The following material has been expanded considerably into a 324 page book “Physics of baseball and softball” published by Springer in March 2011.
Alan Nathan has the world's best Physics of Baseball site at http://go.illinois.edu/physicsofbaseball
Another very interesting Physics of Baseball and Softball site is maintained by Dan Russell at http://www.kettering.edu/~drussell
Lloyd Smith at Washington State University has an excellent site on the physics and engineering of softball, at http://www.mme.wsu.edu/~ssl , including information on how bats and balls are tested and certified for use in the USA.
James Sherwood has a similar site http://m-5.eng.uml.edu/umlbrc describing test procedures used to certify baseball bats and balls at the University of Massachusetts – Lowell.
1. The sweet spot
Batters know from experience that there is a sweet spot on the bat, about 17 cm from the end of the barrel, where the shock of the impact, felt by the hands, is reduced to such an extent that the batter is almost unaware of the collision. At other impact points, the impact is usually felt as a sting or jarring of the hands and forearm, particularly if the impact occurs at a point well removed from the sweet spot.
The sweet spot of a bat exists because bat vibrations are not excited significantly at that spot. The spot is close to the centre of percussion (COP) for a freely supported bat, so it was thought at one time that the sweet spot might be associated with the COP. However, the COP shifts a long way toward the handle when a batter holds the handle, and plays only a minor role in the feel of the bat for an impact at the sweet spot.
These spots are also described in the physics of tennis page. A bat is similar to a racquet but there are some obvious differences. For a tennis racquet only the fundamental mode of vibration is excited by the impact since the impact duration, about 5 msec, is too long to excite higher frequency modes of the racquet. The impact on a baseball bat is shorter (since the bat and ball are harder or stiffer), about 1 msec. As a result, the fundamental and second vibration modes are both excited with about the same amplitude. These modes are shown in the diagram above. Hence, there are two vibration nodes in the barrel. An impact at the fundamental node will not excite that mode, but it will excite the second mode. Similarly, an impact at the node of the second mode will not excite the second mode but it will excite the fundamental mode.
Most people find that an impact between the nodes of the first two modes (node1 and node2 in the diagram) feels best since both modes are then excited only weakly.
For further details, see American Journal of Physics, September 1998 and also American Journal of Physics, 69, 231-232 (2001). The effect of the hands on the centre of percussion is described in American Journal of Physics, 72, 622-630 (2004). Both papers can be downloaded here and here.
2. A Physics of Baseball Project
The experiment described below would be an ideal project on the physics of baseball, for any high school or University physics student. There is enough physics in it to keep a PhD student busy for a few years, but it could also be done by any high school student as a simple introduction to the physics of baseball. The experiment can be done at low bat and ball speeds since the essential physics of the problem does not depend on the actual bat and ball speeds. Rather, the physics depends more on the ratio of the bat and ball speeds, regardless of whether the actual speeds are around 1 mph or 100 mph. Some of the numbers change with actual speed, but the physics doesn’t change. It is not quantum physics or relativity, but a very interesting problem involving the mechanics of colliding objects.
The idea is to mount a bat as a pendulum and strike it with a ball at various spots along the barrel. One question that can be answered this way is "Where on the bat does the ball bounce best?" In theory, the ball bounces best for an impact at the center of mass of the bat, provided we can ignore bat vibrations. But the bat vibrates strongly for impacts near the tip of the bat and near its center of mass. Energy is lost during the collision if the bat vibrates. As a result, the bounce is relatively weak at the tip and near the center of mass.
If the ball is also mounted as a pendulum, the experiment can also be performed in reverse, by swinging the bat at a stationary ball. We can then ask a similar question. Where on the bat is the ball struck with the greatest speed? A first guess would be at the tip of the bat since the tip travels the fastest. But strong vibrations are generated for impacts near the tip. Furthermore, the tip is the lightest and least effective part of the bat.
By combining the results of these two separate experiments, we can then work out what happens when a bat is swung at an incoming ball. The outgoing speed of the ball is just the sum of the two separate parts - the speed when the bat is at rest plus the speed when the ball is at rest.
Some results for an Easton BK7 aluminum bat are shown below in 13 QuickTime movie files, each about 1 Mb. Also shown for comparison is a bounce off a heavy block of granite so that the COR of the ball can be determined. For small swings of a pendulum, the incoming and outgoing speed of the pendulum bob is proportional to its horizontal displacement, so the relevant ball speed ratios can be measured directly off the film with a ruler. The bat is 84 cm long, and has a mass of 849 gram. Its center of mass is 52.6 cm from the knob end. The bat was swung through an axis 143 mm from the knob end. Its moment of inertia about an axis through its center of mass is 0.184 kg.m^2. The impacts were at distances of 2, 7, 12, 17, 24 and 31 cm from the end of the barrel.
The lines behind the bat, passing through the axis, mark out angles of 5, 10, 15, 20, 25 and 30 degrees. The bat was swung through an angle of about 30 degrees to strike the ball. Each impact was recorded on a video camera at 25 frames/sec.
Coefficient of Restitution = Relative speed of bat and ball after collision/ relative speed before collision, where "speed of bat" is the speed of the bat at the impact point.
The COR for the impact on the granite block is about 0.59 in the low speed collision here, but can be as low as 0.45 in a 100 mph collision. The COR for a collision between the bat and the ball is less than 0.59 when energy is lost to vibrations, but can be greater than 0.59 if the bat has a trampoline effect on the ball. A high performance bat has a strong trampoline effect and hence a relatively large COR. The technique here can be be used to measure the performance of any bat, at least at low ball speeds. The swing weight of the bat (that is, its moment of inertia) can be determined from the measured period of oscillation when swung as a pendulum.
The bounce factor found from these and some additional impacts is shown below.
The relevant collision equations are derived in the Physics of Baseball and Softball book (and elsewhere) and are:
1. If the bat is at rest then we can define the ball speed ratio
Bounce Factor = q = (ball exit speed) / (incident ball speed)
q is also called the apparent coefficient of restitution or ACOR. q varies from about zero near the tip of the bat to about 0.2 or 0.3 further along the barrel, and needs to be measured for any given bat since it is usually too hard to calculate. So,
Ball exit speed = q*vin where vin = incident ball speed.
2. If the ball is at rest (when striking a ball off a T ) then
Ball exit speed = (1 + q)V where V = bat speed at impact point just before the impact.
3. If the bat approaches a ball incident at speed vin then
Ball exit speed = (1 + q)V + q*vin
The impact point 2 cm from the tip of the Easton bat is a dead spot where the ball doesn't bounce at all, so q = 0. However, when the bat strikes the ball at the dead spot then the ball exit speed = V = speed of bat just before the collision.
3. Swing speed vs bat weight
Are heavy bats are more powerful than light bats? The answer depends on how fast each bat can be swung. The following three movies show a batter swinging three different bats as fast as possible, filmed at 25 frames/sec. One was a 232 gram broomstick, one was a 871 gram Louisville Slugger wood bat with four holes drilled through the barrel, and one was the 871 gram bat increased to 1333 gram by inserting a steel bolt through each hole. The 232 g bat was swung at 45 radians/sec on average (by the batter in the film). The same batter swung the 871 g bat at 33 radians/sec on average, and he swung the 1333 g bat at 26 radians/sec on average. If you turn the volume up you will hear how the swing speed affects the sound of the “swish” through the air.
The above results are more or less as expected, in that light bats can be swung faster than heavy bats. However, there are many variables in experiments like these. It is not the actual weight that determines the swing speed but the swing weight (ie the moment of inertia). In general, swing weight increases with actual weight, but swing weight also depends on bat length and weight distribution. It has been found that the swing speed for any given batter is proportional to 1/(Swing weight)^n where n is typically about 0.27 for most batters. A similar result is obtained when swinging cylindrical rods, as described in a paper that can be downloaded here.
The weight and construction of a bat also affects the bounce factor q, and it affects the time and position of the bat at which the swing speed is a maximum. The batter might find that he needs to change the way he swings the bat if the swing weight is altered, or he might have a “grooved” swing technique and swing every bat the same way. There is no simple physics answer to the heavy/light bat question here. In practice, batters tend to prefer bats that are lighter, rather than more powerful, since lighter bats are easier to swing and make it easier for the batter to connect with the ball.
A question that is equally interesting is how the swing speed of each body segment varies as the bat speed varies and as the bat swing weight varies. The shoulders reach maximum speed first, then the upper arms, then the forearm and finally the bat. Any two connected segments can be analysed as a double pendulum. The bat and the forearms are locked at right angles at the beginning of the swing, by locking the wrists, otherwise the bat would start to rotate in the wrong direction at the beginning of the swing. Near the end of the swing, the bat swings so fast that the batter can no longer keep his wrists locked, and the centripital force on the bat is about equal to the weight of the batter.
4. How to swing a bat
It is not difficult to swing a bat. It is almost as easy as walking. But how does a batter do it? Specifically, what forces and torques are exerted on the handle, and in what directions do they act? It is very surprising that no one seems to have worked this out before. Adair provides a few answers in his book “The Physics of Baseball” but he does not give the directions or the torques. The diagram below shows the swing of a wood bat filmed from a spot above the batter’s head. The force on the bat can be worked out from the velocity of the center of mass, (CM), and the torques can be worked out from the angular acceleration. The results are very surprising. Initially, the force on the handle is in the opposite direction to the motion of the handle. While the center of mass moves one way (nearly upward here), the handle moves the opposite way (nearly downward). The batter needs to exert a small couple to get the swing started, using equal and opposite forces on the handle, otherwise the barrel of the bat will get left behind. Near the end of the swing, the force is roughly at right angles to motion of the handle since the centripetal force is very large. However, the centripetal force does not act along the axis of the bat, but at an angle, as shown by the orange lines.
The direction of the centripetal force is toward the center of the circle followed by the path of the CM. Since the CM traces out a spiral rather than a circular path, the center of the circle moves, as the bat is swung, along the path traced out by the inner circle of black dots. At any given time, the center of the circle can be found by fitting a circle to three neighboring points, at say time t, and at times t + 0.02 s and t - 0.02 s. This gives the radius, R, of the circle, from which we can calculate the centripetal force MV2/R as well as the force at right angles to that, given by M (dV/dt).
Near the end of the swing, the batter needs to exert a large negative couple on the bat, otherwise the bat will swing around too fast and strike the ball when it is aligned at the wrong angle. The same thing happens when swinging a golf club, but it is not a well-known effect. Rather, most coaches and others think in terms of wrist torque, which is probably much too small to provide the necessary large couple near the end of the swing The couple must come mainly from the two arms, not the wrists. A paper on this subject can be downloaded here.
5. Bounce of a baseball
The impact of a bat and a ball is still not properly understood. A lot has been learned over the years but more experiments need to be done to better understand how the ball bounces and how much spin it acquires when it bounces. The flight of a ball through the air depends strongly on how fast it spins, and in what direction it spins.
QuickTime movie files A and B show a baseball bouncing on a block of granite. The bounce was filmed at 300 fps and is played back at 30 fps in slow motion. The ball is 72 mm in diameter. From that you can work out the speed and spin of the ball before and after it bounces. In principle you could also work out the acceleration due to gravity in Sydney, but not very accurately.
The ball is spinning clockwise before it bounces and hits the granite block with backspin. The ball bounces backward spinning counter-clockwise. The reversal of the spin in each bounce implies that there is some tangential elastic behaviour of the ball in a direction parallel to the block. There is obviously some elastic behaviour in the vertical direction as well, which is why the ball bounces up off the block. Elastic behaviour in the vertical direction is described by a number called the coefficient of restitution (COR) which is about 0.5 for a baseball, depending on ball speed. If the ball didn't bounce at all then COR would be zero. If there was no loss of energy during the bounce then COR would be 1.0
Elastic behaviour in the horizontal direction can be described by a tangential coefficient of restitution, which is about 0.5 or 0.6 for a superball but only about 0.2 for a baseball on a rigid surface. The tangential COR is defined in the same way as the normal COR, but in terms of the horizontal speed of the ball at the contact point rather than the normal (ie vertical) speed of the contact point. The horizontal speed at the contact point depends on the rate at which the ball spins, as well as on the horizontal speed of the centre of mass of the ball. In file A, the ball spins faster than in file B, both before and after the bounce. The ball seems to spin faster when it is fresh and slower after it has been used, but more experiments are needed to determine whether this is in fact correct. In file A, the tangential COR was surprisingly high, about 0.3.
Interesting information on the behaviour of a ball can be obtained by bouncing it on a force plate or on a piezo to measure force vs time. Typical results are shown here for a superball, a baseball and a sorbothane ball. A superball has a COR of about 0.9, a baseball has a COR of about 0.5 and a sorbothane ball has a COR of about 0.1 (ie it hardly bounces at all). The shapes of the force vs time curves are interesting since they tell us how the force on the ball varies with the amount of compression. Since F = m*dv/t, we can integrate the force waveform to measure ball speed vs time, then integrate v = dx/dt to measure the shift in the ball CM vs time.
Superball, 0.5 ms/div
Baseball, 0.2 ms/div
Sorbothane ball 2 ms/div
Graphs of F vs t and x vs t can also be used to generate graphs of F vs x, showing how the force on the ball varies with its compression. Results for a baseball dropped from heights of 1, 5, 10 and 15 cm onto a piezo are shown below. The ball bounces while it is still compressed, and recovers to its original sherical shape after it bounces. Similar results are obtained at much higher ball speeds, the forces and x values being correspondingly much larger. Balls compress along the same F vs x curve, regardless of the ball speed, although the actual compression curve for a given ball will depend on its stiffness.
6. Bat and ball collisions
The collision of a bat and a ball can be understood by treating the bat as a heavy ball whose mass depends on the impact point. The whole mass of the bat is not involved in the collision, unless the collision happens to occur at the centre of mass of the bat. Otherwise, the effective mass of the bat is less than its whole mass. To understand such collisions it is important to first understand the physics of a collision between one ball and another. Movies showing the collision of a baseball with a cricket ball and the collision of two billiard balls are shown in the page on collisions.
7. Hoop Modes
A hollow aluminum bat sounds a lot different to a solid bat since the wall of a hollow bat can vibrate in and out like a drum. Not exactly like a drum, more like a bell or a chime, but the effect is similar. The sound made by a bat can be described as a "ping". The frequency is typically between 1000 Hz and 2500 Hz, meaning that the wall vibrates in and out 1000 to 2500 times in one second. The end result can be a greater ball speed off the bat, in the same way that a tennis ball bounces a lot better off tennis strings than off a hard floor. The effect is known as the "trampoline effect". There is a difference between a bat and a racquet here, in that tennis strings are very light and give back to the ball 95% of their stored elastic energy. The wall of a bat is relatively heavy and returns only some of its stored elastic energy to the ball, keeping the rest to itself so it can ping as loudly as it wants to. That is, a large fraction of the stored elastic energy in the wall is retained as vibrational energy in the bat. If the vibration frequency is less than about 1000 Hz, the end result is that ball comes off the bat at a lower speed than off a wood bat of the same weight and swing weight. For that reason, the walls of a hollow bat need to be relatively stiff so they can vibrate at a frequency larger than 1000 Hz. Thick walls are also needed to help prevent the walls dinting or cracking.
Some hoop mode sounds are recorded in the following QuickTime Movies, using an aluminum bat and several aluminum tubes of various diameters and wall thickness. The tubes were suspended by a length of string so that they could vibrate for a long time without damping. It was found, by measuring the frequencies of lots of aluminum tubes (see photo) that the frequency is directly proportional to the wall thickness and inversely proportional to the tube diameter squared. The stiffness of a cylinder was found to be roughly proportional to its length x (wall thickness / diameter) squared (rather than cubed, since the wall thickness is not negligible compared with the radius for most of these tubes).
Compare sounds here (recorded with a signal generator connected to a speaker)
8. Softball spin speed
It is not easy to find information on the actual rate of spin of a softball in flight, but it is relatively easy to measure, at least at low ball speeds and low spin speeds. I tossed a 12 inch softball at very low speed and filmed the result, using two hands instead of one to increase the spin rate and to make sure the spin axis was horizontal. The result was not a super slow softball spin, neither was it particularly fast. I simply wanted to know whether a hand launched ball spins at around 100 rpm or whether it could be as high as 1000 rpm. The ball rotated more than 1/4 but less than 1/2 of one revolution from one frame to the next, each frame being 0.04 sec apart. Taking the rotation rate as 3/8 revolution / 0.04 sec gives 9.4 revolutions / sec or 560 rpm. It would therefore not be surprising if a softball can be pitched at say 1000 rpm. A rough estimate would be that the ball rotates 1/4 revolution while it is in the pitcher’s hand, and while the hand moves forward say the last 0.2 m. If the hand moves forward at 10 m/s = 22.4 mph then it moves 0.2 m in 0.02 sec, so the spin rate would be 12.5 rev/s or 750 rpm. The hand moves forward faster than this in practice, so the spin rate could easily be around 1500 rpm.
Another simple estimate of spin rate is to throw the ball and catch it with the other hand as soon as it is released. The ball will rotate about 1/2 a revolution from the initial grip to the release point, in about 0.1 sec or less.
The film is shown here, taken at 25 frames/sec.