Objectives
• Describe how the rotational
inertia of an object affects how
easily the rotational speed of
the object changes. (12.1)
1
• List the three principal axes of
rotation in the human body.
(12.2)
THE BIG
IDEA
• Describe what happens when
objects of the same shape but
different sizes are rolled down
an incline. (12.3)
• Explain how Newton’s first law
applies to rotating systems.
(12.4)
• Describe what happens to
angular momentum when no
net external torque acts on an
object. (12.5)
• Explain how gravity can be
simulated. (12.6)
discover!
ROTATIONAL
MOTION
Rotating objects tend to keep
rotating while non-rotating objects
tend to remain non-rotating.
..........
ROTATIONAL
MOTION
I
n Chapter 3 you learned about inertia: An
object at rest tends to stay at rest, and an
object in motion tends to remain moving in
a straight line—Newton’s first law of motion.
In Chapter 8 this concept was extended
when you learned about momentum. In the
absence of an external force, the momentum of an object remains unchanged—conservation of momentum. In this chapter we
extend the law of momentum conservation
to rotation.
12” ruler, clay,
pencil, meterstick
MATERIALS
EXPECTED OUTCOME Students
will observe that an object
whose mass is concentrated at
one end is easier to balance
than an object whose mass is
evenly distributed.
ANALYZE AND CONCLUDE
1. The clay makes the ruler
easier to balance. The
meterstick is easier to
balance than the pencil.
2. A meterstick with balls of
clay near the 40-cm and
60-cm marks would be
easier to rotate.
discover!
What Makes an Object Easy to Rotate?
Analyze and Conclude
1. Try balancing a 12” ruler upright on the tip of
your finger.
2. Mold a large ball of clay around one end of
the ruler.
3. Try balancing the ruler on the tip of your finger with the clay at the top end of the ruler.
4. Now try balancing a pencil and then a meterstick on your fingertip.
1. Observing How did the addition of the clay
affect your ability to balance the ruler on your
fingertip? Which was easier to balance, the
pencil or the meterstick?
2. Predicting Which would be easier to rotate
back and forth, a meterstick held at its center
with balls of clay at the ends or a meterstick
with balls of clay near the 40 cm and 60 cm
marks?
3. Making Generalizations How does the distribution of mass in an object that is easy to balance affect its tendency to remain balanced?
3. The greater the distance
between the bulk of
the mass and the axis of
rotation, the easier it is to
balance an object.
212
212
12.1 Rotational
Inertia
12.1 Rotational Inertia
Newton’s first law, the law of inertia, also applies to rotating objects.
In every case in which an object is rotating about an internal axis, the
object tends to keep rotating about that axis. Rotating objects tend to
keep rotating, while non-rotating objects tend to remain non-rotating. The resistance of an object to changes in its rotational motion is
called rotational inertia (sometimes called the moment of inertia).
The greater an object’s rotational inertia, the more difficult it is
to change the rotational speed of the object.
Just as it takes a force to change the linear state of motion of an
object, a torque is required to change the rotational state of motion of
an object. In the absence of a net torque, a rotating top keeps rotating,
while a non-rotating top stays non-rotating.
think!
When swinging your leg
from your hip, why is the
rotational inertia of the
leg less when it is bent?
Answer: 12.1
Key Term
rotational inertia
Teaching Tip Compare the
concept of inertia and its role
in linear motion to rotational
inertia (sometimes called
“moment of inertia”) and
its role in rotational motion.
The difference between the
two involves the role of radial
distance from a rotational axis.
The greater the distance of mass
concentration, the greater the
resistance to rotation.
Teaching Tip Explain how
the location of an object’s mass
with respect to its axis of rotation
determines its rotational inertia.
The rotational inertia of an
object is a measure of how much
it resists turning.
FIGURE 12.1
Rotational inertia depends
on the distance of mass
from the axis of rotation.
Rotational Inertia and Mass Like inertia in the linear sense,
rotational inertia depends on mass. But unlike inertia, rotational inertia depends on the distribution of the mass. As illustrated in Figure
12.1, the greater the distance between an object’s mass concentration
and the axis of rotation, the greater the rotational inertia. The tightrope walker shown in Figure 12.2 increases his rotational inertia by
holding a long pole, allowing him to resist rotation.
FIGURE 12.2
By holding a long pole, the
tightrope walker increases
his rotational inertia.
CHAPTER 12
ROTATIONAL MOTION
213
213
Demonstrations
Have two 1-meter pipes, one
with two lead plugs in the
center, the other with plugs
in each end. They appear
identical. Weigh both to
show the same weight. Give
one (with plugs in ends) to a
student and ask him or her
to rotate it about its center.
Have another student do the
same with the pipe that has
the plugs in the middle. Then
have them switch. Ask for
speculations as to why one
was noticeably more difficult
to rotate.
Have students try
to balance on a
finger a long upright
stick with a massive
lead weight at one
end. Try it first with
the weight at the
bottom, and then
with the weight at
the top. (Rotational
inertia is greater for
the stick when it is
made to rotate with
the massive part far
from the pivot than
when it is closer.
The farther the
mass, the greater the
rotational inertia—the more it
resists a change in rotation.)
FIGURE 12.3 The short pendulum will
swing back and forth more
frequently than the long
pendulum.
FIGURE 12.4 For similar mass distributions, short legs have less
rotational inertia than
long legs.
FIGURE 12.5 Relate the continuous
adjustments necessary to keep
the object balanced in the
previous demonstration to the
balanced electric scooters in
Figure 11.20 and to the similar
adjustments that must be made
in keeping a rocket vertical
when it is first fired. Amazing!
214
You bend your legs when
you run to reduce their
rotational inertia.
214
A long baseball bat held near its thinner end has more rotational
inertia than a short bat of the same mass. Once moving, it has a
greater tendency to keep moving, but it is harder to bring it up to
speed. A short bat has less rotational inertia than a long bat, and is
easier to swing. Baseball players sometimes “choke up” on a bat by
grasping it closer than normal to the more massive end. Choking up
on the bat reduces its rotational inertia and makes it easier to bring
up to speed. A bat held at its end, or a long bat, doesn’t “want” to
swing as readily. Similarly, as illustrated in Figure 12.3, the short pendulum has less rotational inertia and therefore swings back and forth
more frequently than the long pendulum. Long-legged animals such
as giraffes, horses, and ostriches normally run with a slower gait than
hippos, dachshunds, and mice. The chihuahua shown in Figure 12.4
runs with quicker strides than his longer-legged friend.
It is important to note that the rotational inertia of an object is
not necessarily a fixed quantity. It is greater when the mass within
the object is extended from the axis of rotation. Figure 12.5 illustrates how you can try this with your outstretched legs. Swing your
outstretched leg back and forth from the hip. Now do the same with
your leg bent. In the bent position it swings back and forth more easily. To reduce the rotational inertia of your legs, simply bend them.
That’s an important reason for running with your legs bent—bent
legs are easier to swing back and forth.
FIGURE 12.6
Rotational inertias of various
objects are different.
......
Formulas for Rotational Inertia When all the mass m of an
object is concentrated at the same distance r from a rotational axis
(as in a simple pendulum bob swinging on a string about its pivot
point, or a thin wheel turning about its center), then the rotational
inertia I = mr 2. When the mass is more spread out, as in your leg, the
rotational inertia is less and the formula is different. Figure 12.6 compares rotational inertias for various shapes and axes. (It is not important for you to learn these values, but you can see how they vary with
the shape and axis.)
Teaching Tip Explain why
the same formula applies to
the pendulum and the hoop
in Figure 12.6. (All the mass of
each is at the same distance from
the rotational axis.) State how
reasonable the smaller value for
a solid disk is, given that much of
its mass is close to the rotational
axis. Compare the effort required
to change the rotation of the
various figures.
Rotational inertia
depends very much on
the location of the axis
of rotation. A meterstick rotated about one
end, for example, has
four times the rotational
inertia that it has when
rotated about its center.
CONCEPT How does rotational inertia affect how easily the
CHECK
For some of your students, the
formulas for the rotational
inertia of common shapes may
be seen as a threat. It would
be counterproductive for your
students to think they are
learning any physics at all if
they memorize these formulas.
rotational speed of an object changes?
Ask Supposing the shapes
in Figure 12.6 all have the same
mass and radius (or length),
which would be the easiest to
start spinning about the axes
indicated? The stick about its
CG, as indicated by its small
rotational inertia Which would
be the most difficult to start
(or stop) spinning? The simple
pendulum or the hoop about its
normal axis, as indicated by their
relatively large rotational inertias
discover!
discover!
MATERIALS
What is the Easiest Way to Rotate Your Pencil?
pencil
EXPECTED OUTCOME Students
will find that it is easier to flip
the pencil when it is flipped
about its midpoint rather than
flipped about one of its ends.
But, rotation is easiest when
the pencil is rotated about its
long axis.
1. Flip your pencil back and forth between your fingers.
2. Compare the ease of rotation when you flip it about
its midpoint versus flipping it about one of its ends.
3. For a third comparison, rotate the pencil between your
thumb and forefinger about the pencil’s long axis
(so the lead is the axis).
4. Study the three cases shown in Figure 12.6.
5. Think In which case is rotation easiest?
In this case, is the small rotational inertia consistent with the small r?
Rotation is easiest when
the axis of rotation is the lead.
In this case r is as small as
possible.
THINK
CHAPTER 12
ROTATIONAL MOTION
215
215
......
The greater an
object’s rotational
inertia, the more difficult it is to
change the rotational speed of
the object.
CONCEPT
CHECK
12.2 Rotational Inertia and Gymnastics
There’s plenty of
physics in sports!
Teaching Resources
• Reading and Study
Workbook
• Concept-Development
Practice Book 12-1
Consider the human body. As shown in Figure 12.7, you can rotate
freely about three principal axes of rotation. The three principal
axes of rotation in the human body are the longitudinal axis, the
transverse axis, and the medial axis. Each of these axes is at right
angles to the others (mutually perpendicular) and passes through
the center of gravity. The rotational inertia of the body differs about
each axis.
• Transparency 18
• PresentationEXPRESS
• Interactive Textbook
12.2 Rotational
Inertia and
Gymnastics
FIGURE 12.7 The human body has three
principal axes of rotation.
Section 12.2 usually generates
high interest and shows
applications that your students
can experience for themselves.
Teaching Tip Point out that
the rotational inertia about any
of the axes shown in Figure 12.7
does not depend on the direction
of spin.
Teaching Tip Also emphasize
that the rotational inertia of an
object depends on the choice of
rotational axis.
FIGURE 12.8 216
216
An ice skater rotates around
her longitudinal axis when
going into a spin.
Longitudinal Axis Rotational inertia is least about the longitudinal axis, which is the vertical head-to-toe axis, because most of the
mass is concentrated along this axis. Thus, a rotation of your body
about your longitudinal axis is the easiest rotation to perform. An ice
skater executes this type rotation when going into a spin. Rotational
inertia is increased by simply extending a leg or the arms. The skater
has the least amount of rotational inertia when her arms are tucked
in, as shown in Figure 12.8a. The rotational inertia when both arms
are extended, as in Figure 12.8b, is about three times more than in
the tucked position, so if you go into a spin with outstretched arms,
you will triple your spin rate when you draw your arms in. With your
leg extended as well, as in Figures 12.8c and 12.8d, you can vary your
spin rate by as much as six times. (We will see why this happens in
Section 12.5.)
Transverse Axis You rotate about your transverse axis when you
perform a somersault or a flip. Figure 12.9 shows the rotational inertia of different positions, from the least (when your arms and legs
are drawn inward in the tuck position) to the greatest (when your
arms and legs are fully extended in a line). The relative magnitudes of
rotational inertia stated in the caption are with respect to the body’s
center of gravity.
Rotational inertia is greater when the axis is through the hands,
such as when doing a somersault on the floor or swinging from a
horizontal bar with your body fully extended. In Figure 12.10, the
rotational inertia of a gymnast is up to 20 times greater when she
is swinging in a fully extended position from a horizontal bar than
after dismount when she somersaults in the tuck position. Rotation
transfers from one axis to another, from the bar to a line through her
center of gravity, and she automatically increases her rate of rotation
by up to 20 times. This is how she is able to complete two or three
somersaults before contact with the ground.
Teaching Tip State that just
as the body can change shape
and orientation, the rotational
inertia of the body can change
also. Discuss Figures 12.7 through
12.10 and explain how rotational
inertia is different for the same
body configuration about
different axes. (Refer back to
Figure 12.10 when discussing
conservation of angular
momentum.)
FIGURE 12.9 A flip involves rotation
about the transverse axis.
a. Rotational inertia is
least in the tuck position.
b. Rotational inertia is 1.5
times greater than in the
tuck position. c. Rotational
inertia is 3 times greater
than in the tuck position.
d. The gymnast’s rotational
inertia is 5 times greater
than in the tuck position.
FIGURE 12.10
......
Medial Axis The third axis of rotation for the human body is the
front-to-back axis, or medial axis. This is a less common axis of rotation and is used in executing a cartwheel. Like rotations about the other
axes, rotational inertia can be varied with different body configurations.
CHECK
• Reading and Study
Workbook
• PresentationEXPRESS
• Interactive Textbook
human body?
CHAPTER 12
CONCEPT
Teaching Resources
CONCEPT What are the three principal axes of rotation in the
CHECK
The three principal
axes of rotation in
the human body are the
longitudinal axis, the transverse
axis, and the medial axis.
......
The rotational inertia of
a body is with respect to
the rotational axis. a. The
gymnast has the greatest
rotational inertia when she
pivots about the bar. b. The
axis of rotation changes
from the bar to a line
through her center of
gravity when she somersaults in the tuck position.
ROTATIONAL MOTION
217
217
12.3 Rotational
12.3 Rotational Inertia and Rolling
Inertia and
Rolling
In Figure 12.11, which will roll down the incline with greater acceleration, the hollow cylinder or the solid cylinder of the same mass
and radius? The answer is the cylinder with the smaller rotational
inertia. Why? Because the cylinder with the greater rotational inertia
requires more time to get rolling. Remember that inertia of any kind
is a measure of “laziness.” Which has the greater rotational inertia—
the hollow or the solid cylinder? The answer is, the one with its mass
concentrated farthest from the axis of rotation—the hollow cylinder.
So a hollow cylinder has a greater rotational inertia than a solid
cylinder of the same radius and mass and will be more “lazy” in gaining speed. The solid cylinder will roll with greater acceleration.
Teaching Tip Relate the
acceleration of an object rolling
down an incline to its rotational
inertia. For similar masses, shapes
with greater rotational inertia
(the “laziest”) lag behind shapes
with less rotational inertia.
Ask Predict which will roll
down an incline faster—hoops
or solid cylinders. Try it and
see! Any solid cylinder will beat
any hoop. Mass does not play
a role. The acceleration down
an incline has to do with the
rotational inertia per kilogram.
In effect the mass gets canceled
out.
FIGURE 12.11 A solid cylinder rolls down
an incline faster than a
hollow one, whether or
not they have the same
mass or diameter.
Just as objects of any
mass in free fall have
equal accelerations,
round objects of any
mass having the same
shape roll down an
incline with the same
acceleration.
......
Objects of the same
shape but different
sizes accelerate equally when
rolled down an incline.
......
CONCEPT
Interestingly enough, any solid cylinder will roll down an incline
with more acceleration than any hollow cylinder, regardless of mass
or radius. A hollow cylinder has more “laziness per mass” than a solid
cylinder. Objects of the same shape but different sizes accelerate
equally when rolled down an incline. You should experiment and
see this for yourself. If started together, the smaller shape, whether it
be a ball, disk, or hoop, rotates more times than the larger shape, but
both reach the bottom of the incline in the same time. Why? Because
all objects of the same shape have the same “laziness per mass”
ratio. Similarly, recall from Chapter 6 how the same “weight per
mass” ratio of all freely falling objects accounted for their equal
acceleration: a = F/m.
CONCEPT What happens when objects of the same shape but
CHECK
CHECK
different sizes are rolled down an incline?
think!
Teaching Resources
A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down
an incline. Which will have more acceleration?
Answer: 12.3.1
• Reading and Study
Workbook
• Laboratory Manual 41, 42
Would you expect the rotational inertia of a hollow sphere about its center to
be greater or less than the rotational inertia of a solid sphere? Defend your
answer.
Answer: 12.3.2
• PresentationEXPRESS
• Interactive Textbook
• Next-Time Questions 12-1,
12-2
218
218
12.4 Angular
12.4 Angular Momentum
Momentum
Anything that rotates, whether it be a cylinder rolling down an
incline or an acrobat doing a somersault, keeps on rotating until
something stops it. A rotating object has a “strength of rotation.”
Recall from Chapter 8 that all moving objects have “inertia of
motion,” or momentum. This kind of momentum, which is
called linear momentum, is the product of the mass and the
velocity of an object. Rotating objects have angular momentum.12.4
Angular momentum is defined as the product of rotational
inertia, I, and rotational velocity, ω.
angular momentum rotational inertia (I) rotational velocity (/)
Like linear momentum, angular momentum is a vector quantity
and has direction as well as magnitude. When a direction is assigned
to rotational speed, we call it rotational velocity. Rotational velocity
is a vector whose magnitude is the rotational speed. (By convention,
the rotational velocity vector, as well as the angular momentum vector, have the same direction and lie along the axis of rotation.)
In this book, we won’t treat the vector nature of angular momentum (or even of torque, which also is a vector) except to acknowledge
the remarkable action of the gyroscope. Low-friction swivels can be
turned in any direction without exerting a torque on the whirling
gyroscope shown in Figure 12.13a. As a result, it stays pointed in the
same direction. Similarly, the rotating bicycle wheel in Figure 12.13b
shows what happens when a torque by Earth’s gravity acts to change
the direction of the bicycle wheel’s angular momentum (which is
along the wheel’s axle). The pull of gravity that acts to topple the
wheel over and change its rotational axis causes it instead to precess in
a circular path about a vertical axis. You must do this yourself while
standing on a turntable to fully believe it. Full understanding will
likely not come until a later time.
a
Key Terms
linear momentum,
angular momentum,
rotational velocity
FIGURE 12.12 The turntable has more
angular momentum when it
is turning at 45 RPM than at
1
33 3 RPM. It has even more
angular momentum if a load
is placed on it so its rotational inertia is greater.
Teaching Tip Just as inertia
and rotational inertia differ by
a radial distance, and just as
force and torque also differ by
a radial distance, momentum
and angular momentum differ
by a radial distance. Relate
linear momentum to angular
momentum for the case of a
small mass at a relatively large
radial distance—the object you
previously swung overhead.
Common Misconception
Water drains backward in the
Southern Hemisphere due to Earth’s
rotation.
FIGURE 12.13 The gyroscope is a remarkable device. a. The operation of a gyroscope relies on
the vector nature of angular
momentum. b. Angular
momentum keeps the wheel
axle almost horizontal when
a torque supplied by Earth’s
gravity acts on it.
FACT Test this for yourself.
You’ll find it can flow either
way, depending on the sink’s
structure. The variations in
gravity across the bowl of water
(which produces the “Coriolis
effect”) are much too weak to
affect its direction of flow.
Teaching Tidbit Watch for
the re-emergence of an ancient
way of storing energy—the
flywheel. A flywheel stores the
energy that was used to make it
spin, and it retains that energy as
long as the wheel is free to turn.
Slow down the flywheel, and you
can draw some of that energy
back out. A flywheel takes the
place of a chemical battery and is
more environmentally sound. No
toxic metals or hazardous waste
are involved.
b
CHAPTER 12
ROTATIONAL MOTION
219
219
Teaching Tip For the more
general case, angular momentum
is simply the product of
rotational inertia I and angular
velocity v.
Figure 12.14 illustrates the case of an object that is small compared with the radial distance to its axis of rotation. In such cases as
a tin can swinging from a long string or a planet orbiting in a circle
around the sun, the angular momentum is simply equal to the magnitude of its linear momentum, mv, multiplied by the radial distance,
r. In equation form,
Teaching Tip Rotational KE
is the energy of an object due to
its rotational motion. In keeping
with the spirit of “information
overload,” we don’t treat
rotational kinetic energy in this
book. FYI, KErot 5 1/2 lv2.
angular momentum mvr
FIGURE 12.14 An object of concentrated
mass m whirling in a circular path of radius r with
a speed v has angular
momentum mvr.
Just as an external net force is required to change the linear
momentum of an object, an external net torque is required to change
the angular momentum of an object. Newton’s first law of inertia
for rotating systems states that an object or system of objects will
maintain its angular momentum unless acted upon by an unbalanced external torque.
FIGURE 12.15 ......
Newton’s first law of
inertia for rotating
systems states that an object or
system of objects will maintain its
angular momentum unless acted
upon by an unbalanced external
torque.
CONCEPT
The lightweight wheels on
racing bikes have less angular momentum than those
on recreational bikes, so it
takes less effort to get them
turning.
CHECK
• Reading and Study
Workbook
We know it is easier to balance on a moving bicycle than on one
at rest. The spinning wheels, such as those shown on the bikes in
Figure 12.15, have angular momentum. When our center of gravity is
not above a point of support, a slight torque is produced. When the
wheels are at rest, we fall over. But when the bicycle is moving, the
wheels have angular momentum, and a greater torque is required to
change the direction of the angular momentum. The moving bicycle
is easier to balance than a stationary bike.
• PresentationEXPRESS
CONCEPT How does Newton’s first law apply
......
Teaching Resources
CHECK
• Interactive Textbook
220
220
to rotating systems?
12.5 Conservation
12.5 Conservation of Angular
Momentum
of Angular
Momentum
Just as the linear momentum of any system is conserved if no net
force acts on the system, angular momentum is conserved for systems
in rotation. The law of conservation of angular momentum states
that if no unbalanced external torque acts on a rotating system, the
angular momentum of that system is constant. This means that with
no net external torque, the product of rotational inertia and rotational velocity at one time will be the same as at any other time.
Angular momentum is conserved when no net external torque
acts on an object.
An interesting example of angular momentum conservation is
shown in Figure 12.16. The man stands on a low-friction turntable
with weights extended. Because of the extended weights his overall rotational inertia is relatively large in this position. As he slowly
turns, his angular momentum is the product of his rotational inertia
and rotational velocity. When he pulls the weights inward, his overall
rotational inertia is considerably decreased. What is the result? His
rotational speed increases! This is best appreciated by the turning
person who feels changes in rotational speed that seem to be mysterious. But it’s straight physics! This procedure is used by a figure skater
who starts to whirl with her arms and perhaps a leg extended, and
then draws her arms and leg in to obtain a greater rotational speed.
Whenever a rotating body contracts, its rotational speed increases.
Key Term
law of conservation of angular
momentum
Demonstration
For: Links on rotational
motion
Visit: www.SciLinks.org
Web Code: csn – 1205
With weights in your hands,
rotate on a platform or
rotating stool as shown in
Figure 12.16. Show angular
momentum conservation by
drawing your arms in and
speeding up.
Ask Much of the mass that
flows down the Mississippi River
as mud is deposited in the Gulf
of Mexico. What effect does
this tend to have on Earth’s
rotation? It tends to slow Earth’s
rotation.
Teaching Tip Simulate this
on a rotating table by keeping
one weight held outstretched at
about 45° and then lowering it
toward the horizontal. Students
will see a slowing of your
rotational speed.
FIGURE 12.16
When the man pulls his arms and
the whirling weights inward, he
decreases his rotational inertia,
and his rotational speed correspondingly increases.
Ask When polar ice melts,
the melted water tends to flow
toward the equatorial parts
of Earth, effectively spreading
the mass of ice away from the
polar axis. What effect does
this tend to have on Earth’s
rotation? Again, it tends to slow
the rotation.
Teaching Tip Simulate
this on the rotating table with
two weights held outstretched
overhead and lowered toward
the horizontal. Students will see
even more slowing of rotational
speed.
CHAPTER 12
ROTATIONAL MOTION
221
221
Demonstration
Similarly, when a gymnast is spinning freely, as shown in Figure
12.17, angular momentum does not change. However, rotational
speed can be changed by making variations in rotational inertia. This
is done by moving some part of the body toward or away from the
axis of rotation.
Show the operation of a
gyroscope—either a model
or a rotating bicycle wheel.
You can demonstrate angular
momentum conservation
nicely if you stand on the
turntable or rotating stool and
show different orientations
of the spinning wheel. Begin
with the axes of the spinning
wheel and stationary rotating
platform perpendicular. Then
turn the axis of the spinning
wheel so it is parallel to the
axis of the platform. If you
had no angular momentum
initially, you’ll rotate in a
direction opposite to that of
the spinning wheel to produce
the same zero angular
momentum. Different angles
produce different components
of angular momentum with
interesting effects. (The
angular momentum vector is
along the axis of rotation. Let
the spin be in the direction of
your curled fingers on your
right hand. Then the angular
momentum vector is in the
direction of your thumb.)
FIGURE 12.17 Rotational speed is controlled by variations in
the body’s rotational inertia as angular momentum is conserved during a forward somersault.
Common Misconception
A falling cat will always land on its
feet.
FACT Most often this is true. But
when dropped upside down from
heights less than one foot, ouch!
......
Angular momentum
is conserved when no
net external torque acts on an
object.
CONCEPT
CHECK
Teaching Resources
• Reading and Study
Workbook
• Problem-Solving Exercises in
Physics 7-3
FIGURE 12.18 After being dropped upside
down, the cat rotates so it
can land on its feet.
• PresentationEXPRESS
• Interactive Textbook
222
222
......
The cat shown in Figure 12.18 is held upside down and dropped
but is able to execute a twist and land upright even if it has no initial
angular momentum. Zero-angular-momentum twists and turns are
performed by turning one part of the body against the other. While
falling, the cat rearranges its limbs and tail. Repeated reorientations
of the body configuration result in the head and tail rotating one way
and the feet the other, so that the feet are downward when the cat
reaches the ground. During this maneuver the total angular momentum remains zero. When it is over, the cat is not turning. This maneuver rotates the body through an angle, but does not create continuing
rotation. To do so would violate angular momentum conservation.
Humans can perform similar twists without difficulty, though not
as fast as a cat can. Astronauts have learned to make zero-angularmomentum rotations about any principal axis to orient their bodies
in any preferred direction when floating in space.
CONCEPT What happens to angular momentum when no net
CHECK
external torque acts on an object?
12.6 Simulated
12.6 Simulated Gravity
Gravity
In Chapter 10, we considered a ladybug in a rotating frame of reference. Now consider a colony of ladybugs living inside a bicycle tire,
as shown in Figure 12.19 below. If we toss the wheel through the air
or drop it from an airplane high in the sky, the ladybugs will be in a
weightless condition and seem to float freely while the wheel is in free
fall. Now spin the wheel. The ladybugs will feel themselves pressed to
the outer part of the tire’s inner surface. If the wheel is spun at just
the right speed, the ladybugs will experience simulated gravity that
feels like the gravity they are accustomed to. From within a rotating frame of reference, there seems to be an outwardly directed
centrifugal force, which can simulate gravity. Gravity is simulated
by centrifugal force. To the ladybugs, the direction “up” is toward the
center of the wheel. The “down” direction to the ladybugs is what we
call “radially outward,” away from the center of the wheel.
Simulated gravity usually
generates high interest. If time
is short, this section may be
given as a reading assignment.
Teaching Tip Spur students’
imagination with an assigned
short essay of what life might be
like in a rotating space station.
Ask them to consider such things
as how various sports would
differ, and what kinds of new
games could be played. Tell them
that all answers should be based
on physics principles.
FIGURE 12.19
If the spinning wheel freely
falls, the ladybugs inside
will experience a centrifugal
force that feels like gravity
when the wheel spins at the
appropriate rate.
Teaching Tidbit The law of
angular momentum conservation
is an everynight fact of life to
astronomers.
Need for Simulated Gravity Today we live on the outer surface
of our spherical planet, held here by gravity. Earth has been the cradle
of humankind. But we will not stay in the cradle forever. We are on our
way to becoming a spacefaring people. In the years ahead many people
will likely live in huge lazily rotating space stations where simulated
gravity will be provided so the people can function normally.
Link to ASTRONOMY
Spiral Galaxies The shapes of galaxies such as our Milky Way have much to do
with the conservation of angular momentum. Consider a globular mass of gas in
space that begins to contract under the influence of its own gravity. If it has even
the slightest rotation about some axis, it has some angular momentum, which must
be conserved. As the gas contracts, its rotational inertia decreases. Then, like a
spinning ice skater who draws her arms inward, the ball of gas spins faster. As it
does so, it is flattened, just as our spinning Earth is flattened at its poles. If the
glob has enough angular momentum, it turns into a flat pancake with a diameter far greater than its
thickness, and may become a spiral galaxy.
CHAPTER 12
ROTATIONAL MOTION
223
223
Teaching Tip Discuss rotating
space habitats. Show how g
varies with the radial distance
from the hub, and with the
rotational rate of the structure.
Support Force Occupants in today’s space vehicles feel weightless because they lack a support force. They’re not pressed against
a supporting floor by gravity, nor do they experience a centrifugal
force due to spinning. But future space travelers need not be subject
to weightlessness. Their space habitats will probably spin, like the
ladybugs’ spinning bicycle wheel, effectively supplying a support force
and nicely simulating gravity.
Earth has been the cradle
of humankind; but humans do
not live in the cradle forever.
We may eventually leave our
cradle and inhabit structures
of our own building, structures
that will serve as lifeboats
for planet Earth. Such a
prospect is exciting from both a
technological and a social point
of view.
a
b
FIGURE 12.20 The man inside this rotating space habitat experiences simulated gravity. a. As seen from the outside, the only force
exerted on the man is by the floor. b. As seen from the inside,
there is a fictitious centrifugal force that simulates gravity.
The interaction between the man and the floor of a space habitat,
as seen at rest outside the rotating system, is shown in Figure 12.20a.
The floor presses against the man (action) and the man presses back
on the floor (reaction). The only force exerted on the man is by the
floor. It is directed toward the center and is a centripetal force. As
seen from inside the rotating system, in Figure 12.20b, in addition to
the man-floor interaction there is a centrifugal force exerted on the
man at his center of mass. It seems as real as gravity. Yet, unlike gravity, it has no reaction counterpart—there is nothing out there that he
can pull back on. Centrifugal force is not part of an interaction, but
results from rotation. It is therefore called a fictitious force.
Challenges of Simulated Gravity The comfortable 1 g we
experience at Earth’s surface is due to gravity. Inside a rotating spaceship the acceleration experienced is the centripetal/centrifugal acceleration due to rotation. The magnitude of this acceleration is directly
proportional to the radial distance and the square of the rotational
speed. For a given RPM, the acceleration, like the linear speed, increases
with increasing radial distance. Doubling the distance from the axis
of rotation doubles the centripetal/centrifugal acceleration. At the
axis where radial distance is zero, there is no acceleration due
to rotation.
224
224
Small-diameter structures would have to rotate at high speeds to
provide a simulated gravitational acceleration of l g. Sensitive and delicate organs in our inner ears sense rotation. Although there appears
to be no difficulty at a single revolution per minute (1 RPM) or so,
many people have difficulty adjusting to rotational rates greater than
2 or 3 RPM (although some people easily adapt to 10 or so RPM). To
simulate normal Earth gravity at 1 RPM requires a large structure—
one almost 2 km in diameter. This is an immense structure compared
with the size of today’s space shuttle vehicles. Economics will probably
dictate that the size of the first inhabited structures be small. If these
structures also do not rotate, the inhabitants will have to adjust to living in a seemingly weightless environment. Larger rotating habitats
with simulated gravity will likely follow later. Imagine yourself living
in a rotating space colony such as the one shown in Figure 12.21.
The idea of a rotating
space station to keep
astronauts’ feet on
the floor, wonderfully
shown in the 1968
movie 2001: A Space
Odyssey, and in Arthur
C. Clarke’s 1973 book
Rendezvous with
Rama, is credited to
the Russian scientist
Konstantin Tsiolkovsky
in 1920.
FIGURE 12.21
This NASA depiction of a
rotational space colony may
be a glimpse into the future.
......
If the structure rotates so that inhabitants on the inside of the
outer edge experience 1 g, then halfway between the axis and the
outer edge they would experience only 0.5 g. At the axis itself they
would experience weightlessness at 0 g. The possible variations of g
within the rotating space habitat holds promise for a most different
and as yet unexperienced environment. We could perform ballet at
0.5 g; acrobatics at 0.2 g and lower g states; three-dimensional soccer
and sports not yet conceived in very low g states. People will explore
possibilities never before available to them. This time of transition
from our earthly cradle to new vistas is an exciting time in which to
live—especially for those who will be prepared to play a role in these
new adventures.12.6
CONCEPT
CHECK
......
From within a
rotating frame of
reference, there seems to be an
outwardly directed centrifugal
force, which can simulate gravity.
CONCEPT
CHECK
Teaching Resources
• Reading and Study
Workbook
• Concept-Development
Practice Book 12-2
• PresentationEXPRESS
• Interactive Textbook
How is gravity simulated?
CHAPTER 12
ROTATIONAL MOTION
225
225
REVIEW
Teaching Resources
• TeacherEXPRESS
• Virtual Physics Lab 14
ASSESS
Check Concepts
1
1. An object rotating about an
axis tends to keep rotating
about that axis.
•
2. Yes; when the mass of an
object is concentrated farther
from the axis there is greater
rotational inertia.
•
3. Held closer to the massive
end. There is less rotational
inertia and therefore less
torque is required to make it
rotate.
•
4. A cylinder
5. The short one will swing
quicker in a to-and-fro motion
due to less rotational inertia.
6. Bent legs have less rotational
inertia than long straight legs.
7. Longitudinal axis, transverse
axis, and medial axis
8. By pulling their arms inward
and “balling up”
9. A solid disk because it has less
rotational inertia per mass
and will therefore have the
greater acceleration
10. Linear momentum is the
“strength of motion,” mv.
Angular momentum is the
“strength of rotation,” which
is given by mvr for a particle
in a circle, or more generally,
Iv.
226
REVIEW
Concept Summary
•
•
•
••••••
The greater an object’s rotational inertia,
the more difficult it is to change the rotational speed of the object.
The three principal axes of rotation in the
human body are the longitudinal axis, the
transverse axis, and the medial axis.
Objects of the same shape but different
sizes accelerate equally when rolled down
an incline.
Newton’s first law of inertia for rotating
systems states that an object or system of
objects will maintain its angular momentum unless acted upon by an unbalanced
external torque.
Angular momentum is conserved when
no net external torque acts on an object.
From within a rotating frame of reference, there seems to be an outwardly
directed centrifugal force, which can
simulate gravity.
Key Terms
••••••
rotational inertia (p. 213)
linear momentum (p. 219)
angular momentum (p. 219)
rotational velocity (p. 219)
law of conservation of
angular momentum (p. 221)
226
For: Self-Assessment
Visit: PHSchool.com
Web Code: csa – 1200
think! Answers
12.1
The rotational inertia of any object is less
when its mass is concentrated closer to the
axis of rotation. Can you see that a bent leg
satisfies this requirement?
12.3.1 The cylinders have different masses, but the
same rotational inertia per mass, so both
will accelerate equally down the incline.
Their different masses make no difference,
just as the acceleration of free fall is not
affected by different masses. All objects of
the same shape have the same “laziness per
mass” ratio.
12.3.2 Greater. Just as the value mr2 for a hoop’s
rotational inertia is greater than a solid
cylinder’s ( 12 mr2), the rotational inertia of
a hollow sphere would be greater than that
of a same-mass solid sphere for the same
reason: the mass of the hollow sphere is
farther from the center. A thin spherical
shell has a rotational inertia of 23 mr2,
or 1.66 times greater than a solid sphere’s
1
2
5 mr .
1
11. Both are the same but
expressed differently. Use
Iv for an internal axis of
rotation. Use mvr for an
external axis and when the
size of the object is small
compared with r.
ASSESS
12. When there is no net external
torque
Check Concepts
••••••
Section 12.1
1. What is the law of inertia for rotation?
2. Does the rotational inertia of an object
differ for different axes of rotation?
3. Which is easier to get swinging, a baseball
bat held at the end, or one held closer to the
massive end (choked up)?
4. Which has a greater rotational inertia, a
cylinder about its axis or a sphere about a
diameter if the two have the same mass and
radius?
5. Which will swing to and fro more often, a
short pendulum or a long pendulum?
Section 12.4
10. Distinguish between linear momentum and
angular momentum.
11. The text says that angular momentum is Iω,
then says it is mvr. Which is it?
12. Momentum is conserved when there is no
net external force. When is angular momentum conserved?
Section 12.5
13. What does it mean to say that angular momentum is conserved?
14. If a skater who is spinning pulls her arms in
so as to reduce her rotational inertia to half,
by how much will her angular momentum
increase?
Section 12.2
6. Why does bending your legs when running
enable you to swing your legs to and fro
more rapidly?
7. What are the three principal axes of rotation
for the human body?
8. How does a skater decrease his or her rotational inertia while spinning?
Section 12.3
9. Which will have the greater acceleration
rolling down an incline—a hoop or a solid
disk?
13. Angular momentum of the
system at one time 5 angular
momentum at another time.
Iv or mvr remains constant
when there is no net external
torque.
14. None at all! Angular
momentum is conserved!
15. Her spin rate doubles (while
angular momentum remains
constant).
16. In reading, take care to
answer the question asked!
Angular momentum doesn’t
change (14) but spin rate does
(15).
17. Gravity is simulated by a
fictitious, centrifugal force if
the space station spins.
18. Acceleration is directly
proportional to the radial
distance from the hub, so
larger radii yield greater g for
the same RPM.
15. If a skater who is spinning pulls her arms in
so as to reduce her rotational inertia to half,
by how much will her rate of spin increase?
16. Why are your answers different to the
previous two questions?
Section 12.6
17. How can gravity be simulated in an orbiting
space station?
18. How will the value of g vary at different
distances from the hub of a rotating space
station?
CHAPTER 12
ROTATIONAL MOTION
227
227
Think and Rank
19. C, B, A
20. B, A, C
21. a. A, B, C
b. A, B, C
1
ASSESS
(continued)
22. C, D, B, A
Concept
Think
and
Summary
Rank ••••••
••••••
Rank each of the following sets of scenarios in
order of the quantity or property involved. List
them from left to right. If scenarios have equal
rankings, then separate them with an equal sign.
(e.g., A ⫽ B)
19. Three iron shapes of the same mass rotate
about the axis shown by the circle with the
dot inside. Rank them, from greatest to
least, in terms of the rotational inertia about
this axis.
21. Perky rides at different radial distances
from the center of a turntable that rotates
at a fixed rate. His distances and tangential
speeds at three different locations are as
follows.
(A) r 15 cm, v 7.5 cm/s
(B) r 10 cm, v 5.0 cm/s
(C) r 5.0 cm, v 2.5 cm/s
a. Rank from greatest to least, Perky’s angular momenta.
b. Rank from greatest to least, the amounts
of friction needed to keep Perky from
sliding off.
20. Beginning from a rest position, a solid disk
A, a solid ball B, and a hoop C, race down
an incline. Rank them in order of finishing:
winner, second place, and third place.
22. Students Art, Bart, Cis, and Dot sit on a rotating turntable at different distances from
the center as indicated.
1
4 r.
(B) Bart, m 25 kg, sits at 12 r.
(C) Cis, m 50 kg, sits at 34 r.
(A) Art, m 60 kg, sits at
(D) Dot, m 20 kg, sits at r.
From greatest to least, rank the angular
momenta of the four students.
228
228
Think and Explain
1
23. Agree with Mei Fan. In
general, the farther the mass
concentration of a body
from its axis of rotation, the
greater the rotational inertia.
ASSESS
Think
and
Explain ••••••
••••••
Concept
Summary
23. Mei Fan says that a basketball has greater rotational inertia than a solid ball of the same
size and mass because most of a basketball’s
mass is far from its center. Ashley says no,
that the center of mass of any uniform ball
is at its center, and mass distribution doesn’t
matter. Whom do you agree with?
24. Stand two metersticks against the wall and
let them topple over. Now put a wad of clay
on top of one of the sticks and let them
topple again. Which reaches the floor first?
25. Why is a stick with a wad of clay at the top
easier to balance on the palm of your hand
than an empty stick?
26. At the circus, a performer balances his
friends at the top of a vertical pole. Why is
this feat easier for the performer than balancing an empty pole?
24. The bare stick has less
rotational inertia and hits the
ground first.
30. Jim says that in a race between a can of
water and a can of ice rolling down an
incline, the water filled can will win because
the water inside “slides” down the incline,
while the ice is made to rotate, slowing its
movement down the incline. John now
agrees. Do you agree?
31. Consider two rotating bicycle wheels, one
filled with air and the other filled with
water. Which would be more difficult to
stop rotating? Explain.
32. You sit in the middle of a large, freely rotating turntable at an amusement park. If
you crawled toward the outer rim, would
your rotational speed increase, decrease, or
remain unchanged? What law of physics
supports your answer?
29. Any rolling object takes more time to roll
down an inclined plane than a non-rolling object sliding without friction. Jim says
this is because all the PE of the non-rolling
object goes into translational KE, with none
“wasted” as rotational KE. John doesn’t
think a sliding object slides down an incline
faster than a rolling object. With whom do
you agree?
26. The pole with people at
the top has more rotational
inertia.
27. It increases your rotational
inertia and helps you to
balance by making your body
more resistant to toppling.
28. A bowling ball (solid)—it has
less rotational inertia per
mass.
29. Agree with Jim. All the
energy goes into translational
KE so a non-rolling object will
travel down an incline faster
than a rolling object.
30. Yes, Jim’s analysis is correct.
A can of liquid beats a can of
solid for the reason stated.
31. One filled with water—it has
more rotational inertia.
32. Decrease, in accordance with
the conservation of angular
momentum.
27. If you walked along the top of a fence, why
would holding your arms out help you to
balance?
28. Which will have the greater acceleration
rolling down an incline—a bowling ball or a
volleyball? Defend your answer.
25. The stick with the wad of clay
at the top has more rotational
inertia.
33. A sizable quantity of soil is washed down
the Mississippi River and deposited in the
Gulf of Mexico each year. What effect does
this tend to have on the length of a day?
(Hint: Relate this to a spinning skater who
extends her arms outward.)
34. If all of Earth’s inhabitants moved to the
equator, how would this affect Earth’s
rotational inertia? How would it affect the
length of a day?
CHAPTER 12
ROTATIONAL MOTION
229
33. The soil moves farther
away from Earth’s axis. By
conservation of angular
momentum, like a skater
extending their arms, Earth’s
rotation slows, making the
days slightly longer.
34. More mass at the equator
means that more mass
is concentrated further
away from Earth’s axis
and therefore Earth’s
rotational inertia increases.
By conservation of angular
momentum, Earth’s rotation
would be slower, making the
days slightly longer.
229
35. By conservation of angular
momentum, less mass at the
equator increases the speed
of Earth’s rotation, making
the days slightly shorter.
36. By conservation of angular
momentum, more mass at
the equator slows Earth’s
rotation, making the days
slightly longer.
37. When the train moves
clockwise, the tracks move
counterclockwise. When
the train backs up and
moves counterclockwise,
the tracks move clockwise.
Angular momentum of the
train-wheel system remains
constant. Motion of the track
would be more pronounced.
Motion of the train would be
more pronounced.
38. To conserve angular
momentum; if the small rotor
fails, an external torque acts
on the helicopter, causing it
to rotate (spin).
39. Gravity pulled the particles
closer to the axis of rotation
and therefore the rotational
inertia of the galaxy
decreased. According to
conservation of angular
momentum, the galaxy began
to rotate faster. An increase
in speed throws may stars into
a dish-like shape.
1
ASSESS
(continued)
35. If the world’s
populations move
to the
Concept
Summary
••••••
North and South Poles, would the length of
a day increase, decrease, or stay the same?
36. If the polar ice caps of Earth were to melt,
the oceans everywhere would be deeper by
about 30 m. What effect would this have on
Earth’s rotation?
37. A toy train is initially at rest on a track
fastened to a bicycle wheel, which is free to
rotate. How does the wheel respond when
the train moves clockwise? When the train
backs up? Does the angular momentum
of the wheel-train system change during
these maneuvers? How would the resulting
motions be affected if the train were much
more massive than the track? If the track
were much more massive?
40. An occupant inside a rotating space habitat
of the future will feel pulled by artificial
gravity against the outer wall of the habitat
(which becomes the “floor”). What physics
provides an explanation?
40. The occupant experiences a
centripetal force that provides
support. By Newton’s third
law, the occupant pushes
back against the “floor”
and interprets that push as
gravity.
41. The faster Earth spins, the
more we tend to be thrown
off it, so we don’t press as
hard against a weighing scale.
But in a space station, we are
thrown against the weighing
scale and we weigh more.
230
39. We believe our galaxy was formed from a
huge cloud of gas. The original cloud was
far larger than the present size of the galaxy,
more or less spherical, and rotating very
much more slowly than the galaxy is now. In
this sketch we see the original cloud and the
galaxy as it is now (seen edgewise). Explain
how the law of gravitation and the conservation of angular momentum contribute to
the galaxy’s present shape and why it rotates
faster now than when it was a larger, spherical cloud.
38. Why does a typical small helicopter with a
single main rotor have a second small rotor
on its tail? Describe the consequence if the
small rotor fails in flight.
230
41. Explain why the faster Earth spins, the less
a person weighs, whereas the faster a space
station spins, the more a person weighs.
Think and Solve
1
42. From I 5 mr2, twice the
mass and half the r give I⬘
5 2m(r/2)2 5 l/2, half the
original rotational inertia.
ASSESS
Concept
Summary
••••••
Think
and
Solve ••••••
42. What happens to the rotational inertia of a
simple pendulum when the mass of the bob
is doubled and the length of the pendulum
is halved? (See Figure 12.6.)
43. What happens to the rotational inertia of
a simple pendulum when both the mass
of the bob and the length of the pendulum
are doubled?
44. What happens to the rotational inertia of a
simple pendulum when both the mass
of the bob and the length of the pendulum
are halved?
45. A pair of identical 1000-kg space pods in
outer space are connected to each other by
a 900-m-long cable. They rotate about a
common point like a spinning dumbbell as
shown in the figure. Calculate the rotational
inertia of each pod about the axis of rotation. What is the rotational inertia of the
two-pod system about its midpoint? Express
your answers in kg.m2.
43. From I 5 mr2, twice the mass
and twice the length give
I⬘ 5 2m(2r)2 5 8l, eight times
the original rotational inertia.
46. The two-pod system in the previous question
rotates 1.2 RPM to provide artificial gravity for its occupants. If one of the pods pulls
in 100 m of cable (bringing the pods closer
together), what will be the system’s new rotation rate?
44. From I 5 mr2, half the mass
and half the length give I⬘ 5
(m/2)(r/2)2 5 l/8, one-eighth
the original rotational inertia.
47. Gretchen moves at a speed of 6.0 m/s when
sitting on the edge of a horizontal rotating
platform of diameter 4.0 m. Her mass is
45 kg. Show that her angular momentum
about the center of the platform will be
540 kg.m2/s.
48. If a trapeze artist rotates twice each second
while sailing through the air, and contracts
to reduce her rotational inertia to one-third,
how many rotations per second will result?
46. (Iv)bef 5 (Iv)aft so:
vaft 5 (Iv)bef /Iaft 5
(mr2v)bef /(mr2)aft 5
49. A 0.60-kg puck revolves at 2.4 m/s at the
end of a 0.90-m string on a frictionless air
table. Show that when the string is shortened to 0.60 m the speed of the puck will
be 3.6 m/s.
45. I 5 mr2 5 (1000 kg)(450 m)2 5
2 3 108 kg?m2 (for each pod);
4 3 108 kg?m2 (for two-pod
system)
(4502/4002)(1.2 RPM) 5
1.5 RPM.
47. Angular momentum 5 mvr 5
(45 kg)(6.0 m/s)(4.0 m/2) 5
540 kg?m2/s
48. (Iv)bef 5 (Iv)aft, so vaft 5
(Ibef)(2 rot/s)/(0.33Ibef) 5 6 rot/s.
49. The puck’s angular
momentum is conserved:
mvoro 5 mvnewrnew so
vnew 5 (voro/rnew) 5
(2.4 m/s)(0.90 m)/(0.60 m) 5
3.6 m/s.
Activity
Activity
••••••
50. Gather a selection of canned foods. Predict which will roll faster down an incline.
Compare liquids (which slide or slosh rather
than roll inside the can) and solids. Roll the
cans to test your predictions. Describe your
results.
50. Liquids roll faster because
the PE becomes mainly
translational KE with little
rotational KE.
Teaching Resources
More Problem-Solving Practice
Appendix F
CHAPTER 12
ROTATIONAL MOTION
231
• Computer Test Bank
• Chapter and Unit Tests
231