This article presents exact equations of motion for a rotating bowling ball in a form that explicitly separates contributions due to nonequal principal moments of inertia, center-of-mass offset, and friction between the ball and lane. A computer program that solves the equations demonstrates that all of these factors are important for a realistic analysis of bowling. These factors significantly affect how much balls hook, that is, deflect sideways and approach the pins at an oblique angle. Simulations that approximate real bowling conditions indicate that the largest contribution comes from variable friction along the lane, that is, bowling lanes are generally prepared so that lane friction is higher by a factor of 2 or more along the last one-third of the ball’s trajectory. The analysis supports most (but not all) of the guidelines that bowlers have developed for predicting ball performance.
REFERENCES
1.
The literature is also out of date. Since about 1990, manufacturers began competing to produce balls with much greater variation in core geometry and cover material. This competition caused a proliferation in the varieties of balls available to ordinary bowlers, and has spawned numerous articles by coaches and drillers as they struggle to best utilize these new balls.
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Columns in each issue of the journal Bowling This Month review the properties of balls by all the major manufacturers and discuss how different drilling patterns affect their motion. Two examples of books by drillers that try to explain how ball properties influence performance are B. Taylor, Balance (BT Bowling Products, 1988), and C. Zielke, Revolutions (Revolutions International, Matteson, IL, 1995).
7.
Because the derivative is taken in an inertial reference frame, the angular momentum changes both because the angular velocity changes [first term on the right in Eq. (4)] and also because the moment of inertia tensor I is rotating (second term on the right). See, for example, H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. (Prentice Hall, Englewood Cliffs, NJ, 2002).
8.
ABC/WIBC Equipment Specification Manual (2000).
9.
The rules don’t explicitly specify the length of the lever arm, but all regulation balances appear to have a lever arm of this length.
10.
The limitations on also serve to prevent the ball from losing contact with the lane. In particular, means that for of one mm, ω would have to exceed 100 rad/s or 15 rev/s. Real bowlers seldom if ever exceed ω of about 8 rev/s.
11.
D. Speranza and L. Vezina, C. A. T. S (Computer Aided Tracking System) Information Manual (ABC/WIBC/USA Bowling, 1996).
12.
More precisely, what bowlers want is for the ball’s path to approach the headpin at an oblique angle of perhaps 4°–5°. To accomplish this, the ball must slide further down the lane before deflecting, and then deflect a few more inches.
13.
Equations presenting the first and second derivatives of the Euler angles as a function of and appear explicitly in Ref. 5, or can be easily derived from the equation relating to the Euler angles and their first derivatives which appears in mechanics texts such as L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, New York, 1960).
14.
See Ref. 4, and also
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However, air resistance is not completely negligible. A bowling ball travels with a Reynold’s number of about thus the drag force is approximately where ρ is the air density, A is the ball cross section and is about 0.5. For a bowling ball the drag force is about 1 N, which is sufficient to reduce a ball’s velocity by a few percent over the length of a bowling lane.
16.
However, real bowlers will note that Table III doesn’t include simulations for a large variety of ball properties and initial deliveries; the “realistic” simulations (10–12 and 14–16 in Table III) do not include the full spectrum of label and leverage drilling patterns used by many serious bowlers.
17.
When all radii of gyration are equal, the locus of points on the ball’s surface that are in contact with the lane forms a line (actually, a small circle), visible to bowlers because it leaves a track of oil on the ball. However, precession of axes causes this oil track to smear over a wider area, and reduce friction because there is less oil on the part of the ball in contact with the lane. Bowler’s use the term “track flare’ to describe the widening of the track line when a ball has unequal radii of gyration.
18.
To avoid computational problems, for all the simulations in this paper I assumed that the ball made the transition from sliding to rolling when the slip [Eq. (A2)] became smaller than 1 cm/s.
19.
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© 2004 American Association of Physics Teachers.
2004
American Association of Physics Teachers
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