. The
details of how the force is generated are fairly complex, but the
magnitude of the force F depends on the radius of the ball b,
the spin of the ball s, the velocity V
of the pitch, the density r of the air, and an
experimentally determined lift coefficient Cl.
F = Cl * 4 /3 * (4 * pi^2 * r * s * V * b^3)
where pi is the number 3.14159 .. the ratio of the circumference to
the diameter of a circle.
As the force acts on the ball, it is deflected along its
flight path. If we neglect the viscous forces on the ball, which
slow it down and change the magnitude and direction of the force, we
have a constant force always acting perpendicular (at a right
angle) to the flow direction. The resulting
flight path
is a circular
arc. On the figure, we see the trajectory of the baseball as it moves
from the pitching mound to the plate from right to left. Each white dot
is the location of the ball at .05 second intervals. It gets to the
plate pretty fast! The radius of curvature R of the flight path
depends on the velocity V of the pitch and the acceleration a
produced by the side force.
R = V^2 / a
We can determine this acceleration from
Newton's second law of motion using the force for a spinning ball
and the mass (m) of the baseball (5 oz.).
a = F / m
Since the radius of curvature
depends on the force, all the factors that affect the force will also
affect the trajectory.
Collecting all the information into one equation:
R = (3 * m * V) / (16 * Cl * r * s * b^3 * pi^2)
We can use this equation to make some predictions about the
trajectory of a spinning ball.
Higher spin s produces a smaller radius of curvature R and
a sharper curve.
Higher velocity V produces a larger radius of curvarture and
a straighter curve.
A ball with a smaller mass, like a pingpong ball, has a lower radius
of curvature and curves more.
At higher altitudes,
the density r is lower producing a larger radius of curvature and a
straighter path.
It's very hard to make a ball curve on Mt. Everest, no matter
how much it spins. That is good for golfers and bad for baseball pitchers.
The altitude effect helps to explain the high batting averages
and poor earned run averages for the Colorado Rockies who play 81 games
a year at a high altitude ball park. It is much harder to throw a
curve ball at Coors Field than at Dodger Stadium.
Knowing the radius of curvature and the distance from the pitching
mound to home plate, we can also calculate the distance that the ball
is deflected (Yd) along the flight path.
Here we have a sketch of the geometry of the pitch. There is a
right triangle formed by the radius of curvature R,
the distance to the plate D at the top, and the radius
of curvature minus the deflection distance R  Yd on the
right. We can then use the
Pythagorean Theorem
to relate the sides of this triangle:
R^2 = D^2 + (R  Yd)^2
Now let's do a little algebra:
R^2  D^2 = (R Yd)^2
sqrt(R^2  D^2) = R Yd
Yd = R  sqrt(R^2  D^2)
The pitcher can release the
ball at different distances from the center of the plate (Yp). The
difference between Yp and Yd will give the final location of the
pitch relative to the center of the plate.
You can investigate the effect of
aerodynamics on throwing a curve ball by using the
CurveBall Java Applet.
Have fun !
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