Physics For Scientists And Engineers 6E - part 40

Quick Quiz 6.3

Which of the following is impossible for a car moving in a

circular path? (a) the car has tangential acceleration but no centripetal acceleration.
(b) the car has centripetal acceleration but no tangential acceleration. (c) the car has
both centripetal acceleration and tangential acceleration.

Quick Quiz 6.4

A bead slides freely along a horizontal, curved wire at con-

stant speed, as shown in Figure 6.9. Draw the vectors representing the force exerted by
the wire on the bead at points !, ", and #.

Quick Quiz 6.5

In Figure 6.9, the bead speeds up with constant tangential

acceleration as it moves toward the right. Draw the vectors representing the force on
the bead at points !, ", and #.

Figure 6.9 (Quick Quiz 6.4 and 6.5) A bead

slides along a curved wire.

SECTION 6.2 • Nonuniform Circular Motion

157

Active Figure 6.8 When the force

acting on a particle moving in a

circular path has a tangential

component F

, 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 ! F

(

Passengers on a “corkscrew” roller coaster experience a radial force toward the center

of the circular track and a tangential force due to gravity.

Robin Smith / Getty Images

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 radial component of acceleration, a tangential compo-
nent having magnitude dv/dt. Therefore, the force acting on the particle must also
have a tangential and a radial component. Because the total acceleration is
a ! a

(

, the total force exerted on the particle is

F ! F

(

, as shown in

Figure 6.8. The vector

is directed toward the center of the circle and is responsi-

ble for the centripetal acceleration. The vector

tangent to the circle is responsi-

ble for the tangential acceleration, which represents a change in the speed of the
particle with time.

At the Active Figures link

at http://www.pse6.com, you

can adjust the initial position

of the particle and compare

the component forces acting

on the particle to those for a

child swinging on a swing set.

158

CHAPTE R 6 • Circular Motion and Other Applications of Newton’s Laws

Example 6.7 Keep Your Eye on the Ball

A small sphere of mass m is attached to the end of a cord of
length R and set into motion in a vertical circle about a fixed
point O, as illustrated in Figure 6.10a. Determine the ten-
sion in the cord at any instant when the speed of the sphere
is v and the cord makes an angle " with the vertical.

Solution Unlike the situation in Example 6.6, 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 6.10a, we see that the only forces acting on
the sphere are the gravitational force

g exerted by the

Earth and the force

T exerted by the cord. Now we resolve F

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

This tangential component of the acceleration causes v to
change in time because a

dv/dt.

Applying Newton’s second law to the forces acting on

the sphere in the radial direction and noting that both

and

are directed toward O, we obtain

T ! m

(

cos "

T ' mg cos " !

sin "

sin

" !

What If?

What if we set the ball in motion with a slower

speed?

(A)

What speed would the ball have as it passes over

the top of the circle if the tension in the cord goes to zero in-
stantaneously at this point?

Answer At the top of the path (Fig. 6.10b), where
" !

180&, we have cos 180& ! ' 1, and the tension equation

becomes

Let us set T

top

0. Then,

(B)

What if we set the ball in motion such that the speed at

the top is less than this value? What happens?

Answer In this case, the ball never reaches the top of the
circle. At some point on the way up, the tension in the string
goes to zero and the ball becomes a projectile. It follows a
segment of a parabolic path over the top of its motion, re-
joining the circular path on the other side when the tension
becomes nonzero again.

top

√

0 ! m

top

Investigate these alternatives at the Interactive Worked Example link at http://www.pse6.com.

bot

top

bot

top

(b)

(a)

mg cos

mg sin

Figure 6.10 (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.

Interactive

SECTION 6.3 • Motion in Accelerated Frames

159

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 Newton’s second law is applied by an observer in a non-
inertial frame of reference, that is, one that is accelerating. For example, recall the
discussion of the air hockey table on a train in Section 5.2. The train moving at con-
stant velocity represents an inertial frame. The puck at rest remains at rest, and New-
ton’s first law is obeyed. The accelerating train is not an inertial frame. According to
you as the observer on the train, there appears to be no visible force on the puck, yet
it accelerates from rest toward the back of the train, violating Newton’s first law.

As an observer on the accelerating train, if you apply Newton’s second law to the

puck as it accelerates toward the back of the train, you might conclude that a force
has acted on the puck to cause it to accelerate. We call an apparent force such as this
a

fictitious force, because it is due to an accelerated reference frame. Remember

that real forces are always due to interactions between two objects. A fictitious force
appears to act on an object in the same way as a real force, but you cannot identify a
second object for a fictitious force.

The train example above describes a fictitious force due to a change in the speed of

the train. Another fictitious force is due to the change in the direction of the velocity vec-
tor. To understand the motion of a system that is noninertial because of a change in di-
rection, consider a car traveling along a highway at a high speed and approaching a
curved exit ramp, as shown in Figure 6.11a. 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 by the door on the passenger keeps her from being ejected from
the car. What causes her to move toward the door? A popular but incorrect explanation
is that a force acting toward the right in Figure 6.11b pushes her outward. This is often
called the “centrifugal force,” but it is a fictitious force due to the acceleration associated
with the changing direction of the car’s velocity vector. (The driver also experiences this
effect but wisely holds on to the steering wheel to keep from sliding to the right.)

The phenomenon is correctly explained as follows. Before the car enters the ramp,

the passenger is moving in a straight-line path. As the car enters the ramp and travels a
curved path, the passenger tends to move along the original straight-line path. This is
in accordance with Newton’s first law: the natural tendency of an object is to continue
moving in a straight line. However, if a sufficiently large force (toward the center of
curvature) acts on the passenger, as in Figure 6.11c, she moves in a curved path along
with the car. This force is the force of friction between her and the car seat. If this fric-
tion force is not large enough, she slides to the right as the seat turns to the left under
her. Eventually, she encounters the door, which provides a force large enough to en-
able her to follow the same curved path as the car. She slides toward the door not be-
cause of an outward force but because

the force of friction is not sufficiently great

to allow her to travel along the circular path followed by the car.

Another interesting fictitious force is the “Coriolis force.” This is an apparent force

caused by changing the radial position of an object in a rotating coordinate system. For
example, suppose you and a friend are on opposite sides of a rotating circular platform
and you decide to throw a baseball to your friend. As Figure 6.12a shows, at t ! 0 you
throw the ball toward your friend, but by the time t

when the ball has crossed the plat-

form, your friend has moved to a new position.

Figure 6.12a represents what an observer would see if the ball is viewed while the

observer is hovering at rest above the rotating platform. According to this observer,
who is in an inertial frame, the ball follows a straight line, as it must according to New-
ton’s first law. Now, however, consider the situation from your friend’s viewpoint. Your
friend is in a noninertial reference frame because he is undergoing a centripetal ac-
celeration relative to the inertial frame of the Earth’s surface. He starts off seeing the
baseball coming toward him, but as it crosses the platform, it veers to one side, as
shown in Figure 6.12b. Thus, your friend on the rotating platform claims that the ball

(a)

(c)

(b)

Figure 6.11 (a) A car approaching

a curved exit ramp. What causes a

front-seat passenger to move toward

the right-hand door? (b) From the

frame of reference of the passenger,

a force appears to push her toward

the right door, but this is a fictitious

force. (c) Relative to the reference

frame of the Earth, the car seat ap-

plies a leftward force to the passen-

ger, causing her to change direction

along with the rest of the car.

160

CHAPTE R 6 • Circular Motion and Other Applications of Newton’s Laws

Quick Quiz 6.6

Consider the passenger in the car making a left turn in Fig-

ure 6.11. Which of the following is correct about forces in the horizontal direction if
the person is making contact with the right-hand door? (a) The passenger is in equilib-
rium between real forces acting to the right and real forces acting to the left. (b) The
passenger is subject only to real forces acting to the right. (c) The passenger is subject
only to real forces acting to the left. (d) None of these is true.

does not obey Newton’s first law and claims that a force is causing the ball to follow a
curved path. This fictitious force is called the Coriolis force.

Fictitious forces may not be real forces, but they can have real effects. An object

on your dashboard really slides off if you press the accelerator of your car. As you ride
on a merry-go-round, you feel pushed toward the outside as if due to the fictitious
“centrifugal force.” You are likely to fall over and injure yourself if you walk along a ra-
dial line while the merry-go-round rotates. The Coriolis force due to the rotation of
the Earth is responsible for rotations of hurricanes and for large-scale ocean currents.

▲

PITFALL PREVENTION

6.2 Centrifugal Force

The  commonly  heard  phrase
“centrifugal force” is described as
a force pulling outward on an ob-
ject  moving  in  a  circular  path.  If
you  are  feeling  a  “centrifugal
force” on a rotating carnival ride,
what  is  the  other  object  with
which  you  are  interacting?  You
cannot  identify  another  object
because  this  is  a  fictitious  force
that occurs as a result of your be-
ing  in  a  noninertial  reference
frame.

(a)

(b)

Active Figure 6.12 (a) You and your friend sit at the edge of a rotating turntable. In

this overhead view observed by someone in an inertial reference frame attached to the

Earth, you throw the ball at t ! 0 in the direction of your friend. By the time t

that the

ball arrives at the other side of the turntable, your friend is no longer there to catch it.

According to this observer, the ball followed a straight line path, consistent with

Newton’s laws. (b) From the point of view of your friend, the ball veers to one side

during its flight. Your friend introduces a fictitious force to cause this deviation from

the expected path. This fictitious force is called the “Coriolis force.”

Example 6.8 Fictitious Forces in Linear Motion

A small sphere of mass m is hung by a cord from the ceiling
of a boxcar that is accelerating to the right, as shown in Fig-
ure  6.13.  The  noninertial  observer  in  Figure  6.13b  claims
that a force, which we know to be fictitious, must act in or-
der  to  cause  the  observed  deviation  of  the  cord  from  the
vertical. How is the magnitude of this force related to the ac-
celeration  of  the  boxcar  measured  by  the  inertial  observer
in Figure 6.13a?

Solution According to the inertial observer at rest (Fig.
6.13a), the forces on the sphere are the force

T exerted by

the cord and the gravitational force. The inertial observer
concludes that the acceleration of the sphere is the same as
that of the boxcar and that this acceleration is provided by
the horizontal component of

T. Also, the vertical compo-

nent of

T balances the gravitational force because the

sphere is in equilibrium in the vertical direction. Therefore,

At the Active Figures link

at http://www.pse6.com, you

can observe the ball’s path

simultaneously from the

reference frame of an inertial

observer and from the

reference frame of the rotating

turntable.

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