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
–

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.

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