Gravity (Not the Movie)
One of the great things about baseball is that it’s played in so many different ballparks, each with their own quirks and different dimensions. Much has been written about how different ballparks affect the game: the different distances of the fences, the size of the foul area, the altitude, and even what days the locals hang their laundry outside. These various park factors affect more than just the results of batted balls. They also influence the number of walks and strike outs. I want to take a look at a more esoteric park factor that has to my knowledge been ignored up to this point. Gravity. In high school you were probably told that gravity on Earth was a constant 32 ft/s^{2} (or 9.8 m/s^{2}), which was actually a white lie. To be exact, the Earth’s gravity is 32.1740 ft/s^{2} (or 9.80665 m/s^{2}), but more importantly gravity is not constant.
There are several reasons the Earth’s gravity as we experience it is not constant. First, the Earth is not a perfectly uniform sphere. When mathematically approximating gravity we make the assumption that the Earth is a perfectly uniform sphere. But, since the Earth is not perfectly round and uniform, this assumption leads to a small error in the approximations and does not account for gravitational variations in different locations.
Second, gravity is dependent on your distance from the center of the Earth. Gravity is inversely proportional to the square of the distance between two objects, say between you and the Earth. The further away from the Earth you are, the weaker gravity is, g = g_{0} (r_{e }/(r_{e}+h))^{2 } where r_{e} is the radius of the Earth, g_{0} is gravity at sea level, and h is how high you are above sea level. For example, at Coors Field g=g_{0}(20,925,524.9/(20,925,524.9+5,219.82))^{2 } this equation tells us that gravity is 32.157913 ft/s^{2} at Coors Field, or 0.05% less than gravity at sea level (32.1740 ft/s^{2}).
The third reason why the Earth’s gravity as we experience it is not constant is related to the centrifugal forces caused by the Earth spinning. The fact that the Earth is rotating does not actually change gravity (well this is a lie according to relativity there will be some rotational frame dragging but this effect is extremely hard to detect and surely won’t have a measurable effect on baseball). Centrifugal forces appose gravity and make items feel lighter. These forces are strongest near the equator (where you are the furthest from the Earth’s axis and therefore moving the fastest) and weakest near the poles (where you are closer to the Earth’s axis and rotating more slowly). An easy way to remember this is gravity will be weaker the closer you are to the equator.
Let’s take a break from all this math for a bit. Here is the juicy part, the table below shows the gravity at all the different major league ballparks and the percent increase or decrease in gravity compared to the average gravity at all the ballparks (this is based on EGM2008, made easily available thorough wolfram alpha). Negative percentages indicate a decrease in gravity, while positive percentages indicate an increase in gravity.
Team | g (ft/s2) | % change |
Miami Marlins |
32.11348 |
-0.126% |
Tampa Bay Rays |
32.11936 |
-0.108% |
Houston Astros |
32.12558 |
-0.088% |
Texas Rangers |
32.13392 |
-0.062% |
Arizona Diamondbacks |
32.13474 |
-0.060% |
San Diego Padres |
32.13553 |
-0.057% |
Atlanta Braves |
32.13608 |
-0.056% |
Los Angeles Dodgers |
32.13887 |
-0.047% |
Angeles |
32.14466 |
-0.029% |
Colorado Rockies |
32.14466 |
-0.029% |
Oakland Athletics |
32.15333 |
-0.002% |
Giants |
32.15341 |
-0.002% |
Average |
32.15395 |
0.000% |
St. Louis Cardinals |
32.15517 |
0.004% |
Kansas City Royals |
32.15538 |
0.004% |
Cincinnati Reds |
32.15677 |
0.009% |
Washington Nationals |
32.15742 |
0.011% |
Baltimore Orioles |
32.15886 |
0.015% |
Pittsburgh Pirates |
32.16099 |
0.022% |
Philadelphia Phillies |
32.16119 |
0.023% |
New York Mets |
32.16435 |
0.032% |
New York Yankees |
32.16442 |
0.033% |
Cleveland Indians |
32.16511 |
0.035% |
Chicago White Sox |
32.16655 |
0.039% |
Chicago Cubs |
32.16697 |
0.041% |
Detroit Tigers |
32.1684 |
0.045% |
Boston Red Sox |
32.17023 |
0.051% |
Milwaukee Brewers |
32.17096 |
0.053% |
Toronto Blue Jays |
32.1744 |
0.064% |
Minnesota Twins |
32.17764 |
0.074% |
Seattle Mariners |
32.18997 |
0.112% |
(If you are paying close attention: 1) you might have noticed the average gravity in the table is lower than our conventional constant for gravity, 32.1740 ft/s^{2}. The average gravity in the table above is the average gravity at major-league ballparks only, not the average gravity of all points around the world. 2) The table value for gravity at Coors Field does not exactly match what we calculated earlier. This is because the measure we calculated earlier did not account for centrifugal force or the effects of a non-uniform Earth. The gravity for Coors Field in this table allows for those factors.
The difference between the two most extreme ballparks is 0.07649 ft/s^{2}. Alone this number seems small and is hard to conceptualize. I’ve gone ahead and explored a few different baseball scenarios to illustrate its effects.
So, what does 0.07649 ft/s^{2 }really mean for the game of baseball?
1. Players are measurably lighter at lower gravity ballparks.
CC Sabathia feels just a little lighter while pitching in Miami than when in Seattle, a whole whopping 0.69 lbs lighter! (Perhaps this is why when so many players travel to Florida for Spring Training they report feeling in the best shape of their life…)
2. An outfielder will have slightly longer to catch a fly ball in a lower gravity ballpark.
A fly ball with 4.5 second hang time at an average park would stay in the air 5.7 milliseconds longer in Miami, and in Seattle it would be in the air for 5 fewer milliseconds. That almost 11 millisecond difference in hang time between Miami and Seattle would mean that a running out fielder might cover 2 more inches in Miami, not enough to make any reel difference but interesting nonetheless.
3. Pitches will sink less in a lower gravity ballpark.
Pitches will sink less in Miami than they will in Seattle, but how much less? On a 65 mph slow curve it takes the ball about 0.650 seconds to reach the plate. This ball will drop 0.2 inches lower in Seattle vs. Miami. An average pitch taking 0.45 seconds to reach home plate, will only drop an addition 0.09 inches in Seattle vs. Miami. For comparison the diameter of a baseball bat is 2.6 inches or less. A 0.2 inch difference is 1/13 the diameter of a baseball bat, which is too small of a difference to turn a hit into a swing and miss.
4. Home runs will travel farther in a lower gravity ballpark.
When it comes to home runs one would think differences in gravity would start to play a bigger role. Because home runs are in the air longer, gravity is bound to have a greater effect on them than it does on pitched balls. The hang time of a home run is usually a full order of magnitude longer than that of a pitch. Assuming identical weather conditions, a baseball hit 120 MPH at a 26^{o} angle would travel 13 inches (THAT’S MORE THAN 1 FOOT!) farther in Miami than it would in Seattle. That could make a difference, not in the actual score, but in what seat in the bleachers the ball would land. Although a foot is the largest difference we have talked about so far, practically it doesn’t really matter much for a no-doubt home run that’s traveling over 460 feet.
5. Just for Fun…
On the surface of the Earth if we wanted to look for extremes we would see the highest gravity at the South Pole, which would be 32.26174 ft/s^{2} or 0.335% higher than the average gravity at a major league ball park (this and a few other factors would lead me to believe that playing in the South Pole would really suppress home runs). The other extreme would probably be in Quito, the capital of Ecuador (there is actually a volcano in Ecuador with slightly lower gravity but let’s look at one plausible hypothetical) where gravity is 32.04248 ft/s^{2} or -0.347% below average. In Quito Sabathia would be 1lb lighter than he would at an average ball park and 1.3 pounds lighter than he would in Seattle. That same hypothetical 120 mph home run would go 0.9 feet farther in Quito than it would a an average ball park, and 1.3 feet shorter at the South Pole. This is of course completely hypothetical because we are assuming all other conditions are the same at these two ball parks such as air density and temperature, and this definitely not the case.
Thanks to
National Geospatial-Intelligence Agency for publicly releasing the Earth Gravitational Model EGM2008
Alan Nathan for providing the trajectory calculator tool, which I used to calculate difference in batted ball distances, the calculator can be found on his website http://baseball.physics.illinois.edu/trajectory-calculator.html
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loved this article
Seconded. Great work.
Well, this is amazing. Fantastic work.
Seconded.
Thirded.
Fourthed, or fourtheded.
A very amusing article.
To follow up, while altitude and latitude affect the measured value of the Earth’s gravitational constant at the surface, density of mass below the surface also has a significant affect. Geologist have used variation in “g” to locate oil and certain metals. This means it is not clear if we should put much “weight” in the Earth’s Gravitational Model 2008 when comparing cities of comparable latitude and altitude. A more detailed map of “g” has been recently generated. See: http://phys.org/news/2013-09-gravity-variations-bigger-previously-thought.html
Finally, allow me to correct one notion: centrifugal “force” is not a real force, but rather an effect. We notice it because measurements of “g” are not made in an inertial reference frame.
One more thought – with a lower g value, there is less force of the ground pushing directly up against the player’s foot. This could mean players will not be able accelerate as well – they will have to lean over more to generate the same amount of tangential acceleration. This would have consequences on tracking down flyballs or stealing bags. (Runners might “overshoot” the bag due to less friction.)