Jonathan Kirby finds that hitting a ball off-centre induces a rotation of the mallet head and a slewing of the ball from the direction of swing. This is emphasised in mallets without peripheral (end face) weighting. Don Gugan add a review and analysis at the end.
This afternoon I managed to get to a lawn and conduct some experiments with off-centre hitting. I put down a taut string (boundary string) to give a clear straight line. I swung the mallet along the string, that is, with the central sighting line above the string, hitting a ball placed just to the side of the string, in a position where I would hit the ball near the edge of the face of the mallet. I repeated about 100 times, with 3 different mallets, hitting both with the left edge and the right edge.
I used my mallet: Pidcock (11" x just under 2" square wooden head, with circular lead masses behind the endfaces), an Oakley Woods Brighton mallet (8" x 2" x 2" solid plastic head) and a club mallet (8" x 2" x 2" solid wood head).
I found I could get a consistent result, with the balls from successive attempts rolling full ball into each other. (At least for some strengths of shot. For stronger shots the lawn was not even enough and I couldn't hit consistently enough the same strength to make that happen.) There was no noticeable asymmetry between hitting on the left and on the right. I'll describe hitting on the right edge of the mallet face.
In all trials: During and just after the impact, the mallet twisted clockwise (looking down) in my hands. After the impact, it was travelling not along the line of the string, but in a direction about 20 degrees to the left of that line. (I didn't measure the angle.) The ball went off in a direction a few degrees to the right of the string. There was no noticeable slippage between mallet and ball. At no point could I see any sign that the centre of the front endface moved substantially to the right (although it may have done a little).
Hitting harder reduced the angle at which the ball went off, but I didn't detect a difference in the angle the mallet went in. (I wasn't really looking for the mallet travel angle, and hitting harder obviously made things happen faster.)
The latter two mallets gave similar results. Hitting at a fairly gentle pace, where I got very consistent results, the ball went off at an angle at which it just missed a four foot roquet. The angle the ball went in was approximately halved with my mallet.
Having done the experiments and thought about what I saw, what I think happened is the following (this is technical): In the inertial frame of the ground (which is really what I was directly observing), the mallet rotated about a point near the front endface of the mallet, and at the same time was moving left and forwards. It was obviously accelerating backwards and slightly leftwards during the contact with the ball, and there must also have been an angular acceleration.
In the inertial frame of the mallet at the start of the contact with the ball, the linear acceleration is the same, and I presume the angular acceleration was about the centre of the mallet head.
The transformation between these two frames is one of constant velocity, and I think this simply means that the apparent centre of rotation is shifted between the two frames. (I have not done the calculation to check that this is exactly the case - perhaps someone else would care to). To a reasonable approximation, I think the axis of rotation is the shaft, for the frame of the mallet, and the line through the point of contact, for the frame of the ground.
I think the relevant frame for the Moment of Inertia calculation is the frame of the mallet during the collision. This is a non-inertial frame, but perhaps it can reasonably be approximated by the inertial frame of the mallet at the start of the collision.
In any case, my longer and partially end-weighted mallet was significantly better than the shorter, non-end-weighted mallets.
I hope that sheds some light on the problem.
Don Gugan adds:
Review of "Hitting Off-centre" by Jonathan Kirby – Don Gugan
The physics of striking a ball with a mallet is not simple, and in particular the role of one’s grip is probably impossible to allow for; however, useful upper limits for the effects of hitting off-centre can be obtained by considering the head of the mallet at the moment of impact as hanging freely in space while the ball strikes it and rebounds. The physics is exactly the same in this 'frame of reference' as when the mallet moves to strike the ball, but it is perhaps easier to see what is going on.
The ball delivers a short impulse to the mallet which starts it moving backwards so as to conserve momentum, and if the impact is non-central it also delivers an impulsive torque to the mallet which sets it in rotation at a rate given by
w = 2mvx/I,
where w is the angular velocity of rotation, m is the mass of the ball and v its velocity, x is the perpendicular distance of the line of impact from the centre of rotation, and I is the moment of inertia of the mallet head about its centre of rotation (for a uniform head of mass M, length L, and width a, I = M(L^2 + a^2)/12). We don’t know all the values of these terms for Kirby’s experiments, but we can make reasonable guesses. Substituting M = 3m (which is about right for most mallets), L = 20 cm, and a = 2.5 cm = x in Kirby’s experiments, and if we assume that v ≈ 3m/s, as for a typical 10 yard shot, the rate of rotation, w, is about 14 radians/sec for his two smaller mallets. The critical issue is how far the mallet rotates while in contact with the ball, and since the contact time in a plain hit is always close to a millisecond, the mallet head has turned at most about 14 milliradians during impact, i.e. about 0.8 degrees. The ball rebounds from the mallet while it is turning, so that the average angle during which the mallet exerts a force on the ball will only be about half of this, 0.4 degrees, leading to a deviation of 2.5 inches in a ten yard shot.
The angle of deviation one can infer from Kirby’s results, i.e. just missing a 4 foot roquet, is 4.3 degrees, and is ten times greater than this theoretical value (and this itself is an upper limit of what we should expect); evidently there are things going on which are more important than the moment of inertia of the mallet head.
One expects the mallet head to continue rotating after impact, and to be displaced sideways. The position of the mallet head shown in the diagram must be near the end of the stroke, perhaps half a second after impact, when an unconstrained mallet head would have twisted through about 7 radians - a complete revolution! Evidently the player’s grip and the stiffness of the shaft have long since taken over, and for an observed twist of twenty degrees it appears that the grip has become the dominant factor after about 20 milliseconds.
One also expects the transverse components of momentum to be conserved during the impact; the later positions of the ball and mallet are roughly compatible with this, however, both are about ten times the expected size, and as it is certain that the grip has become important by this time, little can be inferred from these observations.
All in all these results raise more questions than they answer, and give no convincing evidence that increasing the moment of inertia of a mallet head to reduce twist as a result of non-central impact is the secret of straight hitting: it is more likely that it improves the consistency of the initial swing. Kirby’s eleven inch Pidcock mallet with weighted ends would presumably have had a moment of inertia about three times greater than his shorter mallets, and this is is reasonably consistent with his observation that the deviation due to hitting off-centre “was approximately halved”, but there is evidently something else important going on. There is a need for better designed experiments, preferably in a laboratory, and preferably with a camera able to resolve detail at times of less than a millisecond.
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