This is not a question about the game itself but about the physics of hitting a bat with a ball.
My 11-year-old son asked me about this. Sometimes you hit the sweet spot and the ball goes over the fence. Sometimes you hit the ball, but you the bat handle gives you a sting.
I’ve thought about it, and in one case the energy is transferred to the ball, and in the other case I suppose it is somehow transferred to your hands. I’ve further considered that this might have something to so with the resonant frequency and vibrational nodes in the bat; if you hit the ball at an antinode, the bat will flex, acting like a spring and returning energy into the ball. If you hit the ball at a node, on the other hand, the bat will not flex, and the energy would be transferred to your hands.
However, this is all speculation based on whatever knowledge of physics I can scrape together, and no empirical knowledge of baseball bats.
Anybody out there who knows some real facts on this?
The term you’re looking for is center of percussion. If you hit the ball there, on the sweet spot, it won’t impart any bending moment into the bat - it won’t try to pivot around the ball, pushing your hands with it.
I believe what makes the sweet spot sweet is that the resulting vibrations in the bat cancel out where the hitter is holding it. I’m not actually 100% sure that you get maximum velocity from the sweet spot, however…from having played, balls close in on your hands don’t go very far (and can sting like hell) while balls near the end of the bat go farther (and might or might not sting).
The key is in the transfer of momentum. Or applying conservation of momentum. Momentum is mass times velocity. The ball can be treated as a point, but the bat has extent, and it is this extent that creates the sweet spot, and the sting. So, consider a simple bat - one made of a length of dressed wood. Regular in cross section along its length. If the bat is travelling in a straight line as it hits the ball it should be clear that if you hit the ball with the end of the bat you have a situation where the other end of the bat will continue to move, and the end that hits the ball will slow - transferring some momentum to the ball. The free end of the bat will actually start to move in an arc. You only transfer a portion of the bat’s momentum to the ball, the momentum of the now arcing end of the bat will be transferred into the batter through their hands. This may sting a little. Clearly the best place for the ball to hit the bat is at the mid-point - at the bat’s center of momentum. The bat will uniformly slow, and the largest transfer of momentum will occur to the ball.
Now a baseball bat is swung in an arc, and thus does not travel at uniform speed along its length. This means that the center of momentum (being mass times velocity) moves out from the midpoint to somewhere nearer the ouside of the swing. This is now the sweet spot. Hit the ball here and the same thing - the bat slows uniformly and maximum transfer of momentum to the ball, and minimum to the batter’s hands.
We then note that the inner end of the bat is not adding as much to the momentum, moving slower the mass does not contribute nearly as much as mass at the outer end. So a lighter and more efficient bat can be designed where the mass is removed from the inner end (making it easier to hold as well) and moved to the outer end. This moves the sweet spot even further from the mid point, where it is also rather usefully travelling faster, and you end up with the classic design.
Taking the idea to its logical extreme, a bat that is just a long and thin handle with all the mass in a small lump at the end would work very well. Golf clubs basically. Not very practicle for baseball.
IIRC from a Scientific American article a few decades back . . .
There are a few sweet spots on a swing e.g. maximum energy transfer spot is different than the no net torque spot. I believe there is 4 or 5 different spots depending on what result you want.
Modeling the “sweet spot” on a baseball bat was actually one of the problems in this year’s Mathematical Contest in Modeling. The winning solutions will be posted on the linked site soon, so you may find some interesting information there.
It’s not just a matter of momentum transfer to the ball, but that it occurs in a controlled vector; in other words, if you hit at a node which is not oscillating, the ball will fly normal to the axis of the bat, but if you hit at some point advanced or retarded from that point, (where the bat is flexing) it will either impart a linear vector to one side, or an angular momentum causing some of the energy to be imparted as parasitic spin rather than linear momentum. This is compounded by the spin imparted by the pitcher when he throws the ball.
In other words, some of the impulse imparted by the bat as it contacts the ball will be converted to angular momentum of the ball, detracting from the linear momentum that sends the ball flying to the outfield. This is especially true if the bat contacts the ball above or below the axis vertically, causing the ball to fly unpredictably. The flexing of the bat in the horizontal plane during impulse may impart some spin, but I couldn’t say how much without running an explicit dynamic finite element analysis. However, most of the energy loss is acoustical in nature, i.e. interference between the ball during the impulse interval and the natural mode shapes of the bat. The resonances or bending modes of the bat (the natural pseudosinusoidal shape the bat makes upon being struck) are essentially independent of where the bat is held or where the ball strikes it, but if the ball strikes at the inflection point of one of the first three mode shapes it will minimize these losses in the same way that standing at the pivot point of a teeter-totter makes it far easier to maintain balance.
While contributing little directly to the discussion let me reccommend ‘The Physics of Baseball’ by Dr. Robert Adair, official physicist to Major League Baseball, a least at one time.
If it’s cold, composite bats can really sting, and they often come with warnings about use below certain temps. Lots of April complaints about bats around Little League fields, because it is often cold at game time. But these new bats need to be warm to reduce stinging of the hands and to even work properly!
Also… this from Wikipedia and D. Russell…
*Ball players often experience a “sting”[4] in their hands caused by vibrations when the ball does not come in contact with the sweet spot of the bat. The frequency of these vibrations throughout the bat is related to the bending stiffness. Daniel A. Russell of Kettering University has shown that standard aluminum bats have a high bending stiffness that produces vibrational frequencies in the range where most hands are sensitive; therefore, causing a sting. He also has shown that composite materials can lower this bending stiffness without compromising other advantages.[3]
Including the lower bending stiffness, composite baseball bats have a higher damping rate. The damping rate corresponds to how quickly the material lessens the vibrations it is experiencing. Russell also states that composite bats have a damping rate anywhere from 2 to 10 times more than standard aluminum bat.[3] Many ball players therefore refer to composite bats as more forgiving because if they do not make contact with the ball on the sweet spot, they will not feel the vibrations (sting) from the miss hit.[4]*
I believe the recommended MIN temp is 50f on many bats, but these bats get finicky in the low- to mid- 50’s. I don’t feel they are really reliable below 60. That’s half anecdotal (use above 60) and half factual (since bat makers recommend they not be used below 50).
There is some misinformation in this thread dressed up with fancy technical terms.
The sweet spot, however it is defined, has to coincide with the center of percussion of the bat. A ball hit at a place different from the CoP won’t feel solid, no matter what else is going on. The CoP depends primarily on the mass distribution of the bat, but also is affected by the angle the ball strikes at and the way the bat is held. It has nothing to do with the flex-body vibrational modes of the bat.
I don’t believe that the mode shapes or frequencies of the bat can have much to do with the quality of the hit. You’ve seen the high-speed photos of a ball striking a bat, in which the ball is squashed into the shape of an M&M while the bat stays rigid. The vibrations of the bat are too rapid and too small in amplitude to affect the energy transfer to the ball. It is like the difference between dropping a water balloon on a diving board versus dropping it on a concrete floor: not much. The same probably applies to the feel in the hands - the hands and body are so soft compared to the bat that the bat essentially vibrates as a free body.
The feel in the hands is mainly an effect of the damping rather than the mode shapes. When the bat hits the ball, essentially all the bat’s mode shapes are excited except for ones having an antinode at the location of the hit. Damping, especially in the hands themselves, is strongly affected by temperature. Cold hands will transmit the shock of impact much more strongly to the nerves than warm hands. It is possible there might be some small improvement in feel if a bat had an antinode of its fundamental mode coincident with the CoP, but again, these points are dependent not only on the bat but on how it is gripped and swung. This probably accounts for the fact that different batters like different bats.
I think there is enough “spread” in the locations of these various points that any batter can have a solid-feeling hit with any reasonably sized bat. A bad-feeling, or stinging hit, will be more a function of the ball hitting above or below the centerline of the bat and making the bat twist in the batter’s hands, or hitting on the handle, etc.