Moment of Inertia

 

This page discusses some theory about moments of inertia as relevant to croquet mallets.

 

What is Moment of Inertia?

 

The moment of inertia of a solid object is its inherent reluctance to rotate.  It is the rotary equivalent of mass.  For a small object, the moment of inertia is the mass of the object multiplied by the square of the distance from the particle to the axis of rotation.

 

Can it be calculated for whole shapes?

 

Yes. The moment if inertia of a solid block around its centre is given by the formula

 

I = m(C2 + D2)/12

 

where I is the moment of inertia, m is the mass of the object, and C and D are the length and width of the block.

 

So take the case of a typical garden mallet head, 225mm (about 9 inches) long and 60 mm (about 2.5 inches) wide, with a head weighing ūkg (about 1lb 10ozs). The moment of inertia about the axis of the shaft is 3,389 kg mm2.

 

A typical club traditional club mallet fares better.  Assuming a head length of 250 mm (about 10”), and a head weight of 1Kg (about 2lb 2oz), the moment of inertia will be 8,358.

 

Now increase the length of the head to 300 mm (about 12 inches) and modestly increase the weight  to 1.1 Kg (about 2lb 7oz).  The moment of inertia goes up to 7,503 kg mm2,

 

Some commercially available mallets have some peripheral weighting, whereby some limited proportion, say ?, of the overall head weight is comprised of metal weights – often brass – positioned near the end faces. Such a mallet has a MoI of 11,870.

 

But more can be achieved by doing what I do: by drilling large holes in the wooden body of the mallet and using tungsten (which is about half as dense again as lead), a much greater proportion of the weight can be located near the end faces.  By these means, I get the MoI up to 15,736.

 

A further improvement can be achieved by increasing the overall weight of  the head from 1.1 Kg to 1.25 kg; the MoI goes up to 17,975.

 

What is the theoretical maximum moment of inertia for a mallet of such size and weight?  For a 1.1kg head with virtually all the weight concentrated virtually at the corners, the MoI is 23,670.

 

The effect of a single ball

 

At first glance, the equation I = m r2 would suggest that an increase in the overall weight of the mallet head would lead to a proportionate increase in the moment of inertia.

 

In practice, it seems that the mallet head will not rotate around the axis of the shaft, nor the striking surface.  The total impact time appears to be around 2 milliseconds, being about 1 millisecond of compression and about 1 millisecond of decompression. The results of carbon paper tests suggest that there is no lateral slippage between ball and mallet face during an off-centre impact (there may perhaps be a little rolling). Unconstrained objects subjected to torque tend to rotate around their centre of gravity (since this is the axis at which the moment of inertia is least) and, without having conduced tests, it seems likely that the mallet head and ball act during the millisecond of compression as a single body, such the mallet head rotates around the centre of gravity of the mallet head and the ball taken together (the Effective Centre of Gravity).  Where this is will depend on the relative weight of the mallet head and the ball: the lighter the mallet head the more the Effective Centre of Gravity will shift towards the forward end of the mallet head.  Depending on the length and weight of the mallet head, the Effective Centre of Gravity will typically be about a couple of inches forward of the shaft; the shift can be calculated by the formula

 

 

Mmh d = Mb (Lmh/2 – d + Rb)

 

Where Mmh is the mass of the mallet head, d is the distance between the Effective Centre of Gravity and the shaft,  Mb is the mass of the ball,  Lmh  is the length of ythe mallet head and Rb is the radius of the ball.

 

Before considering amount of this effect, we need to consider how much the moment of inertia will go up for an object rotating around an axis that does not pass through its centre of gravity.

Parallel-Axis Theorem:

If you know the moment of inertia about the centre of mass you, can find it about any other pivot point by using the parallel axis theorem, provided the axis of rotation is parallel. The relationship is a simple one: the moment of inertia around that other axis  is given by the formula

I = Icg + M d2

Where Icg is the moment of inertia about the centre of gravity, M is the mass of the object, and d is the distance between the other pivot point and the centre of gravity.

The weight equation

 

Adding weight thus has 2 contrary effects.  The main effect is to increase the MoI.  A subsidiary effect is to shift the Effective Centre of Gravity back towards the shaft, thus decreasing the MoI.  Taking both factors into account, adding 10% weight to a peripherally weighted head adds only 7% to the effective MoI.

 

Comparison graphic

 

 

A summary of the effect of moment of inertia on roqueting

 

Assuming (as appears likely) that the degree of deflection resulting from an off-centre hit is proportional to the effective moment of inertia of the mallet head, the practical result of all this might be as follows:

Take the case of that degree of off-centre hitting that will produce a miss of 12” on a long roquet using a 12” non-weighted head.

 

v     A shorter 10” head would produce a miss of 17”.

 

v     A normal commercial head with some peripheral weighting would miss by 9”.

 

v     My head with more weighting would miss by 7”

 

 

 

See links to experimental data on Testing the Moment of Inertia, and Where the weight goes 

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