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Magnus Force: Pitching a Baseball and Shooting a Puck
The Magnus Force
When we discussed the affect of air drag we ignored the fact an object may also be
rotating
Rotation causes the air to affect the object differently on either side
This difference in force on the object due to air resistance is called Magnus Force
By rotating the air on the side of rotation will be moving more quickly, causing the ball
to feel an extra force making it move horizontally
Consider the motion of air around the ball
The air’s speed near the bottom of the ball is increased by the ball’s rotation and, vice
versa, the air’s speed is lowed down near the top fo the ball due to its rotation
>Hence, the air on the bottom of the ball moves faster than the air on top of the
ball
Therefore, the pressure on the top of the ball is greater compared to the bottom
Recall the Bernoulli Effect: as the air’s velocity increases along a streamline the pressure
decreases
The difference in speed( and hence, pressure) causes the indicated Magnus Force
 The Magnus force acts at right angles to the air resistance and the ball’s velocity
Equation(copy from slides)
 1/2ClpAv2= ClA(1/2pv2
Dynamic pressure= 1/2pv2
Fmagnus is propotional to w*v where w is
Rotation of a Pitched Baseball
The pitch is thrown from a raised mound which is 60ft 6 inches (18.4m) from home
plate and is 10” (25cm) above the infield
A typical ball is thrown at a speed btwn 85100mph (135 to 160km/h)
The batter has about 0.4s to react once the pitch is thrown
Strike Zone
The Ball
Stitching that holds the ball together is very pronounced and helps create resistance and
provide grip
Pitchers put different kings of spin on the ball to use the roughness of the surface and the
seams to alter the Magnus and drag forces on the ball
The drag coefficient for a smooth ball is much greater than a rough ball at the speed of a
pitch (see chart)
The broken line shows where Fdrag=Fgravity

Measurements of the Drag vs. Magnus Force for a Pitched Ball
Magnus force is shown here for a ball rotation of 1800 revolutions per minute (RPM;
that is 30 times/second) a typical curveball rotates 20 to 30 times in a second
Magnus is roughly constant with speed
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Types of Pitches
Rotation of the ball seen by the batter for an overhand pitch throw by a righthanded
pitcher
The direction of the rotation is also the direction of the Magnus Force
*The pitcher sses the ball cruve to the left while the batter sees it curve to the
right
The curve ball is thrown by the pitcher spinning the ball as it leaves his hands
Though the curve ball travels significantly slower than the fast ball, it can be extremely
difficult to hit because of its deflection
The figure shows the path of the ball thrown by a righthanded pitcher (from overhead)
*The ball is rotating counterclockwise
Trajectory of the Curve Ball
 1959: former National Institute of Standards and Technology Director Lymen Briggs
used a wind tunner and professional pitchers to prove a curve ball was not an “illusion”
He found spin was more important than speed in determining the break of the ball
Air Motion Behind a Curve Ball
This ball in a wind tunnel spins 1000 rpm counterclockwise (CCW) at right angles to the
wind
Which way would the Magnus force point in the picture?: pointing up the screen
Brigg’s Measurements
The ball leaves the pitchers hand at 70 mph (113km/h) spinning at 1600 RPM (16
revolutions to the plate) counterclockwise
The ball passes the plate 0.6s later traveling 61 mph (98km/h)
Flight of the Curve Ball
The curve ball follows a nearly constant radius path, but the deflection increases as the
square of the distance
In other words, roughly ¼ of the deflection (roughly ¾”=8.6cm) happens by the time the
ball is ½ way to the plate
By the time the pitch makes it to the plate it has deflected 14.4” (37cm)
Hitting a Curve Ball
A curve ball is hard to hit because during the last 25% of the flight half the deflection
occurs in about 1/6s (167ms)
It takes a batter 1/5s (200ms) to swing the bat, so they have to start their swing before
the ball curves much (and drops as well)
Sagitta
Sagitta: the largers deviations from a straight line drawn from the beginning to the end of
a ball’s flight, here 3.4”
the ball deflects 37cm relative to where it was aimed but only deviates 8.6 cm from its
flight path
Interactive Simulation
We can also study the air flow around the ball using a model developed by NASA
scientists for flow around an airfoil
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