Bat Barrel

where is the sweet spot of a bat?

A baseball bat has a "sweet spot" where a ball can be hit with almost effortless transmission of energy. A careful analysis of baseball dynamics shows that this special spot is located at the point where an applied force would result in pure rotation of the bat about the handle grip.

Determine the location of the sweet spot of the bat shown in the figure . The linear mass density of the bat is given roughly by (0.61 + 3.3x^2) kg/m, where x is in meters measured from the end of the handle. The entire bat is 0.88m long. The desired rotation point should be .05cm from the end of the bat where it is held.

Hint: where is the CM of the bat?

if you could explain, that would be great! thanks!

Bending Modes and the Sweet Spot
There are many definitions for what constitutes the "Sweet Spot" on a baseball bat. From a player's viewpoint, the sweet spot is the place on the bat barrel where the contact between bat and ball results in the best hit - the ball leaves the bat with the greatest speed and the player's hands feel very little vibration from the impact. Most often in the published research the sweet spot is usually refered to as either the center of percussion (COP) or the node of the first bending mode. In my study of baseball bats I prefer to follow the convention used by Rod Cross who defines the sweet zone as the region located between the nodes of the first and second modes of vibration (between about 4-7 inches from the barrel end of a 30-inch Little League bat). Since the vibrational motion of the bat is very small in this region, an impact in this region will result in very little vibration of the bat (no stinging a player's hands) and a very solid hit will result with maximum energy being given to the ball. An impact to the outside (towards the barrel end) or inside (towards the handle) of this zone will result in a much more significant vibration of the bat, often felt as a painful sting. And the ball will not travel as far because some of the energy is now being stored (or dissipated) in the bat's vibration. Howard Brody has convincingly shown that gripping the bat with the hands adds a significant amount of damping, causing the bat's natural vibrations to decay very quickly. However, gripping the bat with the hands has very little effect on the frequencies or the mode shapes. Thus, suspending the bat from rubberbands, as I do in our acoustics lab, to obtain a free-free boundary condition is a legitimate approach to studying a bat's behavior. More........................... http://www.acs.psu.edu/drussell/bats/bend-sweet.html -------- Will the Real Sweet Spot Please Stand Up?
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: the location which produces least vibrational sensation (sting) in the batter's hands

For most bats all of these "sweetspots" are at different locations on the bat, so one is often forced to define the sweetspot as a region, approximately 5-7 inches from the end of the barrel, where the batted-ball speed is the highest and the sensation in the hands if minimized. For the purposes of this paper, I will attempt to examine the sweet spot in terms of two separate criteria. One will be the location where the measured performance of the bat is maximized, and the other will be the location where the hand sensation, or sting, is minimized.
Before we take a look at some of these definitions of the sweet spot, I should mention that bat manufacturers often advertise that aluminum bats have a wider sweet spot than do wood bats, or that a particular model of bat is more forgiving for hits away from the sweet spot than competetor models. This is difficult to verify, because of the variety of sweet spot definitions. One batting cage study found that aluminum and wood baseball bats have similarly sized sweet spots, as defined by the range of locations on the barrel which produced the highest batted-ball speeds. While the widths of the sweet spot regions were the same, the aluminum bats hit balls faster from the sweet spot, and also hit balls faster for impacts away from the sweet spot.
More........................ http://www.acs.psu.edu/drussell/bats/sweetspot.html ----------------- To find one of the "sweet spots" on a baseball bat, hold the bat, hanging down, loosely between your thumb and index finger, just below the knob on the bat's handle. Have a friend tap the bat gently with a hammer, starting at the fat end and moving toward the handle. (You can also do it yourself, although it's easier and more fun with a friend.) You should feel a vibration in your fingers whenever the bat is struck, except when the "node" is hit; then you'll feel nothing. You may also notice a slightly different sound when the node is struck.
When you hit a ball just right, you've hit it on one of the three "sweet spots" of the bat. One of these sweet spots relates to vibration. Whenever an object is struck, it vibrates in response. These vibrations travel in waves up and down the length of the object. At one point, called "the node," the waves always cancel each other out. If you hit the ball on the bat's node, the vibrations from the impact will cancel out, and you won't feel any stinging or shaking in your hand..More............ http://www.exploratorium.edu/baseball/sweetspot.html

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