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Putting Theory

Large Sweetspot - Alias greater Moment of Inertia
By Dr. Leon Seltzer

One of the most popular phrases used in advertising the virtues of metalwoods and heel and toe weighted irons is the "larger sweetspot" possessed by that particular club. The term or word "sweetspot" is well known to all golfers. If the ball is contacted on the sweetspot there will be no angular deflection of the clubface and the ball will carry further than if contacted at any other place on the face of the club.

Technically the sweetspot represents the center of percussion. The definition of the center of percussion is as follows:

Center of Percussion - That point in a body free to move about a fixed axis at which the body may be squarely struck without jarring the axis.

In the case of a golf club, the axis is the shaft of the club. It should be pointed out that a line through the center of percussion does not always pass through the center of gravity. In the case of a clubhead, it does come close to doing so.

Note too that the definition of center of percussion refers to it as a point. This is technically true and a point is no larger than the sharp end of a needle. In dealing with the impact between a golf ball and the face of a golf club, it is not reasonable to think of the golf ball striking a point, the center of percussion, because during impact the ball flattens on the face of the clubhead and has a circular footprint which may be as large as an inch in diameter. The size of the footprint will vary depending on the hardness of the ball and the clubhead speed. Consequently, even on an off-center hit, the part of the ball that is flattened during impact will cover the center of percussion. During a perfect contact situation the center of percussion is at the center of the contact area. Even the slightest deviation from this ideal condition will result in some angular rotation of the face and resultant side spin.

In the case of perfect center hits with a driver, neither the size nor shape of the clubhead will make any difference in the results if the weight of the clubheads and the clubhead velocities are identical.

What then is meant by the expression "larger sweetspot"? This characteristic refers to a clubhead which is more forgiving on off-center hits, a clubhead which has greater resistance to opening and closing on off-center hits. The characteristic that describes the magnitude of the resistance is called Moment of Inertia. The definition of this term is as follows:

Moment of Inertia - A measure of a body's resistance to angular acceleration equal to the sum of the products of each mass element of a body multiplied by the square of its distance from an axis.

In the case of a clubhead, the axis passes through the center of gravity (e.g.) because it is about the e.g. that the club will tend to rotate on off-center hits. Since the moment of inertia is proportional to the square of the distances of the mass elements from the e.g., substantial increases can be achieved by distributing the weight in such a manner as to favor the heel and toe in the case of irons (cavity back) and the perimeter of the clubhead in the case of hollow metal drivers and fairway woods.

Let us look as some common examples of things that ate familiar to all of us in order to have a better understanding of Moment of Inertia, how it works and how it is used. Take, first, the cases of a diver, a gymnast or a circus acrobat When they want to do multiple somersaults, these athletes will bend their knees and grab their ankles, bringing their extremities (the arms and legs) closer to the center of gravity. This reduces the moment of inertia to such an extent that they are able to do as many as three complete somersaults when they would be able to do only one in an extended position.

Another example is that of the figure skater who spins at a high rate on the tips of the skates. In order to accomplish this feat, the skater will hug himself or herself with both arms thereby reducing the moment of inertia about the vertical spinning axis. The skater is able to lower the spin rate and stop the spin by extending both arms straight out to the side. This increases the skater's moment of inertia drastically.

Both of the above cases represented dynamic situations. In a static situation let's take the case of the two-by-eight boards that are used as joists to support a floor. Assume these boards have a cross sectional area of sixteen square inches. You will always see these joists (or it might be roof rafters) installed standing on end with the two-inch dimension at the top and bottom. The reason for this is that the moment of inertia of the board in this position is sixteen (16) times greater than it would be if the eight inch sides were at the top and bottom. This is readily understandable since, by definition, moment of inertia is proportional to the square of the distance of the mass elements from the axis. The moment of inertia for a rectangular beam is

I = bh3/12 where b and h are the base and height dimensions

In the two different positions, the moment of inertia are 85.333 and 5.333.

Next take the example of a sphere which is not that much different in shape from a driver. The moment of inertia of a solid sphere about its e.g. is

I = 2/5 Mr2 where r is the radius of the sphere

In the case of a thin wall sphere, the moment of inertia is

I = Mr2

The solid sphere is analogous to a driver made of solid wood, while the thin wall sphere can be compared to a thin walled metal driver. If the spheres had identical radii and weights, the moment of inertia of the hollow sphere is two and one half (2-1/2) times that of the solid sphere. This means that a hollow metal driver will be that much more effective in resisting angular deflection from off-center hits than a driver made from solid or laminated wood.

Figure 1, Figure 2 & Figure 3

Finally, take one more case, one that has to do with putter design. This is a prime example of the benefit that can be derived from heel and toe weighting, that is, greater moment of inertia. Assume that we have a putter head made from solid square bar stock. The dimensions of the head, in inches, are 4 by 1 by 1, and the volume is 4 cubic inches. Since we are only looking for numbers on which to base a comparison, we will assume unit mass per unit volume. The moment of inertia of this putter head about an axis perpendicular to the top of the putter head passing through the c.g. is 5.668. The putter and the axis used for calculating the moment of inertia is shown in Figure 1.

Figure 2 shows a second putter where Elements A and B have been moved from their central location and placed behind the putter at each end. The moment of inertia of Putter #2 about the axis location is 8.542, an increase of 51 percent.

Figure 3 shows a third putter where Elements A and B have been moved to the ends of the putter, increasing the length of the face to 5 inches. The weight of the putter has not been changed. The moment of inertia of Putter #3 is calculated to be 10.417. This is an increase of 84 per cent over Putter #1 and 21 per cent over Putter #2.

The moments of inertia for all three putters were calculated about axes located at exactly the same distance from the center of the face of the putter. Table 1 is a summary of the results.

Table 1
Putter Number Moment of Inertia % Increase over Putter 1
1 5.668 --
2 8.542 51%
3 10.417 84%

An enlarged sweetspot? No, a greater lateral moment of inertia. What's in a name as long as it benefits the golfer?

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