A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.10 The physics of baseball

K = τ dt. (8.77)

analyze the effect of the collision with the ball on the motion of the bat’s centre

of mass. The centre of mass of the bat acts like a point particle of mass M which

is subject to the two impulses, J and JJ (which are applied simultaneously). If v

is the instantaneous velocity of the centre of mass then the change in momentum

of this point due to the action of the two impulses is simply

M ∆v = −J − JJ. (8.75)

The minus signs on the right-hand side of the above equation follow from the fact

that the impulses are oppositely directed to v in Fig. 82.

Note that in order to specify the instantaneous state of an extended body we

must do more than just specify the location of the body’s centre of mass. In-

deed, since the body can rotate about its centre of mass, we must also specify its

orientation in space. Thus, in order to follow the motion of an extended body,

we must not only follow the translational motion of its centre of mass, but also

the body’s rotational motion about this point (or any other convenient reference

point located within the body).

Consider the rotational motion of the bat shown in Fig. 82 about a perpendic-

ular (to the bat) axis passing through the pivot point. This motion satisfies

dω

I = τ, (8.76)

dt

where I is the moment of inertia of the bat, ω is its instantaneous angular velocity,

and τ is the applied torque. The bat is actually subject to an impulsive torque (i.e.,
a torque which only lasts for a short period in time) at the time of the collision

with the ball. Defining the angular impulse K associated with an impulsive torque

τ in much the same manner as we earlier defined the impulse associated with an
impulsive force (see Sect. 6.5), we obtain

∫t

It follows that we can integrate Eq. (8.76) over the time of the collision to find

I ∆ω = K, (8.78)