Circular Motion and Other Applications of Newton's Laws

166 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws
v
R
mg
R
mg
Figure 6.16 An object falling
through air experiences a resistive
force R and a gravitational force
F g � mg. The object reaches terminal
speed (on the right) when the
net force acting on it is zero, that
is, when R ��F g or R � mg. Before
this occurs, the acceleration
varies with speed according to
Equation 6.8.
v t
where � is the density of air, A is the cross-sectional area of the falling object measured
in a plane perpendicular to its motion, and D is a dimensionless empirical
quantity called the drag coefficient. The drag coefficient has a value of about 0.5 for
spherical objects but can have a value as great as 2 for irregularly shaped objects.
Let us analyze the motion of an object in free fall subject to an upward air
resistive force of magnitude R � Suppose an object of mass m is released
from rest. As Figure 6.16 shows, the object experiences two external forces:
the downward force of gravity Fg � mg and the upward resistive force R. (There is
also an upward buoyant force that we neglect.) Hence, the magnitude of the net
force is
1
2 D�Av2 .
�F � mg � 1
2 D�Av2
(6.7)
where we have taken downward to be the positive vertical direction. Substituting
�F � ma into Equation 6.7, we find that the object has a downward acceleration of
magnitude
a � g �� D�A
2m � v2
(6.8)
We can calculate the terminal speed v t by using the fact that when the force of
gravity is balanced by the resistive force, the net force on the object is zero and
therefore its acceleration is zero. Setting a � 0 in Equation 6.8 gives
g �� D�A
2m � v 2
t � 0
(6.9)
Using this expression, we can determine how the terminal speed depends on the
dimensions of the object. Suppose the object is a sphere of radius r. In this case,
(from A � �r 2 ) and (because the mass is proportional to the
volume of the sphere, which is V � Therefore, vt � √r.
Table 6.1 lists the terminal speeds for several objects falling through air.
4
3 �r3 m � r
).
3
A � r2 vt � √ 2mg
D�A
The high cost of fuel has prompted many truck owners to install wind deflectors on their cabs to
reduce drag.