the CM. For any point P a distance on the opposite side of the CM from the point of impact, the change in velocity of point P is where is the distance of P from the CM. Hence the acceleration at P due to the impulsive blow is: When this acceleration is zero, defines the center of percussion. Therefore, the CP distance, , from the CM, is given by Note that P, the rotation axis, need not be at the end of the beam, but can be chosen at any distance . Length also defines the center of oscillation of a physical pendulum, that 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 beam of uniform density of length , the moment of inertia around the CM is: (see moment of inertia for proof), and for rotation about a pivot at the end, . This leads to: . It follows that the CP is 2/3 of the length of the uniform beam from the pivoted end. Some applications For example, a swinging door that is stopped by a doorstop placed 2/3 of the width of the door will do the job with minimal shaking of the door because the hinged end is subjected to no net reactive force. (This point is also the node in the second vibrational harmonic, which also minimizes vibration.) The sweet spot on a baseball bat is generally defined as the point at which the impact feels best to the batter. The center of percussion defines a place where, if the bat strikes the ball and the batter's hands are at the pivot point, the batter feels no sudden reactive force. However, since a bat is not a rigid object the vibrations produced by the impact also play a role. Also, the pivot point of the swing may not be at the place where the batter's hands are placed. Research has shown that the dominant physical mechanism in determining where the sweet spot is arises from the location of nodes in the vibrational modes of the bat, not the location of the center of percussion. The center of percussion concept can be applied to swords. Being flexible objects, the "sweet spot" for such cutting weapons depends not only on the center of percussion