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Chapter 1

Q. 1.11

A baseball player holds a bat with a centroidal moment of inertia \overline{I} a distance a from the bats mass center. His “bat speed” is the angular velocity with which he swings the bat. The pitched ball is a fastball which reaches the batter with a velocity v. Assuming his swing is a rigid-body rotation about an axis perpendicular to his hands, where should the batter hit the ball to minimize the impulse felt by his hands?


Verified Solution

When the better hits the ball, it exerts an impulse on the bat: call it B. Since the batter is
holding the bat, he feels an impulse as he hits the ball: call it P. The effect of hitting the ball is to change the bat speed from \omega _{1} to \omega _{2}. The impulse momentum diagrams of the bat during the time are shown in Figure 1.26.
Applying the principle of linear impulse and momentum to Figure 1.26 leads to

ma\omega _{1}+P-B=ma\omega _{2}          (a)

Application of the principle of angular impulse and angular momentum about an axis
through the batter’s hands yields

\overline{I} \omega _{1} +ma \omega _{1}\left(a\right) -B\left(b\right) = \overline{I} \omega _{2}+ma\omega _{2}\left(a\right)          (b)

Solving Equation (b) for B, we have

B = \frac{\left(\overline{I}+ma^{2} \right) }{b} \left(\omega _{2}-\omega _{1}\right)        (c)

Substituting Equation (c) into Equation (a) and solving for P leads to

P=\left(\omega _{2}-\omega _{1}\right)\left(\frac{\overline{I}+ma^{2}}{b}- ma \right)           (d)

Thus, P = 0 if

b= a +\frac{\overline{I} }{ma}           (e)

Thus, the angular impulse felt by the batter is zero if b satisfies Equation (e). The location of b is called the center of percussion