Circular Motion and Other Applications of Newton's Laws

Optional Section
6.5
EXAMPLE 6.14
NUMERICAL MODELING IN PARTICLE DYNAMICS 2
6.5 Numerical Modeling in Particle Dynamics 169
Resistive Force Exerted on a Baseball
A pitcher hurls a 0.145-kg baseball past a batter at 40.2 m/s
(�90 mi/h). Find the resistive force acting on the ball at this
speed.
Solution We do not expect the air to exert a huge force
on the ball, and so the resistive force we calculate from Equation
6.6 should not be more than a few newtons. First, we
must determine the drag coefficient D. We do this by imagining
that we drop the baseball and allow it to reach terminal
speed. We solve Equation 6.9 for D and substitute the appropriate
values for m, v t, and A from Table 6.1. Taking the density
of air as 1.29 kg/m 3 , we obtain
� 0.284
As we have seen in this and the preceding chapter, the study of the dynamics of a
particle focuses on describing the position, velocity, and acceleration as functions of
time. Cause-and-effect relationships exist among these quantities: Velocity causes
position to change, and acceleration causes velocity to change. Because acceleration
is the direct result of applied forces, any analysis of the dynamics of a particle
usually begins with an evaluation of the net force being exerted on the particle.
Up till now, we have used what is called the analytical method to investigate the
position, velocity, and acceleration of a moving particle. Let us review this method
briefly before learning about a second way of approaching problems in dynamics.
(Because we confine our discussion to one-dimensional motion in this section,
boldface notation will not be used for vector quantities.)
If a particle of mass m moves under the influence of a net force �F, Newton’s
second law tells us that the acceleration of the particle is a ��F/m. In general, we
apply the analytical method to a dynamics problem using the following procedure:
1. Sum all the forces acting on the particle to get the net force �F.
2. Use this net force to determine the acceleration from the relationship a ��F/m.
3. Use this acceleration to determine the velocity from the relationship dv/dt � a.
4. Use this velocity to determine the position from the relationship dx/dt � v.
The following straightforward example illustrates this method.
EXAMPLE 6.15
D �
This number has no dimensions. We have kept an extra digit
beyond the two that are significant and will drop it at the end
of our calculation.
We can now use this value for D in Equation 6.6 to find
the magnitude of the resistive force:
� 1
2 (0.284)(1.29 kg/m3 )(4.2 � 10�3 m2 )(40.2 m/s) 2
R � 1
2 D�Av2
�
2 mg
�
v 2
t �A
1.2 N
An Object Falling in a Vacuum — Analytical Method
Consider a particle falling in a vacuum under the influence
of the force of gravity, as shown in Figure 6.18. Use the analytical
method to find the acceleration, velocity, and position of
the particle.
2 The authors are most grateful to Colonel James Head of the U.S. Air Force Academy for preparing
this section. See the Student Tools CD-ROM for some assistance with numerical modeling.
2(0.145 kg)(9.80 m/s 2 )
(43 m/s) 2 (1.29 kg/m 3 )(4.2 � 10 �3 m 2 )
Solution The only force acting on the particle is the
downward force of gravity of magnitude F g, which is also the
net force. Applying Newton’s second law, we set the net force
acting on the particle equal to the mass of the particle times