The center of oscillation of a physical pendulum has this interesting property: If an impulse (assumed horizontal and in the plane of oscillation) acts at the center of oscillation, no oscillations are felt at the point of support. Baseball players (and players of many other sports) know that unless the ball hits the bat at this point (called the "sweet spot" by athletes), the oscillations due to the impact will sting their hands. To prove this property, let the stick in Figure a simulate a baseball bat. Suppose that a horizontal force F (due to impact with the ball) acts toward the right at P, the center of oscillation. The batter is assumed to hold the bat at O, the pivot point of the stick.
(a) What acceleration does the point O undergo as a result of F?
(b) What angular acceleration is produced by F about the center of mass of the stick?
(c) As a result of the angular acceleration in (b), what linear acceleration does point O undergo?
(d) Considering the magnitudes and directions of the accelerations in (a) and (c), convince yourself that P is indeed the "sweet spot."