Dr Ian Plummer
Technical
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.
(note original diagram had smooth curves, kinks are due
to the graphic program)
To determine static imbalance, over 60 spuncast and solid moulded balls were
tested in a strong brine solution. The spuncast 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 spuncast, were taken as controls for tests. Amongst the spuncast,
the greatest imbalance found was 1.7gm, and among the solid balls the variation
went up to 5gm. This is understandable, since only the spuncast balls with
their peasized central space need only to be spun reasonably fast to ensure
nearperfect 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 30metre jack.
The gyroscopic stabilisation equations of motion are nonlinear, 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 coordinates 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.
(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 ramplaunched from the same position on
the same doubletrack 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 yaxis 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 Yaxis 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 coordinates 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 Yaxis. In a 30metre 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 hitin. 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 takeoff 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
takeoff with such a draw can be deadly with only a mallethead 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 rollup for an easy
short hoop, quietly, with the last few rolls, turned into a ghastly, nerveracking
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
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