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Dr Ian Plummer

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Bias in Croquet Balls

The problem of bias (as in bowls) and croquet has recently woken us up down here at the bottom end of the World. Early tests conducted on a billiard table soon dispelled the ideas that bias is a myth in croquet. The noticeable 'draw' (lateral motion) on the fine green baize prompted a more careful examination with a view to ultimately testing on grass. The billiards table results are shown on Diagram A, and show only the path of the last few rolls of the ball, because the distance (billiard table) was short. Bias always takes its maximum toll at the end of the run.

ball bias when rolled on flat surface
(note original diagram had smooth curves, kinks are due to the graphic program)

To determine static imbalance, over 60 spun-cast and solid moulded balls were tested in a strong brine solution. The spun-cast balls floated at near constant depth, being weighed on a scale during manufacture. Variations were found in the solid balls, one of which refused to float at all. Curiously enough it was not under size and at 407.2gm only marginally underweight. The specifications for both bowls and croquet balls allow for a considerable latitude in density.

The more unbalanced balls would quickly float with their light side up, and rock or oscillate quite rapidly when disturbed. Two balls were chosen because they had zero imbalance, floating without rolling from any position they were placed in, with nothing but a few dynes of surface tension to stop them. Perhaps a little detergent might have shown up half a gram of imbalance, but these two, both spun-cast, were taken as controls for tests. Amongst the spun-cast, the greatest imbalance found was 1.7gm, and among the solid balls the variation went up to 5gm. This is understandable, since only the spun-cast balls with their pea-sized central space need only to be spun reasonably fast to ensure near-perfect balance.

The mathematical analysis of the whole motion of a wood rolling on a green takes a trigonometrical exponent form, with variables and partials. The closest analogy is to be found in the rolling of a coin on edge, and this case is dealt with by Leonard Meirovitch in his book, 'Methods of Analytical Dynamics', published by McGraw Hill. The round croquet ball and the spheroidal wood make for a more difficult analysis.

During the run of the wood there is first of all an immediate stabilising effect due to the gyroscopic action, causing the wood to rotate about the axis of the maximum moment of inertia, this is the axis of which the bias centre forms one pole. This happens after the first few metres of a 30-metre jack. The gyroscopic stabilisation equations of motion are non-linear, and result in the wood losing its initial launch wobble from almost any offset of the axis of bias: such a shot is used in bowls, and is called a 'narrow' shot. After such correction the wood resumes its trajectory with full bias with the little spot mark of the bias centre rotating without further wobble.

Next comes the main part of the trajectory under stable but biased conditions, and finally the mathematically horrible demise of stability at the end, which mirrors the initial stabilisation, only irregularly.

Back to the testing of croquet balls on a bowling green. The Somerset West Country Club bowlers were kind enough to lend one of their greens for this purpose through the kindness of their President Mr George M. Simpson, who also lent one of his own woods for the 16 metre test runs. Our thanks are hereby recorded in appreciation of his help in this investigation. Diagram B shows all the relevant results of the bowling green tests, the ball positions being given by sets of co-ordinates from the start point as origin: the biased ball was run with the bias on to the left as well as to the right, shown by the two curved paths.

bowling green results
(note original diagram had smooth curves, kinks are due to the graphic program)
Click on image for higher resolution

In each case the croquet ball was ramp-launched from the same position on the same double-track rail, thus as near as possible with constant initial velocity, as well as from the same spot. The bias chosen was 5gm, equal to just under a fifth of an ounce, about 1% of the mass of the ball.

More than 10 runs were made under each of the chosen conditions, to rule out gross experimental error. It was found that the arithmetic mean distance travelled by the unbiased ball along the y-axis was 16.2 metres, while the variation in length was +/- 63cm from the mathematical centre, giving a total variation of 126cm in 1620cm, or just under 8%.

Similarly the distance moved by the biased ball along the Y-axis was just over 16 metres, with the same variation of +/-63cm on the right hand side and +/- 78cm on the left. The arithmetic mean draw was 167cm, being the sum of the right and left bias maximum X co-ordinates divided by two. In the total distance of 16 metres the variation in draw was +/- 23cm on the left side, and +/- 19cm on the right: the mean of these two figures is 21cm, which is 12.5% of the total movement of 167cm. The point marked P is on the last part of the trajectory of the biased ball. The figure of 1.67metres for the croquet ball is not so different from that of the wood, which under test drew 2.01 metres in a 16.2 metres on the Y-axis. In a 30-metre jack a wood will draw between 6 and 9 metres, depending on size and mass.

On a long hard shot the bias will quickly sort itself out to act on the right or left, and even with a gentle shot of 20 metres or so it is plain that an imbalance of well under 0.5gm is sufficient to cause a 100mm draw and thus miss a hit-in. A hard shot will naturally overcome this to a large extent, but it makes one think, does it not? Have you ever made a long take-off and been surprised by the apparent slope on the court, which is not always the same? Maybe it is the nap of the grass, we say. In American croquet, a boundary take-off with such a draw can be deadly with only a mallet-head allowance between the ball and the boundary. Take that 'unlucky' shot which you made on a rather dry bowling green: what looked at first like a perfect roll-up for an easy short hoop, quietly, with the last few rolls, turned into a ghastly, nerve-racking jump shot!

Perhaps we should consider making a definite break with our bowling friends and abandon biased croquet. I would like to suggest that the problem is a real one and deserves some careful attention. A problems of ball specification and testing?

Yours sincerely

R. Le Maitre
Somerset West Country Club

Author: R. Le Maitre
All rights reserved © 2004

Updated 28.i.16
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