# Sweet Spot on a baseball bat

ChinleShale
Hitting a baseball in the sweet spot on a baseball bat gives the best rebound and does not twist the handle of the bat in the hitter's hand. How is this spot and its behavior described in Mechanics?

A similar question: The bumper on a pool table is at a height so that the pool ball does not skid when it hits it. How is this described in Mechanics?

## Answers and Replies

Staff Emeritus
It seems that others have studied the baseball bat sweet spot also. Here's a link
https://www.acs.psu.edu/drussell/bats/sweetspot.html

Fig. 6 Comparison of Batted-Ball Speed and collision efficiency (eA) as a function of position for three slow-pitch softball bats: (red) a good single-walled bat, (blue)a high performance double-walled bat, and (black) a very high performance composite bat.

Edit: Here's a 2nd paper. https://www.acs.psu.edu/drussell/bats/bend-sweet.html

russ_watters and berkeman
A similar question: The bumper on a pool table is at a height so that the pool ball does not skid when it hits it. How is this described in Mechanics?

Let the ball have radius ##r## and initial angular speed ##\omega##. It impacts the table edge at a height ##h## vertically above its centre of mass. Let the ball have mass ##m## and moment of inertia ##K## about its centre.

Suppose that the impulse imparted to the ball is ##\boldsymbol{I} = - I \boldsymbol{e}_x##; thus if the velocity of the centre of mass after the collision is ##\boldsymbol{\mathcal{V}}##, you may write ##m \boldsymbol{\mathcal{V}} = mr\omega \boldsymbol{e}_x - I \boldsymbol{e}_x##, so that ##\mathcal{V}_x = r \omega - I / m##.

The angular impulse imparted to the ball about its centre is ##\boldsymbol{J} = Ih \boldsymbol{e}_z##; thus if the angular velocity after the collision is ##\boldsymbol{\mathcal{\Omega}}##, you may write ##K \boldsymbol{\mathcal{\Omega}} = -K \omega \boldsymbol{e}_z + Ih \boldsymbol{e}_z##, so that ##\Omega_z = -\omega + Ih / K##.

By the rolling condition, you must constrain ##\boldsymbol{\mathcal{V}} + \boldsymbol{\mathcal{\Omega}} \times (-r \boldsymbol{e}_y) = \boldsymbol{0}##, i.e. ##\mathcal{V}_x + r\Omega_z = 0##. This means\begin{align*} r\omega - I/m + r(-\omega + Ih/K) = 0 &\implies -I/m + rIh/K = 0 \\ \\ &\implies h = K / mr \end{align*}

russ_watters
Mentor
Summary:: How can the sweet spot on a baseball bat be described in Mechanics?

Hitting a baseball in the sweet spot on a baseball bat gives the best rebound and does not twist the handle of the bat in the hitter's hand. How is this spot and its behavior described in Mechanics?
Can you please provide a reference for that statement? As a former baseball player (HS level max), I have issues with what you stated. Thanks.

Staff Emeritus
Science Advisor
Education Advisor
The definitive book on this sort of thing is Bob Adair's The Physics of Baseball.

anorlunda and berkeman
Here's a little simplified model I cooked up!

A thin rod of mass ##M## and length ##\mathscr{L}## is pivoted at a point ##\mathcal{O}##. At the lowest point in its swing, it collides with a point of mass ##m## a distance ##\xi## below the point ##\mathcal{O}##, which was initially moving at ##\boldsymbol{\mathcal{u}} = - \lambda_1 \boldsymbol{e}_x## and after the collision was moving at ##\boldsymbol{\mathcal{v}} = \lambda_2 \boldsymbol{e}_x##.

In other words, the ball exerts an impulse ##\boldsymbol{I} = -m(\lambda_1 + \lambda_2) \boldsymbol{e}_x## on the rod. This constitutes an angular impulse ##\boldsymbol{J} = -m \xi (\lambda_1 + \lambda_2) \boldsymbol{e}_z## about the point ##\mathcal{O}##.

Suppose that the angular velocities of the rod before and after the impact are ##\boldsymbol{\omega}_1 = \omega_1 \boldsymbol{e}_z## and ##\boldsymbol{\omega}_2 = \omega_2 \boldsymbol{e}_z## respectively. You can write two equations:\begin{align*} \mathrm{Angular\, \, momentum:} \quad &\frac{1}{3}M \mathscr{L}^2 \omega_1 - m\xi (\lambda_1 + \lambda_2) = \frac{1}{3}M \mathscr{L}^2 \omega_2 \\ \mathrm{Momentum:} \quad &\frac{1}{2} M \mathscr{L} \omega_1 + B - m(\lambda_1 + \lambda_2) = \frac{1}{2} M \mathscr{L} \omega_2 \end{align*}where ##B## is the as of yet undetermined horizontal impulse exerted by the pivot on the rod. This can be easily solved for ##B## by eliminating ##\omega_2##, as so$$\frac{\frac{1}{3} m \mathscr{L}^2 \omega_1 - m \xi (\lambda_1 + \lambda_2)}{\frac{1}{2} m \mathscr{L} \omega_1 + B - m(\lambda_1 + \lambda_2)} = \frac{2\mathscr{L}}{3} \implies B = m(\lambda_1 + \lambda_2) \left[1 - \frac{3\xi}{2\mathscr{L}} \right]$$Qualitatively, the "sweet" spot will occur when the horizontal impulse ##B## exerted by the pivot on the rod is zero (this would correspond to your hands feeling no horizontal impulse from the bat, after hitting the ball). This occurs when ##2\xi = 3\mathscr{L}##, or$$\xi = \frac{2\mathscr{L}}{3}$$

Mentor
Summary:: How can the sweet spot on a baseball bat be described in Mechanics?
Google for “center of percussion”. The Wikipedia article is pretty good.

Doc Al
ChinleShale
ChinleShale
So you have multiple references?
You are confusing me and don't know what you are getting at. The first time I saw these problems was in an old engineering book from the 1920's. That was 40 years ago. Are you asking me to spend a few years trying to find another copy of that book? If you want to go on a jihad why not just close the thread instead of being obnoxious?

I could have just reworded the question with phrases like "I wonder if there is a sweet spot on a baseball bat ... " .

I thought this was an interesting question and remembered it had fascinated me at the time. Pardon me for trying to start an interesting conversation. I could have just looked this up and not bothered with posting here or have figured it out myself. Why don't you look it up and get back to us? Or are you just trying to shut this thread down?

BTW: I have several posts in the Mathematics Forums. There are no references. Please read them and tell me if you "have a problem" with them.

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Staff Emeritus
I thought this was an interesting question and remembered it had fascinated me at the time. Pardon me for trying to start an interesting conversation. I could have just looked this up and not bothered with posting here or figured it out myself. . Why don't you look it up and get back to us?

Here at PF, we try to help people to understand mainstream science. Helping, does not mean doing the research for them, or doing the homework for them. Teachers often consider that unhelpful. The OP should do the research first, then come here to PF for help on the points where he is stuck. The OP can post the link to what he has been reading that is not clear. Or the OP may ask for links to places where he could read more.

ChinleShale
Here at PF, we try to help people to understand mainstream science. Helping, does not mean doing the research for them, or doing the homework for them. Teachers often consider that unhelpful. The OP should do the research first, then come here to PF for help on the points where he is stuck. The OP can post the link to what he has been reading that is not clear. Or the OP may ask for links to places where he could read more.
That is baloney. But fine. Close the thread. Please close all of my posts that have no references e.g. in the Mathematics Forums. For instance, I just posted some exercises in the Differential Geometry Forum that used the definition of a covariant derivative without a reference. It is not my thread but it is called "Christoffel Symbols" or something like that. I also just posted a question about booster shots for the Covid-19 vaccines and did not reference anything.

Or you could cancel me.

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Motore
Mentor
You are confusing me and don't know what you are getting at.
No, not your fault. I think I was the one who was confused. You initially gave a definition of what you thought the sweet spot was, and the reference lists many different versions:
Trying to locate the exact sweetspot on a baseball or softball bat is not as simple a task as it might seem, because there are a multitude of definitions of the sweetspot:
I was just having trouble with the reference since it didn't say there was one definition like the one you started with.

Google for “center of percussion”. The Wikipedia article is pretty good.
It seems like this is the key to answering your question.

ChinleShale
No, not your fault. I think I was the one who was confused. You initially gave a definition of what you thought the sweet spot was, and the reference lists many different versions:

I was just having trouble with the reference since it didn't say there was one definition like the one you started with.

It seems like this is the key to answering your question.

I am really objecting here to censorship. I understand that it is a fine line but people come here to learn and to start dialogues that lead to understanding. It seems to me that a Mentor or whatever you are should consider that first.

By the way answering questions with links e.g. to Wikipedia pages does a disservice to the process of learning through dialogue. There is little point coming to these Forums if that is what it provides.

Unless an OP is wack, which some are, a nurturing approach is what instructs. Many of the answers that I have seen so far in my short time here are perfunctory.

Mentor
I am really objecting here to censorship.
Sorry, what censorship? I'm not understanding your reply.
By the way answering questions with links e.g. to Wikipedia pages does a disservice to the process of learning through dialogue. There is little point coming to these Forums if that is what it provides.
You asked a question, and apparently the Wikipedia article is a good reference for helping to answer your question. Again, I don't see the issue.

Motore
ChinleShale
I just noticed a post on the Differential Geometry Forum asking about the topological meaning of Chern numbers. The question seems a little vague. Still it could lead to interesting explanations and repartee. I am willing to start things off but perhaps you should check it first and demand references maybe ask for Chern's original papers.

Mentor
I just noticed a post on the Differential Geometry Forum asking about the topological meaning of Chern numbers. The question seems a little vague. Still it could lead to interesting explanations and repartee. I am willing to start things off but perhaps you should check it first and demand references maybe ask for Chern's original papers.
Nah, I'm not an expert in mathematics. I'm sure others will ask in the thread if it's appropriate. Certainly when it comes to Engineering subjects (like this thread), I can usually judge if references would help. Like in your OP here when you asserted that the sweet spot was key to keeping the bat from rotating at impact -- I'd never heard (or felt) that before, so that's when a reference is a good idea. And even the reference you linked to did not list that as a property of the sweet spot:

• the location which produces least vibrational sensation (sting) in the batter's hands
• the location which produces maximum batted ball speed
• the location where maximum energy is transferred to the ball
• the location where coefficient of restitution is maximum
• the center of percussion
• the node of the fundamental vibrational mode
• the region between nodes of the first two vibrational modes
• the region between center of percussion and node of first vibrational mode

Science Advisor
Gold Member
Hitting a baseball in the sweet spot on a baseball bat gives the best rebound and does not twist the handle of the bat in the hitter's hand. How is this spot and its behavior described in Mechanics?
I have seen a similar (perhaps a more elementary) version of this question which discusses the best place to put a doorstop, with respect to the line of the hinges and the width of the door, to give least shock to the hinge screws.

Mentor
Unless an OP is wack, which some are, a nurturing approach is what instructs. Many of the answers that I have seen so far in my short time here are perfunctory.
Just the opposite. If you understand the definition of perfunctory, that implies that the responders are trying to minimize their time/effort in responding. The opposite is usually the case here at PF. Responders generally are willing to spend lots of (volunteer/unpaid) time helping sincere OPs, but subscribe to the PF didactic philosophy of helping OPs to "learn how to learn". Spoon feeding responses to lazy OPs does little to actually help them; I hope you agree with that part of the PF philosophy.

So when we ask for reference links, we are both asking for help understanding your level of effort in answering your own question, and trying to gauge your level of understanding of the subject. Time after time in threads that I've been involved in, the request for reference links has helped to resolve the thread quickly.

Like in this case, you could have found the correct answer to your question with a 5 minute web search -- you would have quickly found the concept of "center of percussion" and been happy to have found it on your own. Barring that, you could have started the thread saying "I've searched on these terms..." with no luck so far, and that would have helped us point you to better search terms. But when you disrespect us for posting links to the easy solution to your question, that's not a good look. Have a nice day.

Motore, Klystron, Vanadium 50 and 1 other person
Homework Helper
Gold Member
If the OP is still interested, the bat pivots about the point where the hands are, (approximately). The sweet spot on the bat is where the angular momentum of the bat calculated about this spot is zero. Thereby, the bat can undergo a change in rotation speed from the collision with the ball, with no change in angular momentum,(since the angular momentum of the bat is zero, regardless of the speed of the bat), so that no force (torque) is felt or supplied by the hands when this happens. (The ball also exerts zero torque (calculated from the sweet spot), because the length of the moment arm is zero).

berkeman
Science Advisor
Homework Helper
2022 Award
If the OP is still interested, the bat pivots about the point where the hands are, (approximately).
This was the solution I remember (surprisingly!) long ago from Sears and Zemansky (ancient edition). Looking at the reference supplied by the OP
https://www.acs.psu.edu/drussell/bats/sweetspot.html,
I see the far more reasonable result that the when the ball is contacted the pivot point is actually 2.5 inches off the knob end of the bat. I believe this a much better estimate. Of course the mechanics of finding the Center of Percussion are similar using that pivot.
Perhaps the billiard ball problem is better characterized and the complete solution supplied by @etotheipi shows the rail needs to be 7r/5 above the table if it supplies a purely horizontal impulse.

Charles Link
If the OP is still interested, the bat pivots about the point where the hands are, (approximately). The sweet spot on the bat is where the angular momentum of the bat calculated about this spot is zero. Thereby, the bat can undergo a change in rotation speed from the collision with the ball, with no change in angular momentum,(since the angular momentum of the bat is zero, regardless of the speed of the bat), so that no force (torque) is felt or supplied by the hands when this happens. (The ball also exerts zero torque (calculated from the sweet spot), because the length of the moment arm is zero).

I don't understand the argument. Where exactly can you find a point on the rotating rod about which the total angular momentum of the body is zero?

Further; even if such a point existed it would define a non-inertial frame, so during the collision the centre of mass would experience a fictitous Dirac delta force. Thus an opposing non-zero real torque would need to be provided at the pivot in order to maintain zero angular momentum.

Maybe I misunderstood?

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Science Advisor
Gold Member
so that no force (torque) is felt or supplied by the hands when this happens.
For a bat, it's fundamentally necessary that the batter transfers Energy to the ball during impact so a lot of the above would need additional factors for a meaningful model. There is sure to me more to this than just 'throwing the bat at the ball. If there is no reaction force when the exact sweet spot is used then no Energy will be transferred from the batter during impact. Sports Physics gets more and more complicated, the more you study a problem.

I read, a long time ago and I think it was New Scientist, that the ideal bat for an individual will vary greatly. They said that Babe Ruth used a very heavy bat (he was strong), which allowed him to impart a lot of Energy to the bat - ball. That Energy is Force on the bat times Distance and the batter's 'strength' etc. needs to be matched to the bat. That all goes without saying but the effective point at which the force is actually applied will depend on the batter's hand geometry, arm mass etc. The force is distributed and will also involve some torque from the wrist.

Homework Helper
Gold Member
The same physics applies to a door and a doorstop. For the baseball bat, it is only a very rough approximation, but when you do hit the ball anywhere near the sweet spot, the hands don't feel as much of the impact.

berkeman
That part is okay, I gave some analysis in #6 to a similar conclusion.

My objection to #20 was that there does not exist a point on the bat about which the total angular momentum is zero, and even if one did, it would be an accelerating point so you'd also need to take into account inertial forces and impulses acting at the centre of mass.

So what Sears and Zemansky wrote sounds wrong, if that is an accurate reflection of their argument!

Gold Member
Back in grad school, a friend of mine became interested in the center of percussion and hammers. He couldn’t find one on the market that took advantage of the effect, so he designed one, patented it, and tried to sell it to a hardware company.

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Mentor
Back in grad school, a friend of mine became interested in the center of percussion and hammers. He couldn’t find one on the market that took advantage of the effect, so he designed one, patented it,
Interesting. What did it look like?

Homework Helper
Gold Member
That part is okay, I gave some analysis in #6 to a similar conclusion.

My objection to #20 was that there does not exist a point on the bat about which the total angular momentum is zero, and even if one did, it would be an accelerating point so you'd also need to take into account inertial forces and impulses acting at the centre of mass.

So what Sears and Zemansky wrote sounds wrong, if that is an accurate reflection of their argument!
The calculation is done simply at the moment of impact, and it is assumed the bat pivots at a point near where the hands are. The speed of the bat is in the same direction all along the bat, and is ## v=\omega r ##. It is the same thing as the door and the doorstop.

etotheipi
Gold Member
Interesting. What did it look like?
This was 20+ years ago, so memories are a bit fuzzy. He purposefully designed it to look like a normal hammer. Basically, he added internal weightings to adjust things. I think he also added some weight above where the hammer head connected to the shaft, but nothing that would stand out under a quick inspection. I believe he also found a couple of other physics related improvements, but I cannot remember what they were.

Homework Helper
Gold Member
It sounds like he designed the hammer so that the center of percussion of the hammer, when pivoted about a typical hand position, was right in the center of the head of the hammer, where it meets the nail.

The calculation is done simply at the moment of impact, and it is assumed the bat pivots at a point near where the hands are. The speed of the bat is in the same direction all along the bat, and is ## v=\omega r ##. It is the same thing as the door and the doorstop.

It is still insufficient. Consider a point on the bat which is a distance ##\xi## from the pivot. The angular momentum about such a point is of magnitude$$L = M \omega \left( \xi - \frac{\mathscr{L}}{2} \right)^2 + \frac{1}{12} M \mathscr{L}^2 \omega$$Note that this is strictly > 0. Hence, there is no point on the bat about which there is zero angular momentum.

Further, it does not matter that the calculation is only performed within an interval ##[t_c - \epsilon, t_c + \epsilon]## around the collision. If the coordinate system has its origin connected rigidly to the bat then you must still account for an inertial impulsive Dirac delta force through the centre of mass.

Gold Member
It sounds like he designed the hammer so that the center of percussion of the hammer, when pivoted about a typical hand position, was right in the center of the head of the hammer, where it meets the nail.
That was the initial idea. As he got into it, he found other physics improvements.

Charles Link
Homework Helper
Gold Member
Angular momentum ## L= \int \delta \, \vec{v} \times \vec{x} \, dx ##, where ## \vec{x} ## is measured from the sweet spot, and ## \delta ## is the mass per unit length. The cross product will have opposite signs on either side of the sweet spot to make ## L=0 ##, when the proper spot is picked for the origin.

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etotheipi
Ah, you are considering instead an inertial frame with an origin fixed at the position through which the sweet spot passes through at the lowest point in its swing! Okay, yes the argument works, we understood each other.

I thought you had meant a coordinate system with body-fixed origin, in which case angular momentum is as written in #31.

Charles Link
Gold Member
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berkeman and Charles Link