Physics 199BB The Physics of Baseball

1
Physics 199BBThe Physics of Baseball

• Fall 2007 Freshman Discovery Course
• Alan M. Nathan
• 403 Loomis
• 333-0965
• a-nathan_at_uiuc.edu
• Week 4

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Forces on a Baseball in Flight

• Gravity
• Already discussed
• Drag (air resistance) Force
• Already discussed
• Magnus Force
• Now we do this

3
The Magnus Force
Courtesy, Popular Mechanics

• ball deflects air
• air must deflect ball in opposite direction
• (Newtons 3rd Law)
• size of FM proportional to ?

4
Recall our definitions

• ? is angular velocity
• a measure of how fast the ball is spinning
• units are rad/s or rev/min (rpm)
• to convert from rad/s to rpm
• multiply by 60/(2?)
• to convert from rpm to rad/s
• divide by 60/(2?)
• ? has a direction

5
The spin axis is the line connecting the south to
north pole (right-hand rule)
spin axis
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The Magnus ForceThe magnitude of FM

• FM ½CL?Av2
• CL is the lift coefficient
• CL CM(R?/v)
• FM ½CM?AR?v
• CM is the Magnus coefficient
• A dimensionless number

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The Magnus ForceSome numerology
FM ½CM?AR?v plug in know values of ?,A,R
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The Magnus ForceNumerical Examples

• v50 mph, ? 1800 rpm, CM 1
• FM/mg 0.276
• v100 mph, ? 1800 rpm, CM 1
• FM/mg 0.552
• See Adair, Fig. 2.2, page 12
• (more on this later)

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The Magnus ForceThe direction of FM

• Force acts in the direction
• another right-hand rule
• force is perpendicular to both v and ?
• force acts in the direction that the leading edge
  of the ball is turning

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Some Qualitative Effects of the Magnus Force

• Backspin makes ball rise
• hop of fastball
• undercut balls increased distance, reduced
  optimum angle of home run
• Topspin makes ball drop
• 12-6 curveball
• topped balls nose-dive
• Breaking pitches due to spin
• Cutters, sliders, etc.

11
Additional Effects
Balls hit to left/right curve toward foul pole
12
Additional Effects

• Tricky trajectories of popups
• --popup behind home plate with lots of backspin

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Incorporating Magnus into Excel

• We now have a 3-dimension problem
• For our first examples, we will only consider
  2-dimensional problems
• topspin or backspin, but no sidespin

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Case 1 Backspin

• FMx -FM sin(?)
• FMy FM cos(?)
• aMx -2.09 x 10-6 CM?vg sin(?) -2.09 x 10-6
  CM?gvy
• aMy 2.09 x 10-6 CM?vg cos(?) 2.09 x 10-6
  CM?gvx

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Case 2 Topspin
y

• FMx FM sin(?)
• FMy -FM cos(?)
• aMx 2.09 x 10-6 CM?gv sin(?) 2.09 x 10-6
  CM?gvy
• aMy -2.09 x 10-6 CM?gv cos(?) -2.09 x 10-6
  CM?gvx

?
?
FM
x
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Now look at the filetrajectory-drag-Magnus-2da.x
ls

• Batted balls
• Low initial angles
• range increases
• angle for maximum range decreases
• trajectory more asymmetric
• Higher initial angles
• range decreases
• trajectory more symmetric
• cusps and loops

17
Now look at the filetrajectory-drag-Magnus-2db.x
ls

• Pitched balls
• Backspin reduces drop (fastball)
• Topspin increases drop (curveball)

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How do we know what CM is?

• An Experiment Done At UIUC

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Tracking The Trajectory
Motion Capture System--Beckman
20

• Motion Capture System
• 10 cameras
• 700 frames/sec
• 1/2000 shutter
• very fancy software
• www.motionanalysis.com

• Pitching Machine
• project horizontally
• 50-110 mph
• 1500-4500 rpm

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Typical Data
22
Trajectory data (top) for one of the pitches
projected, where y and z are the coordinates of
the dot on the ball in the coordinate system
shown in the inset. The ball is projected at a
slight upward angle to the z direction and is
spinning clockwise (topspin) about an axis
perpendicular to y-z plane. Solid curves are
least-square fits to the data using Eq. 4,
resulting in CD 0.44 and CL 0.33. The long
dashed curve is the center-of-mass trajectory for
the y coordinate, which is consistent with a
downward acceleration of 1.58g due to the
combined effects of gravity and the Magnus force.
The short dashed curves are the
center-of-masscoordinates for both y and z with
both CD and CL set to zero, indicating that the
data are very sensitive to CL but not to CD.
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Results for Lift Coefficient CL
FM 1/2?ACLv2 Sr?/v CM CL/S 100
mph, 2000 rpm ?S0.17
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Results for Drag Coefficient CD
FD 1/2?ACDv2
Conclusion Major disagreements for v 70-100
mph
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Data Do Not Agree with Adair

• Experimental Data CM ? 1 for S0.1-0.3
• For 2000 rpm, S0.1-0.3 corresponds to 57-171 mph
• For 1000 rpm, range is 85 to 255 mph
• So, most of interesting range is covered
• Adair (see p. 24)
• For 2000 rpm
• CM0.8 at 50 mph-agrees with data (0.8)
• CM0.4 at 100 mphtoo low (1.1)
• I have written a paper about this (see web site)

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Now lets include sidespin

• z is third dimension, points to pitchers right
• Lets look at pitched ball only
• Spin axis lies in y-z plane
• ?0 for backspin, 180 for topspin
• ?90 or 270 for pure sidespin

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Here are the formulas
y
spin axis
FM

• FMx FM sin (?)vz/v-cos(?)vy/v
• FMy FM cos (?)vx/v
• FMz -FM sin(?)vx/v
• aMx2.09x10-6 CM?g sin(?)vz-cos(?)vy
• aMy2.09x10-6 CM?gcos(?)vx
• aMz-2.09x10-6 CM?gsin(?)vx
• Notes
• when ? is 0 or 180, these formulas are identical
  to the ones previously used
• Since v?vx, FMx?0
• FMy is responsible for up/down break
• (max when ?0 or 180)
• FMz is responsible for left/right break
• (max when ?90 or 270)
• 5. FM makes angle ?90 with z axis

?
z
batters view