The Physics of the Gyro Pitch

We need to have a heart-to-heart about the flight of pitches and the spin of the ball. You already know the spin on the ball induces the air around the ball to exert a Magnus force on the ball. That is, a ball with backspin feels an upward Magnus force keeping the ball from falling as fast as gravity requires, while topspin does the opposite. Sidespin results in a pitch veering left or right depending on which way the ball spins.

Here’s the problem with that thinking. The topspin, backspin, and sidespin are all measured with respected to the velocity vector (direction of motion of the ball). The trouble is, the direction of motion of a pitch is changing on its journey from the mound to the plate. So, the direction of motion is changing.

So, the question is, does the spin stay oriented with respect to the motion of the ball as it drops or veers to the side or does the spin stay fixed with respect to the field? So that you really understand what I’m getting at here, let’s think about gyrospin instead of topspin or sidespin.

Gyrospin is the rotation of the ball about the direction it is moving. Below is a sketch of a ball moving toward you, spinning around as indicted by the blue arrow. Gyrospin causes no change in motion of the ball. That is, there is no Magnus force due to gyrospin. So, imagine a pitch thrown only with gyrospin. The previous question could be rephrased, “As the ball drops on the way to the plate, does all the gyrospin remain gyrospin?” The answer is no. Some of the gyrospin becomes sidespin. Amazing!

I can’t think of an easier way to understand this than to spend some quality time explaining how physicists think about spin. Spin, like velocity, is a vector. That is to say, spin has a direction in space. Look at the sketch above of a ball coming at you with gyrospin. You might think it is impossible to assign a single direction in space to the spin because the ball is going around in circles.

It turns out you actually can, if you follow the “Right-Hand Rule.” Raise your right hand and repeat after me. I will bend my fingers around the ball in the direction of its rotation, and my thumb will point in the direction of the spin vector. Perhaps this needs more explanation.

Below is a photo of the hand position you’ll need to master the rule. You don’t actually need the ball in your hand; you can just imagine it. Take your right hand, stick out your thumb, and curl your fingers as if they are wrapped around a ball.

Notice you can keep your bent fingers and your thumb straight in the same relative position even if you rotate your wrist. To find the spin direction for the gyrospin above, rotate your wrist as needed until your fingers wrap around the image of the gyroball in such a way that your fingers point along the blue arrow that represents the rotation. Your thumb will now be pointing directly out of this page–toward you. According to physics, the direction of the spin is…out of the page. Crazy, huh? Crazy, but useful.

By now, anyone in the room with you while you’re reading this is wondering what the heck you’re doing with your right hand. Forget that noise, let’s see if you can follow the Right-Hand Rule for top and back spin. Below is a sketch of a ball with topspin on the left and a second sketch with backspin on the right. Are you able to convince yourself that the resulting green spin vectors are consistent with the Right-Hand Rule?

Just to solidify your understanding of rotation and spin vectors, below is the situation for sidespin. Have you mastered the Right-Hand Rule?

This is pretty confusing, so let’s summarize,

I know this seems pretty frickin’ goofy, but now we can understand the behavior of the gyroball. That is, how gyrospin can become sidespin during the flight of a pitch. Below is an example of an 86 mph slider thrown by Dylan Bundy on April 5, 2017. Look at the close-up of the pitch between 53s and 60s on the video. Examine the “dot” on the slider, from which you can see that nearly all the rotation of the ball is gyrospin.

Notice the very late break on the ball to the pitcher’s left (catcher’s right). As an expert in spin directions, you know this break to the catcher’s right is the result of sidespin with an upward spin vector. Our challenge now is to explain where the sidespin came from given that ball was released with only gyrospin.

If we make the reasonable but incorrect assumption that the initial gyrospin stays oriented relative to the velocity of the ball, then the situation would look like the sketch below as the ball travels. The velocity vector (red) and the spin vector (green) are shown. In this incorrect case, the spin vector maintains its alignment with the velocity.

The pitch above never develops any sidespin during the flight of the pitch. So, it is hard to explain the late break. Compare that with the situation below where the initial spin of the ball stays oriented with respect to the field, not to the velocity of the ball.

As the velocity vector starts to tip downward, the spin vector stubbornly remains pointed straight toward the backstop. That is, the gyrospin is staying constant with respect to the field not the velocity of the ball. If you need a fancy physics term to explain this behavior it is the Law of Conservation of Angular Momentum. This law explains why a spinning top stays upright and stable even though the top just tips over when it is not spinning.

Above is a close-up of the last image of the gyroball getting near the plate. The original gyrospin is now labeled “no longer just gyrospin” because this spin has some portion along the velocity, in other words true “gyrospin,” and a portion that is perpendicular to the velocity that is “sidespin.” Get out your right hand and figure out which way the ball will bend due to the sidespin. Hopefully, you agree the ball will veer toward the catcher’s right, in agreement with Dylan Bundy’s pitch.

You might notice the amount of sidespin that develops depends upon how much the ball is dropping. The slower the speed of the pitch, the more the ball drops on its way to the plate, the more velocity vector points downward, and the larger the sidespin becomes. The sideways Magnus force depends upon both the speed and the sidespin, so there is likely an optimum speed for a gyroball.

To get a sense of the size of the effect when the gyrospin contributes some sidespin, I looked at a pitch thrown at the same initial speed as Bundy’s, about 85 mph. I compared the motion of a ball thrown with no spin to a ball thrown with 1500 rpm of gyrospin. The horizontal position at home plate varied by half an inch. No, not a lot, but enough to convert a homer into a pop fly.

I know a lot of this was difficult to follow; after all, I’ve tried teaching this to undergrads for many decades. I’m just too limited to figure out a better way to explain spin. However, I did give it my best, so if you didn’t like this article, you can just use the Right-Hand Rule to stick your thumb…[editorial deletion].

References & Resources

• This article is based upon a wonderful conversation at SaberSeminar 2017 that included Kelvin Yeo and Seth Daniels of Rapsodo Baseball as well as Alan Nathan of the University of Illinois.
• David Kagan, The Hardball Times, “The Physics of the Force on Spinning Baseballs”
• Sal Khan, Khan Academy, “Angular momentum”