Environmental Engineering Reference
In-Depth Information
without producing much vibration in the handle. In this section we look at sudden
collisions that cause changes to both the linear and angular momenta of a rigid
body, as happens when you strike a ball with a bat or when a door slams against a
doorstop. In the event of such a collision there can be large forces exerted, just like
in the example of the tennis player's serve in Section 2.3.5. Usually, we do not know
the details of how the forces depend on time but we can nevertheless make progress
if we know the impulse imparted by the collision, i.e. the change in momentum p .
Likewise, the collision will generally produce a time-dependent torque but since we
don't know the time-dependence of the force we speak about the angular impulse:
t 2
L
=
r
×
p
=
r
×
F (t) d t,
(4.44)
t 1
where the collision exerts a force F (t) at a position r from time t 1 to t 2 .
Let us try to compute the position of the sweet spot in the collision between a
bat and a ball. For simplicity we will ignore the effect of gravity by considering a
bat at rest on a frictionless horizontal surface as shown in Figure 4.10. The centre
of mass of the bat is at C , a distance h from the handle of the bat H . A ball strikes
the bat, imparting an impulse p a distance b from the centre of mass as shown in
the figure. We are interested in the subsequent motion of the bat and in particular
the motion of the handle H immediately after the impulse has been delivered. The
sweet spot ought to correspond to the special value of b such that H does not
move in the split second after the impact. After the collision the bat constitutes
an isolated system, so in an inertial frame the momentum of the centre of mass
and angular momentum of rotation about the centre of mass are both constant. The
final linear momentum is just the impulse:
=
p
M V c ,
(4.45)
where V c is the velocity of the centre of mass and M is the total mass of the bat.
Likewise, the final angular momentum (about C ) is the angular impulse:
|
L
|=|
r
×
p
|=
bp.
(4.46)
V C
h
b
H
C
w
p
Figure 4.10
An impulse p causes translation and rotation of a rigid body.
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