# Can Wobble Rob(ble) Hitters? Fly Ball Distance and Baseball Precession

In the chase to break the story of the “smoking gun” behind the recent surge in MLB home runs, many a gallon of digital ink hath been spilt exploring possible modifications to the MLB balls, home-run-optimized swing paths, and even climate change. In my field of Earth Science (atmospheric chemistry, to be more exact), it’s rare that a trend in observations can be easily attributed to a single causal factor. Air quality in a city is driven by emissions of pollutants, wind conditions, humidity, solar radiation, and more; this typically leads to a jumble of coupled differential equations, each with a different capacity to impact overall air quality. To my untrained eye, agnostic to the contents of the confidential research commissioned by MLB and others, this problem is no different: a complex mixture of factors, some compounding each other and some canceling others, is likely fueling the recent home-run spike.

This article will examine the potential for a change in the MLB ball minimally explored thus far: reduction of precession due to decreased internal mass anisotropy. What a mouth full! “Precession” and “anisotropy” don’t have the same ring as “juiced ball” or “seam height” (though they may be on par with “coefficient of restitution”). But these words can be replaced with a more familiar (though funny-sounding) word: **wobble**. This wobble can occur for many reasons, but the most probable explanation in baseball is that the internal baseball guts are slightly shifted from the center of the ball. This could be due to manufacturing imperfection, or in the course of a game, contact-induced deformation of the ball.

Precession, in general, occurs when the rotational axis of an object changes its own orientation, whether due to an external torque (such as gravity) or due to changes in the moment of inertia of the rotating object (torque-free). Consider a spinning top: the top spins about its own axis (symmetrically spinning about the “stem” of the top) while the rotational axis *itself *(as visualized by the movement of the stem) can trace out a coherent pattern. If imparted with the same initial “amount” of spin in different ways, the total **angular momentum **(from both rotation and precession) of the top will be the same whether it’s spinning straight-up or precessing (wobbling) in an elliptical path.

**Figure 0: Perhaps the most hotly debated spinning top in the world**

As with other potential explanations relating to a physical change in the ball, a change in mass distribution could have occurred unintentionally due to routine improvements in manufacturing processes. By getting the center of mass (approximately, the cork core of the baseball) closer to the *exact geometric* *center* of the ball, backspin originally “lost” to precession (in the form of wobble-inducing sidespin) could remain as backspin while conserving total angular momentum; increased backspin has been shown to increase the “carry” of a fly ball, therefore increasing the distance (potentially extending warning-track shots over the fence). A deeper discussion of angular momentum can be found in any mechanics textbook or online resource (such as MIT OCW handouts), but the key takeaway when considering a particular batted fly ball is that productive backspin gets converted to non-productive precession (roughly approximated as sidespin in one axis) when mass is not isotropically (uniformly from the center in every direction) distributed. This imparts a *torque-free* precession on the spinning ball, causing the rotational axis to trace out a coherent shape.

Precession in baseball has not been deeply studied; in fact, when explicitly mentioned in seminal baseball physics resources, it is noted as a potential factor that will be ignored to simplify the set of physical equations. Together, dear reader, we shall peek behind the anisotropic veil and explore how precession might impact fly-ball distance, and by extension, home-run rates.

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For those of us with some experience throwing a football, even just in the park, we can picture the ideal “backyard Super Bowl” pass: a tight spiral that neatly falls into the outstretched hands of the intended receiver. The difficulty of executing such a perfect throw is evident in the number of nicknames for imperfect throws that wobble (precess) on their way up the field short of their intended target (see “throwing ducks” re: Peyton Manning). In football, the wobbly precession of a ball in flight is typically blamed on the passer or credited to a defender for deflecting it (or in some cases, allegedly, a camera fly wire). It’s not as easy to imagine such behavior in baseball: even in slow-motion video shots of fly balls, the net spin of the ball is dominated by backspin. In addition, the nearly-spherical shape of a spinning baseball has significantly different aerodynamics than the tapered ellipsoid used in football. However, even a small amount of precession has the potential to shave yards off the distance of a football pass; therefore, impacts of precession are certainly worth exploring in the game of baseball.

As a sometimes-teacher (I have taught two laboratory classes at MIT), I strongly believe in the power of simple physical models to qualitatively inform trends in the not-so-simple real world. Therefore, for the first step of exploring the effect of ball precession in the game of baseball, I have turned to the wonderful Trajectory Calculator developed by Dr. Alan Nathan. The Calculator numerically solves the trajectory of a batted ball by computing key physical properties in discrete time steps. While many physical attributes of the ball are calculated in the various colored fields, any of them can be overwritten with custom values.

**Figure 1: Fly Ball Distance with Nathan Trajectory Calculator defaults, conversion of backspin to sidespin**

In Figure 1, I use the Trajectory Calculator to explore the effect of sidespin conversion on a single fly ball with the same initial contact conditions as the default (100mph exit velocity, 30-degree launch angle, default meteorological conditions), with the total spin set to 240 radians per second. Backspin is not converted to sidespin in a one-to-one fashion: because of the Pythagorean relationship between these factors, total spin is equal to the square root of the sum of the squares of sidespin and backspin. Therefore, to conserve angular momentum, a 10% reduction in backspin (216 rad/s) yields 104.6 rad/s of sidespin, which together lead to a ~1% decrease in fly ball distance from 385.3 ft to 381.3 ft.

With all of the assumptions made here, notably that introduction of precession can be simulated as pure conversion to sidespin to conserve angular momentum, the effect of precession on the flight path is clear but rather modest in this simple approach. However, the Calculator results show that by reducing the “wobble” in a ball’s trajectory, it will carry further. A league-wide reduction in precession would mean that balls would, on average, travel further, leading to an uptick in home runs. If decreased precession would also decrease the effective drag the ball experiences in flight, the effect of increased fly-ball distance could be even further enhanced.

A more realistic exploration of precession will require further modification to the modeling tools at hand. Following Brancazio (1987), which studied the effects of precession on the trajectory of a football, and additional follow-on work, a precession-only physical model can be developed to explore more complex aspects of the problem posed here. Elements of this precession-only model can be fed back into the Nathan Trajectory Calculator, but without a full understanding of some unconstrained physical constants and mechanical aspects of the pitch-contact-trajectory sequence, a tidy figure in the style of Figure 1 will be difficult to produce.

Again, as I mentioned above, I find simple models to be effective tools for teaching concepts. Therefore, let’s consider a “perfect” baseball to be a completely uniform, isotropic sphere, as in Figure 2. This perfect ball is axially symmetric and should not have any precession in its trajectory due to changes in its moment of inertia (I). Now, let’s add a small “spot mass” (that doesn’t add roughness to the surface) on the surface of the ball along the axis of rotation corresponding to pure backspin (the x-axis here). This ball with a spot mass should approximately represent an otherwise-perfect sphere whose center of mass is slightly shifted in the x-direction.

**Figure 2: (A) real baseball, (B) perfect sphere, (C) sphere with a point mass at the surface, and (D) sphere with slightly offset center of mass approximately equivalent to (C)**

If the model ball has a mass *m1* that is isotropically distributed through the entire sphere, and a point mass with mass *m2* that is located on the surface along the x-axis, the moment of inertia can be calculated in each direction, summing the contributions from the bulk mass *m1 *and the point mass *m2* (Figure 3).

**Figure 3: Moments of inertia for isotropic ball (mass m1) with a point mass (m2) at the surface**

Of course, the mass of a real baseball isn’t isotropically distributed, and there is no such thing as a “point mass” in reality; however, by exploring different combinations of m1 and m2 that sum to to mass of an actual MLB baseball (5.125 oz, as used in the Nathan Trajectory Calculator), the ball can be distorted in a controlled manner to explore the effects on precession and fly-ball distance. Using a set of equations derived from Brancazio (1987) Equation #7, the initial backspin of a ball (omega_x0) can be calculated given an initial total spin (omega), the variable B (the “spin-to-wobble” ratio indicating the number of revolutions about the x-axis per precession-induced “wobble”, a function of the moments of inertia I_x and I_yz), and the angle of precession (built into the variable C, with theta being the angle between the x-axis and the vector of angular momentum when precessing, similar to the angle between a table and the “stem” of a spinning top).

**Equation Block 1: Derivations from Brancazio (1987) used in a simple model of baseball precession**

The limitation of this approach is that in order to explore the theta-m2 phase space, we must prescribe *a priori* an angle theta at which the precession occurs. By instead solving for theta from **equation 5** above (Figure 4), we can get a sense of the possible values for theta by prescribing the fraction of omega that is converted to precession (the variable A, a mixture of omega_y and omega_z, also called “effective sidespin”).

**Figure 4: Contour plot of theta (degrees) with respect to ranges of m2 and variable A (effective total sidespin)**

Figure 4 shows that angles between 0 and 6 degrees are reasonable for the conditions explored using the approach from Brancazio (1987) as translated to baseball. So let’s turn to **equation 6**, using a range of angles from 0 to 6 degrees, to explore the effects of precession on backspin omega_x (Figure 5).

**Figure 5: Contour plots of backspin (omega_x) and effective sidespin (variable A) with respect to m2 (as % of m) and theta (degrees)**

Great, the effect of a point mass along the x-axis of the ball can be quantified in this model! The effect is modest, but has the potential to slightly decrease the distance of an identically struck isotropic ball. But there is one major limitation to the model as currently shown: when the angle theta is chosen *a priori*, there is no capacity of the model to correct to a more physically stable angle. In fact, along the entire x-axis of the plots in Figure 5, where m2 = 0, the ball should be completely isotropic and therefore no precession would occur; a small initial theta would likely be damped out over a small number of time steps. In addition, the contours of constant omega_x in Figure 5a curve in the opposite sense than might be expected: increasing m2 should lead to more pronounced procession. On the other hand, this very simple model does not take into account the possible effects of *torque-induced* precession caused by gravity (extending the effect of mass anisotropy alone), nor does it account for additional drag impacting a precessing ball. More study is needed to further elucidate the possibility of precession having a considerable impact on fly-ball distance; however, unlike the sometimes-empty calls for “further exploration” of minimally promising leads in academic journal articles, I intend to execute such investigation.

All of these limitations are inherent in the fact that, without outside data to constrain the physics of precession as it applies to baseball, the problem we are trying to solve with this simple model is an *ill-posed problem* in which there is not a unique solution for a given set of initial conditions. Luckily for us, we live in the Statcast age where **position, velocity, and spin** of the baseball are all continuously measured (if not fully publicly available). In addition to benefits gained from Statcast data, this problem can also be further constrained by experimental data on MLB balls. Finally, an opportunity to put my skills as an experiment-first, computational-modeling-second scientist, to use! Stay tuned to these pages for follow-up experiments and data analysis in this vein.

The conspiratorial allure of an intentional ball modification directly induced by Commissioner Rob Manfred is visible on online comment sections far and wide; however, many of the most credible explanations for ball changes are benign in Commissioner intent and perhaps attendant with improvements in ball-manufacturing processes. In any case, there are likely multiple facets to the current home-run surge. Ball trajectory effects due to precession have traditionally been ignored to simplify the problem at hand; this initial exploration shows that due to the difficulty of the problem, that was likely a good trade-off given the data available in the past. In the future, however, past work in diverse areas from planetary dynamics to mechanics of other sports can be used alongside new and emerging data streams to help determine the impact of precession on fly-ball distance.

*Python code used to generate Figures 4-5 can be found at https://github.com/mcclellm/baseball-fg*

*Special thanks to Prof. Peko Hosoi (MIT) and Dr. Alan Nathan for providing feedback on early versions of this idea, which was born on a scrap of paper at Saberseminar 2017.*

Michael is a PhD Candidate in Atmospheric Science in the Department of Earth, Atmospheric, and Planetary Science (EAPS, Course 12) at MIT. Along with three other sports-inclined MIT friends, he co-founded the baseball toolBOX (www.btoolbox.org) research venture, which aims to use analytical methods from fields such as atmospheric science and electrical engineering to discover actionable insights in baseball physics and mechanics. You can reach him with comments, questions, and juiced-ball conspiracy theories at michael@btoolbox.org

It would be as interesting to look at anisotropic effects on the pitched ball that might either increase or decrease pitch movement, break or wobble, as well as ball flight off the bat. At least in that case, some semblance of experimentation might yield reproducible results. I wonder as well if the coefficient of restitution of a compressed ball coming off a bat is effected meaningfully by anisotropic behavior. As I recall, those are rather complex tensor described relationships that may be too complex to describe easily in an analytical function.

What initially sparked this idea was actually throwing BP in the MIT summer softball league. We had a softball that had a huge goose-egg deformation, nearly busting the seams of the ball. Somehow, despite me throwing it near the trash can multiple times, it kept making its way back into the ball bag. It took this wild corkscrew path as I threw it to the batter.

Only later did I sit down with Alan Nathan at Saberseminar and work out the feasibility of this idea. I’m planning some experiments in the near future because, like you said, the difficulty of solving this issue analytically (even just identifying the time-dependent moment of inertia tensors of the ball) will be prohibitive.

But thanks for the suggestions–I would imagine that the pitch is affected, especially with the amount of sidespin and topspin intentionally imparted by the pitcher. I also imagine that mass anisotropy would affect the COR (since the ball would compress in different areas) but it’s not immediately clear to me whether there would be a measurable difference in game-relevant conditions.

It’s convenient that today there are pitch machines that can ‘throw’ a baseball that curves or sinks in a reproducible way at any chosen velocity. I’d suspect that strobe photography or perhaps newer technology would allow for discrete analysis of trajectory as well. It would be a very interesting experiment to run, IMO. All to often the intricacies of batted ball dynamics get dissected without similar focus on the pitched ball, which IMO is perhaps even more important.

BTW, fascinating insights.

Thank you very much! I’m truly approaching this problem with the wonder that led me to atmospheric science in the first place. Awe at the physics that lead to towering thunderstorms isn’t that far removed from the amazement at all of the physical factors that impact the game we all love.

It’s not too hard to imagine replicating the experiment Alan Nathan presented at Saberseminar (https://www.fangraphs.com/tht/fly-ball-carry-and-the-home-run-surge/) to drill into the potential effects of precession on a fly ball trajectory.

I also happen to very strongly agree that there is a ton left to be learned on the other side of the bat-ball contact: the flight of the pitch itself. Because the contact is so brief and location-dependent, I would imagine that very small changes in the trajectory of a pitch will have an outsized influence compared to very small changes in the trajectory of a fly ball.

I look forward to reading more from your efforts.

This is really cool research. I do wonder, though, if the additional complexities that you alluded to in your introduction would make the effect of anisotropy negligible. For example, when accounting for real distributions of parameters such as launch angle, exit velocity, and backspin, does the effect of anisotropy have a significant impact? What are reasonable estimates for the degree of anisotropy? (I’d guess that bat-ball impact required to generate a home run would be a more likely source of anisotropy than manufacturing, but it would be interesting to find out.) Also, you mentioned torque due to gravity, but I think a much larger term would be torque due to air friction.

One other question I have is about the way you treat loss of backspin. It seems that you are assuming the backspin has decreased uniformly throughout the trajectory, but wouldn’t it vary with time as the angular velocity vector rotates? For that matter, do you recall the rate of precession?

Even if the effects on home run distance aren’t very large, this is interesting work.

Additional work should help clarify whether this is a negligible factor, I hope! That’s a very interesting idea–maybe I should make friends with one of the notorious ball hawks out there and measure the degree to which hard-hit foul balls are deformed.

It certainly would vary with time as the angular velocity vector rotates, especially if the “point mass” leading to precession is off the x-axis. I went back and forth between waiting to write something up that was more physical, but I ultimately decided that if this piece can serve as a toehold for others (including me) to start looking into this, it was worth the “risk” of getting “scooped.” (note the *heavy* air quotes on those two terms)

The precession rate is drawn from Equation 3 (the “spin-to-wobble ratio” B), and the terms neatly cancel so that it only depends on the ratio of m1:m2. B = 1+(5/2)(m2/m1) for this example; when m2 is 10% of m1 (which would be pretty large), B= 1+(5/2)(0.1) = 1.25. So for every 5 full spins, there are 4 full precessions. It gets a little funny at the edges, where there might be one precession for every spin at m2=0, but with a “magnitude” of precession relaxing to 0 after an initial perturbation due to the isotropy when m2=0. But I haven’t started (yet!) exploring the decay or amplification of precession in different physical cases.

But since I believe in openness with data analysis like this, I’m very happy to have people play around with the simple Python code I wrote following Brancazio (1987). And thanks for the kind words and suggestions!

Thanks. So then (perhaps I missed it) is the effective sidespin the average over a full cycle of precession?

As for the torque, I seem to recall seeing some limited data on moment coefficient versus spin rate. I built my own trajectory calculator, which I assume is similar to Dr. Nathan’s though I think I used a different source for lift coefficient. I currently treat angular velocity as constant throughout the trajectory, but I think it would be interesting to solve for it.

Side note: I’d be interested to see your Python code. I’ve been intending to learn it for several months, but never have the time. Perhaps this will give me motivation and a starting point!

Effective sidespin is sqrt(A) from Equation 1, which is a combination of both off-axis components (everything that contributes to omega that is NOT omega_x, backspin). That’s not the best way to treat that, necessarily, but it was much easier to compute than actually solving for omega_y and omega_z.

I just pushed to my github repository https://github.com/mcclellm/baseball-fg a beginner’s Python tutorial I wrote for the class I taught at MIT (Experimental Atmospheric Chemistry). You’ll need a Python distribution (I recommend anaconda with Python 2.7, downloadable at continuum.io) and the test.csv data file also pushed to the same repo. You should be able to download the files as a zip, rather than cloning the repo, if that’s easier.