Thursday, March 22, 2012


Swerve only occurs while the ball is skidding. Once pure rolling has begun, the ball will move in a straight line - assuming that the ball has no bias and the alley is level. There are three components of spin, only one of which directly affects a skittler's ability to swerve the ball:
  • About a vertical axis - will not affect the ball's trajectory along the alley but may come into play on collision with pins.
  • About a horizontal axis across the alley (ω2 radians/s) - this affects the transition from skidding to pure rolling and the ball's speed down the alley but has no direct impact on swerve.
  • About a horizontal axis along the alley (ω1 radians/s) - this is the component that creates swerve. Ceteri paribus, the greater the speed of delivery, the longer the length of time over which swerve can occur.
If a ball of radius r released at speed V is spinning on release, the angle through which the ball swerves (ψ) and the transverse movement (x) are given by C.B. Daish (Chapter 14 of The Physics of Ball Games) as follows:

tanψ = 2ω1r / (7V - 2 (V - ω2r))

x = ½ (µg sinθ) T2

where: tanθ = ω1r / (V - ω2r)
T = 2U / 7μg
and U2 = (V - ω2r)2 + (ω2r)2

As an example of the magnitude of the swerve which can be achieved, consider a 4½ inch diameter ball, delivered at 8 m/s along a 10 m alley, spinning at 1 revolution per second about an axis along the alley with a coefficient of friction of 0.3. Skidding will cease after about 5½ m where the transverse movement will be about 4 cm, increasing to 12 cm when the pins are reached. The overall swerve is only 1° but the potential impact of spin can mean the difference between hitting a pin and missing all.


Lessons from evaluating realistic ranges of speed of delivery and spin:
  • Late swerve may be a reality but it is most likely to be due to a bias in the ball or an imperfectly level alley as the pins are approached. A vertical mismatch between boards is likely to have a significant effect.
  • Attempting to spin a slow delivery causes swerve in the early stages of travel down the alley making control of the trajectory more difficult. For a given spin the transverse movement is inversely proportional to the speed of delivery, i.e. is disproportionately greater for slower deliveries. If the spin is proportional to the speed of delivery then transverse movement is approximately independent of delivery speed.
  • If you have a natural tendency to spin the ball clockwise along the length of the alley don't stand on the right side to launch it! Likewise avoid the left side if you have a tendency to anti-clockwise spin. Otherwise the ball will tend to straighten on its way down the alley and impacting the pins at an angle will not be facilitated.
  • In view of these conclusions, avoid deliberate spin in matches until your delivery is well practised!

Wednesday, March 07, 2012

Slide-roll transition

For pure rolling, the velocity of that part of the ball which is in instantaneous contact with the alley must be zero. This requires that the velocity of the ball along the alley should be perfectly balanced by the rotational velocity of the ball's surface at the contact point. Any imbalance will give rise to skidding, in which case a frictional force will operate, the ball's linear speed will decrease and its rotation will increase until the velocity of the contact point is zero and pure rolling is occurring.

The speed of delivery and initial spin are major factors in both time from release and distance down the alley before pure rolling is established. It is therefore also a dominant inluence on the ability to swerve the ball.

Assuming that the ball is not rotating on release, the time taken (T) and distance travelled (S) before pure rolling are given by C.B. Daish (Chapter 14 of The Physics of Ball Games) as follows:

T = 2V / 7µg ; S = 12V2 / 49µg

where V is the speed of delivery (m/s),
µ is the coefficient of sliding friction,
and g is gravitational acceleration (m/s2).

The coefficient of sliding friction (µ) for wood on wood is between 0.2 and 0.5 depending on the conditions of the ball and alley surfaces. The lower value is appropriate to pristine balls on a newly laid and polished alley. A value towards the upper end of the range is more likely to apply to the conditions on our home alley. The gravitational acceleration (g) is relatively constant in the Thornbury area with a value of about 9.81 m/s2 (although there is allegedly a significant mascon in the vicinity of Tytherington!) Observation of Geordies suggests that speed of delivery (V) is typically in the range 1 to 5 m/s perhaps as high as 10 m/s, occasionally 15 m/s, for Farmer Hall.

The following figure illustrates behaviour for a friction coefficient of 0.3. A ball launched at 5 m/s will slide for 2 m and will roll thereafter. A ball launched at 10 m/s under these conditions will be sliding for 8 m, i.e. most of the length of the alley.

Once pure rolling is achieved the linear velocity will be about 0.7V. The coefficient of rolling friction is between 0.002 and 0.05 and will not slow the ball significantly before pins are hit (or not).

In preparation for studying swerve it is convenient to write all equations in terms of dimensionless distances and times normalised to the point of transition from skidding to pure rolling.

The velocity (v) of the ball and its distance travelled as a function of time are given by:

v/V = 1 - 2t/7T when t<T; v/V = 5/7 when t>T
s/S = 7/6 (t/T) - 1/6 (t/T)² when t<T; s/S = 1/6 + 5/6 (t/T) when t>T