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**Unformatted text preview: **r 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. n bot
Top A ntop
mg mg (b) (c) Bottom
(a) a magnitude n top
Fr ntop ntop m ntop mg mg v2
r mg (2.70 mg. Applying Newton’s second law yields
m v2
r mg v2
rg 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. 1 (225 m/s)2
103 m)(9.80 m/s2) 1 0.913mg Answer nA 1.913mg directed to the right. 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. QuickLab
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? Figure 6.9 6.2 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 6.2 Nonuniform Circular Motion 159 Some examples of forces acting during circular motion. (Left) As these speed skaters round a
curve, the force exerted by the ice on their skates provides the centripetal acceleration.
(Right) Passengers on a “corkscrew” roller coaster. What are the origins of the forces in this
example? Figure 6.10 When the force acting on a particle moving in a circular path has a tangential component Ft , the
particle’s speed changes. The total force exerted on the
particle in this case is the vector sum of the radial force
and the tangential force. That is, F Fr Ft .
F
Fr Ft particle must also have a tangential and a radial component. Because the total acceleration is a ar at , the total force exerted on the particle is F Fr Ft , as
shown in Figure 6.10. The vector Fr is directed toward the center of the circle and is
responsible for the centripetal acceleration. The vector Ft tangent to the circle is responsible for the tangential acceleration, which represents a change in the speed of
the particle with time. The following example demonstrates this type of motion. EXAMPLE 6.8 Keep Your Eye on the Ball A small sphere of mass m is attached to the end of a cord of
length R and whirls in a vertical circle about a ﬁxed point O,
as illustrated in Figure 6.11a. Determine the tension in the
cord at any instant when the speed of the sphere...

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