**Unformatted text preview: **is v and the
cord makes an angle with the vertical. Solution Unlike the situation in Example 6.7, the speed is
not uniform in this example because, at most points along the
path, a tangential component of acceleration arises from the
gravitational force exerted on the sphere. From the free-body
diagram in Figure 6.11b, we see that the only forces acting on 160 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws
vtop mg Ttop R
O O
T mg cos θ T bot θ v bot mg sin θ θ Figure 6.11 (a) Forces acting on a sphere
of mass m connected to a cord of length R and
rotating in a vertical circle centered at O.
(b) Forces acting on the sphere at the top and
bottom of the circle. The tension is a maximum at the bottom and a minimum at the top. mg
mg
(a) (b) the sphere are the gravitational force Fg m g exerted by the
Earth and the force T exerted by the cord. Now we resolve Fg
into a tangential component mg sin and a radial component
mg cos . Applying Newton’s second law to the forces acting
on the sphere in the tangential direction yields
Ft mg sin at g sin mat This tangential component of the acceleration causes v to
change in time because at dv/dt.
Applying Newton’s second law to the forces acting on the
sphere in the radial direction and noting that both T and ar
are directed toward O, we obtain
Fr T T m mv2
R mg cos
v2
R Special Cases At the top of the path, where
have cos 180° 180°, we
1, and the tension equation becomes
Ttop m v2
top
R g This is the minimum value of T. Note that at this point at 0
and therefore the acceleration is purely radial and directed
downward.
At the bottom of the path, where
0, we see that, because cos 0 1,
v2
bot
Tbot m
g
R
This is the maximum value of T. At this point, at is again 0
and the acceleration is now purely radial and directed upward. Exercise At what position of the sphere would the cord
most likely break if the average speed were to increase? g cos Answer At the bottom, where T has its maximum value. Optional Section 6.3 MOTION IN ACCELERATED FRAMES When Newton’s laws of motion were introduced in Chapter 5, we emphasized that
they are valid only when observations are made in an inertial frame of reference.
In this section, we analyze how an observer in a noninertial frame of reference
(one that is accelerating) applies Newton’s second law. 6.3 4.8 161 Motion in Accelerated Frames To understand the motion of a system that is noninertial because an object is
moving along a curved path, consider a car traveling along a highway at a high
speed and approaching a curved exit ramp, as shown in Figure 6.12a. As the car
takes the sharp left turn onto the ramp, a person sitting in the passenger seat
slides to the right and hits the door. At that point, the force exerted on her by the
door keeps her from being ejected from the car. What causes her to move toward
the door? A popular, but improper, explanation is that some mysterious force acting from left to right pushes her outward. (This is often called...

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