Mallets: End-weighting and the Moment of Inertia
Some New Measurements
By Brian Russ (Exeter Croquet Club)
The spur to contributing this article was finding Towlson's article [ref. 1] on this website after having been exposed to various claims and expositions of the benefits of end-weighting of mallets. [See also ref. 2] Briefly, the Moment of Inertia (MoI) is the counterpart to 'ordinary' inertia in the context of circular, rather than, linear motion; i.e. it expresses the tendency of a body to resist twisting (or spinning). If you have ever sat on a rotating stool you will have noticed that, to spin rapidly, you need to draw your arms and legs towards your body. This reduces your MoI by moving mass towards the axis of rotation. Conversely, if, once spinning, you extend your limbs outwards (increasing your MoI) you slow down - you have 'end-weighted' your body. Likewise, during the swing of a mallet any tendency to deviate from a straight line - whether by inadvertent twist of the wrists, say, or hitting the ball off-centre - is resisted more effectively in a mallet head which has its mass concentrated near the faces. Most mallets are symmetrical about the shaft so the mass near the striking face is no more important than that at the rear.
Of particular relevance to this article are firstly: Towlson's unconventional, though convenient, choice of units - where I shall follow him - which puts the magnitudes of MoI into the approximate range 0-200; and, secondly, the concept of comparing a mallet's MoI to a notional maximum possible value even though that ignores practical constraints (of construction and usage, to say the least). Although this seems fatuous it does provide the only firm basis for comparisons between mallet designs. Bow [ref. 3] refers to the efficiency of a mallet of given weight and dimensions which is the ratio of measured (or calculated) MoI to that of Towlson's 'two point-mass' head - even though an efficiency of 100% is impossible. A better term could be the end-weighting effectiveness; the effectiveness of the design in securing the best MoI. This has relevance to the choice of mallets from a manufacturer who offers a range of head lengths, for example, while keeping to the same design. As Towlson shows, extending the length of a head while keeping a constant weight will increase the MoI but not necessarily the efficiency/effectiveness.
Useful though these articles were, my ideal was a numerical comparison of mallets available on the market together with the ability to find the MoI of a homemade mallet - either my own or that of other DIY-ers.
The only published account of a mallet's MoI measurement of which I'm aware described employing a gramophone turntable [ref. 3]. This seemed to require removing the head from the shaft and thus restricting the field of application. Many undergraduate physics texts and exercises on MoI can be found on the internet; they involve using a 'bifilar' (or torsion) pendulum. This differs from the familiar clock pendulum in having two equal, parallel strings and the 'bob' is replaced by a cradle (platform). The object to be tested (in our case a mallet, upside-down) is placed in the cradle and given a twisting, rather than swinging, motion. The oscillation period and the dimensions of the apparatus yield the MoI.'
Torsion pendulum clocks do exist - sometimes they have a glass cover that reveals the twisting/rotating movement but they have no strings [ref. 4]. This type of clock can be suitably compact. I found that for a mallet however, the two filaments need to be as long as possible in the interests of precision. This determined the design of my apparatus; allowing for the inclusion of the mallet shaft also implied a tall construction.
Much of the design derives from the formula: [see ref. 5, somewhat mathematical]
See the diagram. It is about 120cm tall while the filaments may need to be longer. The apparatus normally sits on a table edge so that the mallet can be placed in the cradle - about 6-7cm wide to hold any normal mallet head - the shaft's end nearly reaching the floor. Even so, the top of the apparatus may near the ceiling.
As W, R and L are fixed for a given trial, only T remains to be measured by experiment. To make T of a convenient magnitude (and hence, precision) L has to be as large as possible and R as small as possible. However, the latter condition militates against precision and the minimum is value of R is limited to the width of the cradle.
The apparatus was built of simple, readily-available materials, using simple tools and techniques. One unexpected difficulty was that of the selection of the filament. Fine wires seemed an obvious option but they could not be straightened satisfactorily. Cotton was abandoned because of its built-in spin/twist while a monofilament such as nylon or fishing line stretched too much (and slowly). A fishing braid brought the best compromise.
Only a micrometer was purchased - for better performance than a ruler.
The cradle was made of balsa wood for lightness, hence low MoI - though see below.
Before starting any experiments with mallets the apparatus was first tested with some 'standard' samples - flat, rectangular sheets of lead whose weight and dimensions could be readily measured enabling a calculation of the MoI with confidence. Several samples were tried and the linearity of the results assessed. It was found that the cradle had a negligible MoI but there was noticeable stretching of the filaments while using weights up to 1.5kg. Hence, L had to be measured after the (slow) stretching had occurred.
Once the mallet is put in the cradle it is left to come to rest and any adjustments made to ensure: the verticality of the apparatus, and of the mallet; the equality of the filaments; and the siting of the mallet shaft along the centre-line of the suspension. Typical values of 2 - 5 seconds for an oscillation were found; hence a stopwatch was used to record the time taken for 5 or 10 periods. A few initial oscillations were used to check that, as far as possible, rotation is the only movement; though it is not known how much, if at all, concurrent swinging affects the timing. It is well-known that the simple pendulum formulae can be applied only for small oscillations of less than ~5° amplitude.
Experiments are usually subject to systematic errors: these arise from the design rather than the measurements. In this case the derivation of the formula has to ignore, for example, a number of inherently difficult factors such as: air resistance acting upon the filaments and the mallets, the finite thickness of the filaments, the correction needed for appreciable amplitudes of oscillation, the effects of thermal expansion due to temperature variation. Of these only the effect of air resistance was apparent here - while comparing the persistence of the cradle's oscillations when empty and when loaded (imagine a pendulum bob being replaced by a ping-pong ball). This puts the determination of the cradle's MoI in some doubt. However, there were other difficulties encountered with this design: the axis of rotation needs to pass through the centre of mass of the test-piece for the formula to apply. Furthermore, without a suitable starting mechanism it is not easy to ensure that rotation is the only motion - 'normal' pendulum swinging all too easily occurs as well.
Briefly: with W about a kilogram, measured to the nearest gram; L about 120cm, measured to the nearest millimetre; five or ten periods of oscillation, measured to within a few milliseconds; the imprecisions here (I prefer this term in order to avoid the hint of 'mistake' that might be inferred from 'error') are of the order 1 part in 1000. But as R is about 7cm. it can be measured with much less precision, not made any easier by the lack of rigidity of the filaments as suspended; let us say, 1 part in 200. For an experimental MoI calculated as, say, 100 the implied imprecision would be ~1 (even a calculated imprecision has its own imprecision!)
The MoI of the shaft should be considered also in thinking about complete mallets as opposed to mallet heads. As Towlson shows, any long thin body has a small MoI about its long axis. For a typical wood shaft with diameter approx. 2.5cm this can be calculated to be less than 1 and an experimental value bears this out. When comparing mallets with MoI in the range, say, 100 ± 50 this is clearly not decisive so there is no benefit to be had in modifying the shaft. It should be noted that for a compound body, such as a mallet, the MoI for the separate parts can be simply added together. When dealing with the relevance of the MoI to the striking of a ball it is the MoI of the head and shaft together that counts.
The figures obtained for the metal samples - all of them flat sheets - give confidence in the method. The agreement between calculated and experimental values is perhaps flattering because the sheets have much less air resistance than the mallets. Their linearity suggests that the mallets, though heavier, have also yielded useful results.
The experiments have demonstrated that a mallet's Moment of Inertia can be determined to a level of precision useful for purposes of comparison. The effect of end-weighting has been quantified though the magnitude of the effect has not yet been related to the mechanics of the contact between ball and mallet. Refinement of the apparatus and procedure could probably yield more precise results but their value might be limited.
This apparatus can be, and has been, modified (in the cradle) to measure the MoI of croquet balls. The laws specify carefully their dimensions, weight and resilience without any mention of the MoI, possibly an important factor in the behaviour of a ball - including that phase of its motion after a mallet contact as its translational energy is partly changed into rotational energy. There may well be no need to specify the MoI of a ball except as to its consistency with other balls in play. In this context it is worth noting that a ball has a MoI of approximately 3; thus differences between balls would be more difficult to spot.
1. http://www.oxfordcroquet.com/tech/towlson/ Retrieved 11/1/2015.
2. http://www.insearchoftheperfectmallet.com/moi.htm Retrieved 11/1/2015.
3. http://www.insearchoftheperfectmallet.com/Testingmoi.htm Retrieved 11/1/2015.
Note that the units of the MoI quoted here are 100 times larger than in Towlson's article
4.http://farside.ph.utexas.edu/teaching/301/lectures/node139.html Retrieved 11/1/2015.
5. https://www.youtube.com/watch?v=m9iHEanmNWc Retrieved 11/1/2015, http://www.rose-hulman.edu/~moloney/PH314/bifilar_pendulum_05.doc. Retrieved 11/1/2015.
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