Thermal Expansion of Croquet Balls
It is important to know how much a croquet ball might expand in the heat when deciding on sensible settings for the clearance between balls and hoops.
Measurements were made on Dawson 2000 and Sunshiny Tournament balls and the Coefficient of Linear Expansion, α, was calculated.
These figures do not appear to have been presented elsewhere. To put the expansion into context, 1/64" = 0.015625".
An approximate take-home message is that a rise in temperature of:
These magnitudes of temperature rises occasionally occur in the UK climate. However, as will be shown in another article, radiant heating from direct sunlight causes large surface temperature rises. The ambient temperature produces smaller expansions.
Consider a dutiful championship referee selecting the largest Dawson ball out of a set (it could be any colour) early in the morning when the temperature could be 15 ºC. His intention is to set the hoops at 1/64" clearance - the standard minimum clearance for a championship event; 1/32" - 50%.
The weather is wonderful and by mid-afternoon the air temperature is 30 ºC. Assuming the Dawson balls to be at this temperature throughout, they will have expanded by approximately (30 - 15) / 31.1 x 0.022" = 0.106", around 3/256th inch. The clearance on the biggest ball has gone down from 1/64th (= 4/256") down to 1/256". Basically very bad news if you select the largest ball.
Further, the Balls Specification and Approvals allows "The maximum and minimum diameters of balls in a set must not differ by more than 3/64 inch (1.2 mm)". What does it look like? The diagram below is to scale showing a Red ball with 1/64" clearance and the accompanying permitted minimum size Yellow ball and its clearance (1/32"). The white bar represents a standard 5/8" hoop upright.
The diagram above shows the difference in clearance between the permitted largest and smallest balls when the clearance is 1/64" say at 15 ºC. If we have a 15 ºC temperature rise, both balls will expands by ~3/256". For the largest ball however the clearance is now reduced to 1/256" whilst for the smallest it is 1/32 - 1/256" which is still ~1/32". I would want to be playing with the smaller ball!
As a quick aside what is happening to the gape of the hoop with this temperature rise? "Steel" is quoted as having a linear coefficient of expansion of ~1.2 x 10-5 / ºC - an order of magnitude less than has been found for the plastic balls. Consequently it is ignored in these arguments. What is happening to the strength of the soil around the carrots as the temperature rises is open to speculation. It will also depend on whether the carrots are pulled together by the soil to produce the clearance or pulled apart.
We know that the balls will expand by ~1/64" when they get 15-30 ºC hotter, which represents viable air temperature increases likely to be found in summer.
There are two useful results:
What has not been discussed however is the change in properties which accompanies a temperature rise. The balls' bounce (coefficient of restitution) is likely to decrease making them 'deader' to play with. Additionally their surface properties may change - they may become more rubbery/grippy. These ideas are explored a companion paper.
It was considered easier to cool balls down and let them warm whilst taking temperature measurements rather than heat them up, e.g. in a water bath, and watch as they cooled. (The balls would have to be left in a thermostatic water bath for a long time to get warmed through - much easier to dump them in a freezer overnight). A reference on 'Castable Polyurethane Polymers' (IR Clemitson) gives the following general guide for polyurethanes:
Consequently the balls should not be permanently damaged in boiling water nor by -80 ºC.
Dawson and Sunshiny balls were left in a freezer for at least 12 hours. The internal temperature of the freezer was around -18 to -20 ºC. It was assumed that the entire ball would be chilled to this temperature after 12 hours. Ambient temperature on the day of diameter measurement was ~16 ºC offering measurement over a 34 ºC range.
A ball measuring jig was set up, consisting of a ¾" thick steel base containing a 3/8" vertical hole. This hole was used to fix the ball's position. A standard dial gauge (0.001" divisions), supported by a magnetic holder, was set vertically above the hole and set so that the dial was approximately mid-range when a room temperature ball was inserted between the gauge and the hole in the plate.
The ball was transferred to the plate using oven gloves. In addition to the test ball there was a room temperature ball beside the rig as a check for the temperature reading device and to determine when the test ball had reached room temperature.
Temperatures were taken using a non-contact IR thermometer (photograph). This had previously been used on a set of Dawsons and showed that similar temperatures were recorded from all balls when they were left to equilibrate to room temperature. This demonstrates that the emissivity of the balls is similar.
In the minute that was taken to get the balls from the freezer to be centred on the hole in the test rig the surface temperature increased markedly.
It was appreciated that having the ball centred in a 3/8" hole would mean that not the whole diameter of the ball would be measured. A calculation (Appendix D) showed that only 0.27% of the ball's diameter (≈ 1/128") lay within the hole and that would not hugely affect the expansion measurements.
The imperial dial gauge was marked in 0.001" increments and included a revolution counter. It was confirmed that a reading of 250 corresponded to ¼". The worktop was lightly drummed before making a reading to allow the pointer to settle, overcoming any stickiness of the mechanism. A larger dial gauge reading indicated a larger diameter.
The key measurements are the first and the last; it can be assumed that the ball starts off at a uniform temperature from the freezer and given enough time ends at room temperature. Clearly the ball gains heat through mainly its surface hence the surface may be warmer than the core during the warming stage.
Dawson Green Ball
*4th decimal place is estimated between dial gauge marks
Thermal expansion, α, is given as:
Where ∆L is the change in length, L0 the original length (3 5/8" = 3.625") and ∆T, the temperature change.
For the Dawson Brown ball:
For the Dawson 2000 Green ball:
For the Sunshiny Tournament Blue ball:
The croquet ball sat in a 3/8" hole on a thick metal plate; the height change above the metal plate was measured using a dial micrometre. Technically some of the plastic lies in the hole and we only measure the height above the plate (see photograph above)
So rather than measuring the expansion of 3.652" of plastic, we measure slightly less. Also as the ball chills, its diameter will change and the dimensions above and below the hole will change marginally.
How significant is the bulge into the hole?
Using the product of segments theorem
a1 * a2 = b1 * b2 ;
a1 = a2 = 3/16 = 0.046875"
b1 + b2 = 3 5/8" = 3.625"
How big is b1? Plugging the numbers in and solving the quadratic yields:
b1 = 0.00972436226801"
b2 = 3.61527563773"
To put things into context 1/64" = 0.015625"
b1/( b1 + b2) is 0.268%, hence will be considered negligible.
1. Note this is at variance with the current (2017) Tournament Regulations which allows: "BALL ROUNDNESS. The diameters of all balls used on a court may differ by no more than 1/32" for Championship conditions, or 1/16" otherwise."
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