Center of oscillation

The Center of Percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is struck at the center of percussion. The center of percussion is often discussed in the context of a bat, racquet, door, sword or other extended object held at one end.

The same point is called the center of oscillation for the object suspended from the pivot as a pendulum, meaning that a simple pendulum with all its mass concentrated at that point will have the same period of oscillation as the compound pendulum. In sports, the center of percussion of a bat or racquet is related to the so-called "sweet spot", but the latter is also related to vibrational bending of the object.

Imagine a rigid beam suspended from a wire by a fixture that can slide freely along the wire at point P, as shown in the Figure. An impulsive blow is applied from the left. If it is below the center of mass (CM) it will cause the beam to rotate counterclockwise around the CM and also cause the CM to move to the right. The center of percussion (CP) is below the CM. If the blow falls above the CP, the rightward translational motion will be bigger than the leftward rotational motion at P, causing the net initial motion of the fixture to be rightward. If the blow falls below the CP the opposite will occur, rotational motion at P will be larger than translational motion and the fixture will move initially leftward. Only if the blow falls exactly on the CP will the two components of motion cancel out to produce zero net initial movement at point P.

When the sliding fixture is replaced with a pivot that cannot move left or right, an impulsive blow anywhere but at the CP results in an initial reactive force at the pivot.

For a free, rigid beam, an impulse ${\displaystyle Fdt}$ applied at right angle at a distance ${\displaystyle b}$ from the center of mass (CM) will result in the CM changing velocity ${\displaystyle dv_{cm}}$ according to the relation:

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