Testing Moments of Inertia

It is easy to calculate the theoretical moment of inertia of ideal objects, like infinitely small points of mass, and given the appropriate formula, of any known body whose constituent parts are homogeneous and known.  But what of a real object like a weighted mallet head that one cannot look inside?

A relatively straightforward and reliable way to test a mallet head’s moment of inertia is to put it on a record deck.  I have used a Garrard Quartz DDQ650, which is fitted with a stroboscope which verifies that the turntable is turning at a given speed – I have used 45 rpm.

Calibration

The deck is first calibrated by using small weights of known mass at various measured distances from the centre, and comparing the number of revolutions that it takes the deck to stop turning with the known moment of inertia.  My calibration table looked like this:

Test Results

I have then tested 4 of my heads, and RPM Series 2000 head marked with the words “Peripheral Weighted”, and then a Dawson Millennium, and also a block 60 mm wide, with no holes or weighting other than a G10 striking face glued at either end.  In each case, I have not only observed the absolute moment of inertia, but also the MoI efficiency, ie the observed MoI expressed as a percentage of the theoretical MoI if the head consisted of nothing but a point mass at either end (see note).  In practice, of course, that theoretical maximum could never be achieved using real materials.

A thin block would have an MoI efficiency of 33.3%.  Increasing the width of the black to the proportions of a typical mallet gets the figure up a shade: I got 35% for my block.

The results show that the RPM head is indeed peripherally weighted, but not by much.  Its MoI efficiency is 38%, ie a bit more than the solid block, but not dramatically so.  Other RPM mallets might perform better.

The Dawson Millennium head (I removed the shaft for testing purposes) looks as if it might be peripherally weighted, but surprisingly came in at 35%, the same as the unweighted block.  The centres of the weights on that head are a fair way back (52mm) from the end faces, which is only 6mm better than the 57% minimum (see note), and the relatively heavy metal side pieces also end a fair way back (about 12mm) from the ends. As far as I am aware, Dawson does not advertise this mallet as peripherally weighted, and different results might be obtained with different weights fitted (the head is designed so as to be able to change the weights pretty easily).

At the other end of the scale, the mallet that is probably more peripherally weighted than any other – Alan Pidcock’s round-section graphite 2001 model – comes in at 73% according to his own calculations (see note).

My heads came in at around 60%.  In other words, I am getting more than half as much MoI again as an RPM or Dawson mallet relative to the weight and length of the mallet head, but not quite as much as the Pidcock.

The raw data of these tests is on a spreadsheet here.

The Proof of the Pudding

Is there any way of testing the theory that increasing the moment of inertia really decreases the directional error in shot hit off centre?  This is not such easy as it may seem, since, as others have pointed out, a really good player can achieve consistently