WASHINGTON -- With the crack of a bat, the ball disappears in a blur over the third-baseman's head. "It's going ... going ... gone!" the radio announcer says. Mark McGwire has just hit a record-breaking home run. Beyond the obvious drama of a home run, there is a world of physics surrounding the collision of a smooth, rounded stick and a small sphere of tightly wrapped yarn. Titanic forces are involved forces that can deform and compress a baseball to half its original diameter or snap a sturdy bat as though it were kindling.
Those forces -- and their potential effects -- were the subject of an invited talk by University of Illinois experimental nuclear physicist and baseball enthusiast
Alan M. Nathan at this year's meeting of the American Association for the Advancement of Science. Nathan presented his talk during the AAAS symposium on the science of baseball.
"As a physicist, there are few things more satisfying than being able to figure out and explain the various phenomena we encounter in our everyday lives," Nathan said. "The game of baseball is rich in such phenomena -- things like the break of a curve ball, the flutter of a knuckleball or the flight of a McGwire home run. The goal in studying the physics of baseball is to better understand these and other aspects of the game."
The collision between bat and ball is surprisingly violent, with forces reaching in the thousands of pounds and lasting less than one-thousandth of a second. In an instant, the ball's forward motion is reversed and it is sent flying along a new trajectory. For a well-hit ball -- whether it's a home run or a line drive -- the speed of the ball leaving the bat is crucial.
"Dynamically, you can think of the ball as a simple spring," Nathan said. "The force between the bat and ball compresses the spring, converting the kinetic energy of motion into potential energy. At the point of maximum compression, the ball momentarily comes to rest. Then, the spring expands, converting potential energy back into kinetic energy."
Because the collision is highly inelastic, not all of the kinetic energy is restored, however. A significant fraction of the energy is "lost" as frictional heat. Energy is lost to the ball in other ways, as well.
"Newton's law of action-reaction tells us that whenever the bat exerts a force on the ball, the ball exerts an equal and opposite force on the bat," Nathan said. "So it is reasonable to ask what effect this reaction force has on the bat."
When the bat strikes the ball, the bat will naturally recoil. That recoil force is energy that is also lost as far as the ball is concerned. But the heavier the bat, the less the recoil.
"All other things being equal, you want as heavy a bat as possible," Nathan said. "But of course all things are not equal, because you also have to swing the bat. Therefore, a compromise must be reached for each player."
When the ball hits the bat at its center of mass, the bat will simply recoil. Collisions occurring elsewhere will cause the bat to rotate about its center of mass, conserving angular momentum. Both recoil and rotation tend to reduce the energy that goes into the rebounding ball, lowering its exit speed.
"Treating the bat as a perfectly rigid body that can only do two things -- recoil and rotate -- is called rigid-body kinematics," Nathan said. "But a real bat does more than that. It can also vibrate, as anyone who has mishit a ball and felt the resulting sting, or seen a broken bat, has witnessed."
Whenever the impact is not on the "sweet spot" -- a vibrational node of the bat and the best place to hit the ball -- the collision creates vibrations that propagate back and forth along the bat, much like the vibrations on a guitar string. In general, any energy that goes into exciting vibrations is energy that did not go into propelling the ball from the bat.
While the vibration-free zones of good hitting on most wooden bats are similar, those on aluminum bats are different. Aluminum is harder to bend, making an aluminum bat about twice as stiff as its wooden counterpart.
Aside from its greater strength, an aluminum bat's hollow cylindrical shape is more rigid than a solid wooden bat containing the same mass. And because the mass is more uniformly distributed along an aluminum bat, its moment of inertia is increased, which induces less rotation. An important consequence is that the sweet spot is larger for an aluminum bat, allowing more room for error.
"A mishit ball will travel farther off an aluminum bat, with less vibration and stinging of the hands," Nathan said. "In particular, an aluminum bat is far more forgiving for an inside pitch. But, aluminum bats are not allowed in the major leagues."
The collision between bat and ball is so short -- typically about half a millisecond -- that the ball only "knows" about what is happening at the working end of the bat at the moment of impact. In fact, on the time scale of the collision, the ball does not even know that the far end of the bat exists.
While the bat does deform slightly under the impact -- like a stiff spring -- it takes time for that pulse of energy to travel down the length of the bat and back up again, Nathan said. By the time the pulse has returned to the site of impact, the ball is long gone. Any clamping action of the hands will have no effect.
"So, it's a myth that some batters can 'muscle' the ball," Nathan said. "It may seem counterintuitive, but if the batter could let go of the bat just prior to hitting the ball, there would be no noticeable effect. The ball would respond in exactly the same way."
Whether the collision is as simple as a bat hitting a ball, or more complex like a high-energy subatomic particle bombarding a nucleus, the same physical principles apply, Nathan said. "Knowing what those principles are and how they operate are important keys to understanding what is happening in this fascinating world around us."
CONTACTS: Alan Nathan can be reached during the conference, Thursday (Feb. 17) through Sunday (Feb. 20), at the Normandy Inn, (202) 483-1350. After the conference, he can be reached at the UI at (217) 333-0965. Nathan's e-mail is a-nathan@uiuc.edu and his Web site is www.npl.uiuc.edu/~a-nathan, where he has posted his AAAS talk. Jim Kloeppel, UI physical sciences editor, can be reached during the conference at the conference press room, (202) 319-1757, or at the Marriott Wardman Park, (202) 328-2000; after the conference, he can be reached at (217) 244-1073.