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Physics 199BB The Physics of Baseball

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Week 4. 7. The Magnus Force: Some numerology. FM = CM ... 9. The Magnus Force: The direction of FM. Force acts in the direction. another right-hand rule ... – PowerPoint PPT presentation

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Title: 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

2
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
6
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

7
The Magnus ForceSome numerology
FM ½CM?AR?v plug in know values of ?,A,R
8
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)

9
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

10
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

13
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

14
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

15
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
16
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)

18
How do we know what CM is?
  • An Experiment Done At UIUC

19
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

21
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.
23
Results for Lift Coefficient CL
FM 1/2?ACLv2 Sr?/v CM CL/S 100
mph, 2000 rpm ?S0.17
24
Results for Drag Coefficient CD
FD 1/2?ACDv2
Conclusion Major disagreements for v 70-100
mph
25
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)

26
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

27
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
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