A. Introduction

Since originally writing this article the Croquet Association (CA) has produced a specification for croquet balls. It does not however go anything like far enough to tie down the parameters of the balls. Be aware that the specification for Championship balls is tighter than that for balls for general play. The tournament specifications are set on a country by country basis.

The main metrics concerning croquet balls are:

• Absolute size 3 ⁵/₈" plus or minus ¹/₃₂", in diameter
• Mass 16 oz  plus or minus ¹/₄ oz
• Coefficient of restitution (bounce) 35-45" rebound when dropped from 60" (Laws of Croquet Appendix 2).

These limits are defined in Law 3c the Laws of Croquet. The Croquet Association Fixtures Book in the past has defined:

• Roundness 3 ⁵/₈" +¹/₃₂", - ¹/₁₆", but this is included or omitted on a year to year basis!

Note the term roundness has only been encountered in the Fixtures book so far.  The variation of these parameters with temperature is also of considerable interest but is not specified. Other non-specified parameters are also important as they affect the playing characteristics:

• Centre of mass
• Rotational moment
• Surface finish (milling patterns)
• Durability
• Light absorbency and heat capacity
• Stability

To paraphrase; no one wants a smooth soft ball with bias which does not roll like a standard ball and warms up unlike the others if left in the sun but not the same every time!

When reviewing a manufacturer's set of balls, all the balls of a series must be tested individually, i.e. it is not sufficient just to test all the primary colours if secondary are also going to be sold. The pigments used to produce a particular colour may degrade the properties.

B. Equipment for testing balls

The following equipment is recommended for testing balls:

• 1" thick steel plate set rigidly in concrete set in firm ground - as per Appendix 2.
• Thermometer.
• Ball gauges and callipers.

Now for the novel bits ...

• A small aquarium tank complete with heater, thermostat and thermometer.
• Large bag of salt.
• Thermocouple thermometer (small thermocouple).
• Standard approved croquet ball.
• Smooth ramp and stop watch.

To put you out of your suspense the aquarium serves two purposes - it allows the balls to be heated to a known temperature - they can be sunk in the water and left to warm to the water temperature. They then can be removed and either bounce tested or have their dimensions checked. Secondly by making a concentrated salt solution the balls can be made to float. This allows their centre of mass to be checked.

C. Coefficient of restitution (bounce)

A ball is dropped from a known height (60") onto the horizontal steel plate and its rebound height is recorded. To satisfy the regulations it must rebound between 30-45". The rebound heights of a set of balls to be used in a game must not differ by more than 3 inches. A number of drops is required on each ball to average out variations. Ideally the bounce characteristics should be recorded at a number of temperatures. If the balls have been brought from say a car boot or club hut to the test site they should be left for a considerable time to equilibrate to the ambient temperature. The temperature of the ball should be noted when the readings are made.

The method of measuring the bounce to date has been to put 6 chalk crosses or other marks on the surface of the ball where three orthogonal axes centred on the ball's centre emerge from the ball. These are numbered and the ball is dropped onto the steel plate with each one of these uppermost in turn. Provided that the ball is released without imposing rotation it should ensure that a different part of the ball is sampled on each drop. The average of the rebound heights is taken. Testing balls is best done by two people - one to drop and record the results and the other to estimate the rebound height.

Measuring points. A small point, if a ball is dropped 60", then all measurements should be made to the bottom of the ball from the surface of the steel plate. If someone is watching how high a ball rises they will see the measurement to the top of the ball most easily and then the diameter of the ball must be subtracted from this value.

Release methods. Generally the ball is held in the fingers and released. With practice the ball can be dropped with little rotation. For a permanent test site a more reproducible method may be devised but the precision of the rebound measurement, which is likely to be modest, should be considered before investing too much effort.

A promising release method is to considera see-saw with a counter weight balancing the croquet ball.  A second, supported, heavy weight is hung from the arm bearing the ball but closer to the pivot. All weights fall at the same rate, however because of the mechanical advantage, if the supported weight is released it will yank away the support from under the ball faster than the ball is decending.  Obviously there are variations , e.g. where the pivoted platform supporting the ball is held by a magnet rather than being counter-balanced.  (See diagram)

Rebound measurement. The trusty eye of the beholder is the usual detector. Care must be take to avoid parallax - e.g. squinting at an angle. The viewer must have their eye at approximately the rebound height so they may sight horizontally to the scale behind the ball. Again there is scope for using video recorders and the like, but it seems a trifle sophisticated. A good way to avoid parallax is to place a mirror by the scale behind the ball and, by moving your eye, attempt to get the ball and its image coincident when making the reading.

Note that we cannot presume that the coefficient of restitution is indepenent of force (drop height). Thus if the ball rebounds 60% of the drop height at 60" drop height you cannot expect the same percentage if it is dropped from 20" or 80". Similarly most materials become less bouncy as they get warmer. Prof. Alan Pidcock reports that the coefficient of restitution did not change for drop heights lower than 60"

The bounce test equates to hitting a ball moderately softly. If all the potential energy of the ball at 60" was converted to kinetic energy it would equate to a ball travelling at ~5.5 m.s^-1. As the ball only rebounds to ~²/₃ this height, ¹/₃ of the energy is lost in the bounce thus 4.49 m.s^-1 is the velocity of the ball. (Energy is proportional to the squar of the velocity). As an indication if a ball was hit it this rate across the width of the croquet lawn (assuming no losses) it would take ~5.8 seconds to cross - quite slow.

Temperature range. Croquet is played throughout the year and all over the globe. The temperature range for testing balls is not stated. The range 0º-30ºC would cover most ambient temperatures in which croquet is played, although not necessarily cover the temperatures that the balls reach - a dark ball in the sun will heat up above ambient.

Quick calculations indicate that the temperature effect on the dimensions of the ball is minimal. Most plastics expand by a factor of 1/1000^th per degree Celsius; hence the diameter of a croquet ball would change by a little more than ¹/₄ mm for a 30ºC change. It is the effect on the coefficient of restitution which is probably the most significant.

D. Absolute size

The size of balls is most easily verified with pass/no-pass ball gauges. These are normally a pair metal plates with an accurately machined hole in them. One hole is the minimum permitted diameter and the other the maximum. For these gauges to be accurate you should look for thick plates (e.g. 4mm) with at least ¹/₂" of metal around the outside of the hole. If there is less metal around the hole it is likely that the metal will have deformed when cut and the hole ends up non-circular. The Laws state "A ball must be 3 ⁵/₈" inches, plus or minus ¹/₃₂ inch, in diameter"

Obviously the test ball should pass through the maximum gauge and not through the minimum.

What is the diameter of a croquet ball? When the surface is textured  where do you measure to - the ball could have dimples like a golf ball or spines like a hedgehog?

E. Roundness

The roundness is a measure of the deviation from being a perfect sphere. The constraints for roundness are not part of the Laws but appear in the Croquet Association Fixtures Book. It should be noted that in 1997 the roundness was defined as 3 ⁵/₈" ± 1/₃₂", but in the 1998 Fixtures Book, this was 3 ⁵/₈" +¹/₃₂, - ¹/₁₆.

There is a possible confusion about the regulations for ball size and roundness. A ball may be as small as 3 ¹⁹/₃₂" under the size regulation and be a perfect sphere apart from a hollow an extra ¹/₁₆" deep. Is this ball acceptable? Perhaps the sensible interpretation is that 'within the absolute limits of size the roundness is 3 ⁵/₈" ± 1/₃₂"'.

Given the above interpretation the same gauges as were used to check for basic size can also be used for the roundness test. The ball is presented to the gauges in all orientations and still should satisfy the pass/no-pass test.

F. Centre of mass

A ball may be perfectly spherical but have an uneven distribution of weight within its bulk. This will cause it to deviate from a straight path like a bowls ball. The centre of mass can be found by floating the ball and seeing if it has a marked preference for settling in one orientation. (See the article by Richard Le Maitre).

One problem is to float the ball. A croquet ball will sink in water, its density should be 1.1089 g.cm^-3, water is 1.00g.cm^-3. You need a denser medium. Calculations indicate that you would need a 15% by weight salt solution. That can be prepared by dissolving ~166g of salt per litre of water at 20ºC (unchecked!). Add more salt to increase the floatation, or water to reduce it.  The degree of bias can be measured by attaching a weight to the surface of the ball until the bias is countered.  For really sensitive measurements it has been suggested that some surfactant (detergent) should be added to the brine to lower the surface tension. Reports indicate that some balls require as much as a 5g weight.

G. Rotational moment

Spheres of the same size and mass (weight) can roll in completely different ways. As an example if a solid ball and a hollow ball, which is composed of a heavy shell, are rolled down a slope then the solid ball will travel down the slope faster. Balls can be tested against a 'standard ' ball or the time taken to roll down the slope measured. Although you may be able to demonstrate that the ball has different rotational properties there are no specifications that must be met.

The older Jaques Eclipse balls were essentially solid and some of the newer 'solid' plastic ones have a void in the middle. The size of the void is around the size of a grape.

H. Surface finish

Milling patterns. The Laws say nothing about the surface appearance of balls, but they are addressed in the CA's ball specification. The specification is only applicable in the domain of the CA, but is not 'hooked' from the Laws, it just exists. Balls could be as smooth as ball bearings or prickly like hedgehogs. The rougher a ball then the more 'pull' is experienced and, after being struck the sooner it stops skidding and starts to roll on the grass. These are significant effects in the game. There are two main problems; firstly specifying a surface finish and secondly, coping with the inevitable wear on a ball. Unless a standard is defined any surface finish is currently acceptable.

A smooth or dimpled croquet ball (like a golf ball) would be least susceptible to wear.  The current design where the ball is covered in tiny pyramids is probably the worst design!  A dimpled surface also would not allow the 'meshing' of balls which is thought to contribute to pull.

Surface properties. A second consideration is the hardness and friction of the surface material - it could be like steel or soft and gripping like synthetic rubber. Again there is no standard.  The old Jaques Eclipse ball used a cork/rubber composite core that gave the ball its bounce with a thin hard shell giving durability and allowing a surface finish. The modern plastic balls appear to use a single compound to supply the bounce and wear characteristics.  Most players notice differences when doing cut rushes or when the ball catches the hoop uprights.

I. Durability

No one wants a ball which takes permanent deformations readily, or splits. The outer shell (or bulk if appropriate) should not take nicks if they are hit by the edge of a mallet or hit a sharp object off the court. The test of durability is probably a practical one - does it last a season or two!

J. Light absorbency and heat capacity

The heating effect of sunlight when a ball is left out in the sun has often been referred to but not measured. You can plainly feel a higher surface temperature on a black ball than on a yellow ball which has been in the full sun. The ability of a ball to absorb heat (its heat capacity) will determine how quickly it gains or loses heat. These measurements are peripheral to the effects of temperature on the other physical properties as if the bounce does not change with temperature then there is no need to worry about the temperature. Small thermocouples placed at the centre of the ball and at its periphery would allow this effect to be monitored.

K. Stability

Most plastic compounds degrade with time due to slow chemical reaction and exposure to light and air. Consequently one would want to test whether the properties of a croquet ball change with time. It has been suggested that early Barlow balls are less bouncy as they get older. A re-test of the balls after a year or so would be needed to determine whether the properties are changing.

Appendices

Density

Density is mass over volume. Volume is 4/3 pi r^3=⁴/₃.p.r³

Miniumum density = Largest and lightest sphere  = 446.90625/419.3797 = 1.065636 g.cm^-3
Maximum density  = smallest and heaviest sphere = 461.09375/398.2386 = 1.157833 g.cm^-3

Thus the density can vary by ~8%

Ball Velocities

The main equation is Energy (E) is a half the Mass (m) times the square of the Velocity (v): E=¹/₂.m.v²

As an example for a ball dropped 60" on to a steel plate.  The Potential Energy,  E_v is given by the mass times the height (h) times the standard gravity constant (g): E_v=m.g.h

1"=.0245 meters, hence h =  60"=  1.524 meters
The ball is m=16 oz=0.454 kg and the standard gravity constant g=9.806 m.s^-2
Hence Energy =1.524 * 0.454 * 9.806=  6.784 Joules Combining the above terms for energy: E=¹/₂.m.v² =m.g.h
This can be rearranged for v,  eliminating m: v²=2.g.h  hence v=sqrt( 2.g.h ).  This pleasingly predicts that the falling velocity is independent of mass (neglecting air resistance).

The velocity of the ball when it hits the plate is hence: v=sqrt (2 * 9.806 * 1.524)=  5.47 m.s^-1

Friction Coefficients

The following were measured by Stan Hall of Australia - see accompanying article.

Coefficient of Sliding Friction, m_s

fairly fast lawn (normal, cut to 6mm)=0.48
fairly heavy lawn=0.53 Coefficient of Rolling Friction, m_r

fast lawn (cut to 4mm)=0.055
normal lawn=0.065
heavy lawn=0.1

Contact Times

The following were measured by Stan Hall of Australia - see accompanying article.

Mallet - Ball contact times

Single ball strokes=~1ms
Croquet stroke=  ~2.9ms

Ball - Ball contact times
approximately 1.8ms