Center of percussion
Encyclopedia
The center of percussion is the point on an object where a perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

 impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point, so that the pivot will not be moving momentarily after the impulse.
The same point is called the center of oscillation for the object suspended from the pivot as a pendulum
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...

.

Centers of percussion are often discussed in the context of a bat, racquet
Racquet
A racquet or racket is a sports implement consisting of a handled frame with an open hoop across which a network of cord is stretched tightly. It is used for striking a ball in such games as squash, tennis, racquetball, and badminton...

, sword
Sword
A sword is a bladed weapon used primarily for cutting or thrusting. The precise definition of the term varies with the historical epoch or the geographical region under consideration...

 or other long thin objects. The center of percussion may or may not be the "sweet spot
Sweet spot
Sweet spot may refer to:*Sweet spot *Sweet spot *Sweet spot...

" depending on the pivot point chosen.

Further explanation

Imagine a beam is suspended from a wire by a U-bolt so that it can move freely along the wire. A blow to the beam below the center of gravity will cause the beam to rotate around the CoG (center of gravity) and also cause the CoG to move away from the blow. If the blow falls above the CP (center of percussion) the movement of the CoG will cause the U-bolt to move away from the blow since the effect of the translational acceleration will outweigh the effect of the rotational acceleration. If the blow falls below the CP the opposite will occur, rotational acceleration will outweigh translational acceleration and the U-bolt will move towards the blow. Only if the blow falls exactly on the CP will the two cancel out to produce no net movement of the U-bolt.

Calculating the center of percussion

For a free, rigid beam, a force F applied at right angle
Right angle
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...

 at a distance b from the center of gravity
Center of gravity
In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity...

 (CoG) will result in the CoG moving at a velocity V according to the relation:


where M is the mass of the beam. Similarly the torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

 exerted will be as per the relation:


where I is the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

 around the CoG and is the angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

.

For any point P on the opposite side of the CoG from the point of impact, the velocity of point P is


where A is the distance of P from the CoG. Hence:


The velocity v is then given by:


The axis of rotation is situated where and the corresponding center of percussion is at distance b from the CoG, with


This is also the center of oscillation of a physical pendulum of the same mass M, hung at the pivot point. (The center of oscillation is the position of the mass of a simple pendulum that has the same period as the physical pendulum.)

Center of percussion of a uniform beam

For the special case of a free beam of uniform density of length L the moment of inertia around the CoG is:
(see moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

 for proof)

Hence:

Relation to the sweet spot

The sweet spot
Sweet spot (sports)
A sweet spot is a place where a combination of factors results in a maximum response for a given amount of effort. In tennis, baseball, or cricket, a given swing will result in a more powerful hit if the ball strikes the racquet or bat on the latter's sweet spot.The sweet spot is often believed to...

 on a baseball
Baseball
Baseball is a bat-and-ball sport played between two teams of nine players each. The aim is to score runs by hitting a thrown ball with a bat and touching a series of four bases arranged at the corners of a ninety-foot diamond...

 bat is generally defined as the point at which the impact feels best to the batter (it is also occasionally defined as the point at which the maximum velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

 is imparted to the ball, but this may not be the same point).

Although it has long been believed the center of percussion and the sweet spot are the same, recent practical observations have indicated that the point many batters feel is "sweetest" corresponds to a pivot point in the arm, beyond the handle of the bat.

Application to swordmaking

The center of percussion of a sword
Sword
A sword is a bladed weapon used primarily for cutting or thrusting. The precise definition of the term varies with the historical epoch or the geographical region under consideration...

 is the point on the blade
Blade
A blade is that portion of a tool, weapon, or machine with a cutting edge and/or a pointed tip that is designed to cut and/or puncture, stab, slash, chop, slice, thrust, or scrape animate or inanimate surfaces or materials...

 where cutting produces the least hand shock. It is also the division between the weak and middle sections of the blade.

Like the center of balance
Center of balance
Center of balance is a point with respect to which the object in question is balanced with respect to applied forces. In particular areas the term may have specific meaning and special discussion, and may refer to one of the following definitions:...

 of a sword, the center of percussion can be moved by employing a heavier pommel or changing the mass distribution of the blade.

One of the vibrational nodes of the second harmonic of a vibrating sword (the node closest to the tip) is also often (mistakenly) referred to as the center of percussion. The significance of the vibrations about this mode have been contested as having little relevance to sword physics.

So-called “blade harmonics” are a commonly misunderstood concept, especially by sword enthusiasts and their internet communities. The common belief is that a sword must be “harmonically balanced” in order to cut properly, because the vibrations would otherwise interrupt the line and power of the cut. As explained above, this proposition is false: the vibrations caused by a sword cut are almost unnoticeable except as a mild stinging to the hands even in blades that lack this quality. It has also been demonstrated that the object the sword cuts through serves to further reduce the intensity of any vibration, making it even less noticeable.

Many experts speculate that harmonic balance is merely a byproduct of proper construction and balancing, rather than an intentional quality added to weapons. Unfortunately, some sword vendors advertise "secret techniques" of harmonic balancing in an attempt to "prove" the superiority of their products. This only serves to amplify the false impressions of the value of harmonic balance by seeming to lend them legitimacy.
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