# And why?

Dave Tutelman  --  June 28,2017

Yes, it weighs just under 46 grams. You probably knew that.

Here are the details of  why it does, and how much under 46 grams.

## Why would we expect it?

The Rules of Golf say that the ball must not exceed 46 grams in weight. Well, it's a bit more precise than that. "The weight of the ball must not be greater than 1.620 ounces avoirdupois (45.93 g)." But 46 grams is good enough for what we need to know.

If the USGA is setting a limit, I have to believe there is a performance advantage to exceeding the limit. So we would expect manufacturers to want to build balls right up to that limit without exceeding it.

I grabbed a bag full of golf balls and took them to my basement workshop, where I weighed them. I used an inexpensive digital scale with a resolution of a tenth of a gram. Here is the distribution I observed.
• It's pretty close to a bell curve, considering the size of my sample (28 balls).
• Only one ball was over 46 grams in weight, and that one only by a tenth of a gram.
• All but two of the balls were in the half-gram range from 45.5g to 46.0g.
• The two outliers were out by 0.1g, one at each end of the distribution.
The balls were everything from inexpensive distance rocks to premium balls like the Pro-V1, ChromeSoft, and B330. There did not seem to be a bias in the distribution by make or model, but that was just by eyeball; I didn't run any statistical tests, and I doubt there were enough balls of any one model to draw a valid conclusion. All in all, that suggests that the ball makers work to keep ball weight as close to 46 grams as they can without exceeding it.

Here's another fun fact for you. One pound is 454 grams. That means 10 golf balls weigh almost exactly one pound. If you have a bag of balls and want to know how many are there, just weigh the bag in pounds and move the decimal point one place to the right.

At this point, we have answered both questions in the title. The rest of this article is an analysis and "lab experiment" that looks more closely at "why?" Specifically, it asks what advantage there is to a ball that exceeds the limit. It doesn't come up with information that is particularly useful for a golfer or a consumer; I just want to record it here in case the question arises again.

## Why the rules limit?

What is the advantage that the rules try to take away by limiting balls to 46 grams? Usually, when the USGA limits what you can do with equipment, the first suspect is driving distance. So let's investigate how driving distance varies with the weight of the golf ball.

### Computer simulation

The TrajectoWare Drive computer program accurately simulates carry distance of a drive (as long as spin does not exceed 4000rpm), and permits varying lots of conditions -- including the weight of the golf ball. So I tried varying the ball weight and seeing how the distance varied. I expected that the distance would go up markedly as weight increased. Much to my surprise, it didn't. At the 95mph clubhead speed and 10.5° loft I used, the peak distance occurred at just about 46 grams of ball weight. That isn't the sort of performance that usually motivates the USGA to set rules limits. Let's look further.

I tried varying the clubhead speed. I set up the driver loft at 12° to give maximum distance at about 100mph, used "neutral" numbers like 200 grams for clubhead weight and zero angle of attack, and varied the ball weight.

This gave a much more interesting result. For clubhead speeds at 100mph or more, there is a distance advantage to a heavier golf ball.

How much variation are we talking about? The curves on the graph are pretty flat, and it's not that easy to see. So I tried displaying them a different way.
I normalized all the curves so each would show the distance relative to its distance with a 46-gram ball. This shows very clearly the value of departing from the limit set by the rule.

And there is very real value there. A ball weight of 52g, 6 grams over the limit, can get another 13 yards of carry (the orange curve). And notice that the curve is still going up sharply, suggesting that a ball closer to 60g would do even better. That is certainly reason for the USGA to limit the ball weight.

But you have to be a very big hitter for that much gain; you would need 120mph of clubhead speed. At a PGA Tour average clubhead speed of 113mph, you would see about 8 extra yards for the same 52g ball. And the potential gain at 100mph is less than a yard for any ball weight.

But wait! It also works the other way. For clubhead speeds of 90mph or less, you lose distance with a heavier golf ball. No rule needed for them. But that raises an interesting question: can a slow swinger get extra yardage from a lighter ball? And if so, why didn't the USGA put both upper and lower limits on the weight of the golf ball?

Yes, there certainly is distance to be gained from a lighter ball. At 70mph (probably typical for a women's golf league, or at a seniors-only course), a 40g ball will get 8 extra yards -- and the curve says there is more to be had from still lighter balls.

Before we get to the question of USGA intent, let's see if we can explain the surprising result -- heavy balls give more distance for big hitters but less distance for slow swingers -- in terms of physics.

### Physical explanation

Why should a heavier ball go farther than a lighter one? This is surprising on the face of it. After all, the heavier ball will have lower initial ball speed, all other things being equal. Perhaps even more surprising is that the lighter ball goes farther with a low clubhead speed.

The answer is aerodynamic forces. Here are the forces -- the only forces -- on a ball in flight.
• Drag is an aerodynamic force exerted exactly opposite to the direction the ball is traveling.
• Lift is an aerodynamic force exerted exactly perpendicular to the direction the ball is traveling. It is usually upwards, hence the name "lift".
• Weight is a gravitational force always directed straight down.
Let's look more closely at each of these forces.
 Force Direction Proportional to: Helps or hurts distance Drag Opposite ball's path Square of ball speed ... other things ... Hurts Lift Perpendicular to path, generally upwards Square of ball speed Spin ... other things ... Usually helps Weight Straight down Mass of ball Hurts

The reason I emphasized "square" is that square-law variation makes these forces substantially different for the different clubhead speeds. When we compare 120mph with 70mph, the ratio is 1.7.When we compare their squares, the ratio is 2.9, almost 3. That's very substantial. If lift and drag on the big hitter's ball is three times that on the slow swinger's ball, we should expect big differences in the trajectory.

Don't believe it? Let's look at what happens in the absence of lift and drag.

Here are the trajectories of two balls hit identically, except one travels in a vacuum and the other at sea level air density. You might have expected drag to cause the ball not to go as far in full atmosphere. But in fact, lift plays an even bigger role than drag. Lift keeps the ball in the air, working against the weight force which pulls the ball to the ground. So lift will make the ball go farther, because it keeps the ball moving forward through the air longer. The ball may slow down because of drag, but it is still going forward at a significant pace, so keeping it in the air will increase the carry distance.

So lift helps. But drag hurts; it slows the ball. How much? Back to basics: Newton's F=ma can be solved for acceleration: a=F/m. Any slowing down of the ball is deceleration, or negative acceleration. Why negative? The drag force (see the diagram above) is exactly opposed to the direction the ball is traveling. So it is a negative force in that it is opposite the direction of motion. If the force is negative, then the acceleration will also be negative -- slowing the ball. For a given amount of drag, you can reduce the deceleration by increasing the mass of the ball. So we can count on mass to work against drag and limit how much the ball slows down.

Let's get back to those big differences between the big hitter and the slow swinger, and how that factor of 3 difference in lift and drag will affect their ball flight and hence their distance.

 Big hitter: clubhead speed = 120mph Slow swinger: clubhead speed = 70mph Drag The big hitter is helped a lot by the extra mass of a heavier golf ball. Since the drag is only 1/3 that of the big hitter, the advantage of extra mass is not nearly as great. Lift The big hitter has three times the lift, due to the extra ball speed. But it's even more than that; the lift advantage is probably over 4 when you factor in the extra spin due to higher clubhead speed. With all that lift, a heavy ball isn't much of a problem. The slow swinger, with only a quarter of the lift, has trouble keeping the ball in the air. Remember, lift is working against gravity. Too little lift and gravity wins earlier; the ball tumbles from the sky and stops its forward progress. Any extra weight on the ball is going to be a problem here. Excessive Lift If the lift force is so large as to actually be greater than the weight, the ball is going to curve upwards. This is called "ballooning", and it hurts distance. Remember, lift is mostly upwards, not completely upwards. It is perpendicular to the path of the ball. If the path of the ball slopes upward more, then the perpendicular lift slopes backwards more. That backwards component is another force decelerating the ball from getting downrange. Ballooning hurts distance, and extra weight helps the big hitter by reducing ballooning. Excessive lift? What "excessive"? Give me more of it! I don't have enough lift. If you can't give me more lift, then at least give me less weight in the ball.

So ball weight counters drag, and the big hitter has a much bigger drag problem. Ball weight also counters lift. But lift is good for the slow swing and probably bad for the big hitter. Bottom line: it is no longer surprising that a big hitter wants more ball weight and a slow swinger wants less.

Let's take another look at that graph of relative distance. We made a mistake there, and now we're in a position to understand how and why. The mistake was to use the same loft for all the clubhead speeds. The graph assumes the same driver loft for all clubhead speeds. But the ideal loft will be greater for lower clubhead speeds.  Here is a table of the best loft for each clubhead speed.

 Clubhead speed 70 80 90 100 110 120 Optimum loft 18° 15.5° 13.5° 12° 11° 9.5°

So let's assume that each of the golfers at each of the speeds was properly fitted for their driver, and plot relative carry distance vs ball weight again.

The general shape is still the same; big hitters get more distance from a heavier ball, and slow swingers more distance from a lighter ball. But the gains and losses from ball weight are much smaller.
• A 52g ball now only gets 4 extra yards for a big hitter, compared with the previous 13 yards. At least as important, the curve looks like 5 yards is the maximum it can ever improve.
• A 40g ball only gets 2 extra yards for a slow swinger, compared with the previous 8 extra yards. And it looks like 2 yards is the biggest the improvement is ever going to get.
The physics should tell us why. The slow swinger's fitted driver will have 18° of loft to the big hitter's 9.5° -- almost twice the loft. Loft does two things -- increases launch angle and increases spin -- that the slow swinger needs. If they didn't get it from added driver loft, they'd have to make up for it with a lighter ball. So the difference in loft makes up for most of the difference in ball weight. The shape of the curve for the fitted driver is similar to that for the fixed-loft driver, but the amplitude of the curve is less than a third the size.

### Balls for slow swingers

Still, if you don't have much clubhead speed, there is some small advantage to a lighter ball than 46 grams. Do the ball manufacturers take advantage of it and make, say, a 40g ball for seniors and women? Well, most of the ball companies do make one or more women's model golf balls. They didn't show up in my first set of measurements because I didn't have any around my "lab". (My wife gave up golf more than 10 years ago because she wasn't physically up to it.) But if there is an advantage in practice and not just on a graph, you would expect that women's golf balls would be the place it would show up.

So how do I get a variety of them to weigh?

Colonial Terrace GC, my town's municipal course, has a women's league that plays on Tuesday mornings. So the next Tuesday, I showed up at 7am and got permission to hang out at the registration desk and ask to weigh women's golf balls. (Get that snide grin off your face. I got enough of that on Tuesday.) Here are things I learned from that effort.
• Only half the women in the league use (or even have in their bag) a women's model ball.
• Of those that didn't, only half were even aware that women's golf balls exist.
• The balls were lighter than the men's golf balls. But hardly at all; there was only a 0.3g difference in the average weight.
Here are the statistics for the weights that I found.
 Men's model Women's model Average 45.71 gram 45.41 gram Std. Dev. 0.16 gram 0.22 gram Median 45.7 gram 45.5 gram
The display on the graph is an old-school way to visualize the relationships of empirical probability distributions. The horizontal bar is the mean (average) and the rectangle behind it extends one standard deviation each side of the mean. Note that the rectangles have a bit of overlap between them (not much at all), and the standard deviation of neither distribution extends all the way to the mean of the other. That suggests a small but probably significant difference in the underlying distribution.

A curious result, which leaves us a dilemma. The women's balls are lighter (probably with statistical significance), but not nearly enough lighter to give a physical advantage. So the golf ball manufacturers seem to be doing it deliberately, but not for the purpose of greater distance. How can we explain this? There are a few possibilities:
1. There is no performance need to make them lighter. Golf ball manufacturers can increase lift by playing with the dimple pattern and thereby affect the aerodynamic lift force. This is probably easier and even more effective for performance than making a whole line of lighter golf balls.
2. The only performance aspect we've mentioned so far is distance. How about wind resistance? It is a good thing for a ball's flight to be less affected by a crosswind or headwind. And a heavier ball is more wind-resistant. So let's keep that aspect of performance.
3. But the women's model balls are indeed lighter, if not enough to affect performance. Why? Notice that the standard deviation is larger. Perhaps each manufacturer decided to reduce costs by widening the limits of quality control (larger standard deviation), but still stay under 46 grams by designing to a lighter center of the distribution (lower mean weight). That's a strong statement considering what a small sample size I have, so it is speculative at best.
In any event, there is some small distance advantage to be had by making lighter balls for lower clubhead speeds. I haven't seen any indication that ball manufacturers are taking advantage of it.

## Conclusion

Golf balls are manufactured to be just under 46 grams in weight. Men's or gender-neutral golf balls tend to be within a half gram of that, women's balls maybe a third of a gram lower than men's.

The rules set an upper limit of 46 grams, and there is a distance advantage that big hitter gain with heavier balls. But there is a small distance advantage for slow swingers to use lighter balls. So the rule seems less interested in keeping things equal than "protecting" golf courses from big hitters.