Circular Motion and Other Applications of Newton's Laws

158 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws
A
Top
Bottom
(a)
a magnitude n top � mg. Applying Newton’s second law yields
�F r � n top � mg � m v2
r
ntop � m v2
r � mg � mg � v2
rg
n top � mg �
QuickLab
(225 m/s) 2
(2.70 � 103 m)(9.80 m/s2 � 1� �
)
Hold a shoe by the end of its lace and
spin it in a vertical circle. Can you
feel the difference in the tension in
the lace when the shoe is at top of the
circle compared with when the shoe
is at the bottom?
� 1�
Quick Quiz 6.3
A bead slides freely along a curved wire at constant speed, as shown in the overhead view of
Figure 6.9. At each of the points �, �, and �, draw the vector representing the force that
the wire exerts on the bead in order to cause it to follow the path of the wire at that point.
6.2
0.913mg
�
n bot
mg
n top
In this case, the magnitude of the force exerted by the seat
on the pilot is less than his true weight by a factor of 0.913,
and the pilot feels lighter.
Exercise Determine the magnitude of the radially directed
force exerted on the pilot by the seat when the aircraft is at
point A in Figure 6.8a, midway up the loop.
Answer nA � 1.913mg directed to the right.
NONUNIFORM CIRCULAR MOTION
In Chapter 4 we found that if a particle moves with varying speed in a circular
path, there is, in addition to the centripetal (radial) component of acceleration, a
tangential component having magnitude dv/dt. Therefore, the force acting on the
mg
(b) (c)
�
�
Figure 6.8 (a) An aircraft executes
a loop-the-loop maneuver as
it moves in a vertical circle at constant
speed. (b) Free-body diagram
for the pilot at the bottom
of the loop. In this position the
pilot experiences an apparent
weight greater than his true
weight. (c) Free-body diagram for
the pilot at the top of the loop.
Figure 6.9