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ROTATIONAL MOTION

In 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 lineNewtons first law of motion. In Chapter 8 this concept was extended when you learned about momentum. In the absence of an external force, the momen-tum of an object remains unchangedcon-servation of momentum. In this chapter we extend the law of momentum conservation to rotation.

Rotating objects tend to keep rotating while non-rotating objects tend to remain non-rotating.

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What Makes an Object Easy to Rotate?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 fin-ger with the clay at the top end of the ruler.

4. Now try balancing a pencil and then a meter-stick on your fingertip.

Analyze and Conclude1. 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 distri-bution of mass in an object that is easy to bal-ance affect its tendency to remain balanced?

discover!

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ROTATIONAL MOTION

Objectives Describe how the rotational

inertia of an object affects how easily the rotational speed of the object changes. (12.1)

List the three principal axes of rotation in the human body. (12.2)

Describe what happens when objects of the same shape but different sizes are rolled down an incline. (12.3)

Explain how Newtons 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!MATERIALS 12 ruler, clay, pencil, meterstick

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 The clay makes the ruler easier to balance. The meterstick is easier to balance than the pencil.

A meterstick with balls of clay near the 40-cm and 60-cm marks would be easier to rotate.

The greater the distance between the bulk of the mass and the axis of rotation, the easier it is to balance an object.

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2.

3.

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CHAPTER 12 ROTATIONAL MOTION 213

12.1 Rotational InertiaNewtons 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-rotat-ing. The resistance of an object to changes in its rotational motion is called rotational inertia (sometimes called the moment of inertia).

The greater an objects 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.

Rotational Inertia and Mass Like inertia in the linear sense, rotational inertia depends on mass. But unlike inertia, rotational iner-tia depends on the distribution of the mass. As illustrated in Figure 12.1, the greater the distance between an objects mass concentration and the axis of rotation, the greater the rotational inertia. The tight-rope walker shown in Figure 12.2 increases his rotational inertia by holding a long pole, allowing him to resist rotation.

think!

FIGURE 12.1Rotational inertia depends on the distance of mass from the axis of rotation.

FIGURE 12.2By holding a long pole, the tightrope walker increases his rotational inertia.

When swinging your leg from your hip, why is the rotational inertia of the leg less when it is bent?Answer: 12.1

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12.1 Rotational Inertia

Key Termrotational 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 objects 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.


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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, doesnt want to swing as readily. Similarly, as illustrated in Figure 12.3, the short pen-dulum 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 illus-trates 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 eas-ily. To reduce the rotational inertia of your legs, simply bend them. Thats an important reason for running with your legs bentbent legs are easier to swing back and forth.

FIGURE 12.3 The short pendulum will swing back and forth more frequently than the long pendulum.

FIGURE 12.4 For similar mass distribu-tions, short legs have less rotational inertia than long legs.

FIGURE 12.5 You bend your legs when you run to reduce their rotational inertia.


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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!

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 inertiathe more it resists a change in rotation.)

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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 com-pares rotational inertias for various shapes and axes. (It is not impor-tant for you to learn these values, but you can see how they vary with the shape and axis.)

CONCEPTCHECK ...

... How does rotational inertia affect how easily the rotational speed of an object changes?

What is the Easiest Way to Rotate Your Pencil?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 pencils 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 iner-tia consistent with the small r?

discover!

FIGURE 12.6Rotational inertias of various objects are different.

Rotational inertia depends very much on the location of the axis of rotation. A meter-stick rotated about one end, for example, has four times the rotational inertia that it has when rotated about its center.

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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.

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.

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!MATERIALS pencilEXPECTED 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.

THINK Rotation is easiest when the axis of rotation is the lead. In this case r is as small as possible.

discover!MATERIALS pencilEXPECTED 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.

THINK Rotation is easiest when the axis of rotation is the lead. In this case r is as small as possible.


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12.2 Rotational Inertia and GymnasticsConsider 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.

Longitudinal Axis Rotational inertia is least about the longitu-dinal 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.)

Theres plenty of physics in sports!

FIGURE 12.7 The human body has three principal axes of rotation.

FIGURE 12.8 An ice skater rotates around her longitudinal axis when going into a spin.


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The greater an objects rotational

inertia, the more difficult it is to change the rotational speed of the object.

T e a c h i n g R e s o u r c e s

Reading and Study Workbook

Concept-Development Practice Book 12-1

Transparency 18

PresentationEXPRESS

Interactive Textbook

12.2 Rotational Inertia and Gymnastics

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.

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Transverse Axis You rotate about your transverse axis when you perform a somersault or a flip. Figure 12.9 shows the rotational iner-tia 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 bodys 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.

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 rota-tion and is used in executing a cartwheel. Like rotations about the other axes, rotational inertia can be varied with different body configurations.

CONCEPTCHECK ...... What are the three principal axes of rotation in the

human body?

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 gymnasts rotational inertia is 5 times greater than in the tuck position.

FIGURE 12.10The 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 somer-saults in the tuck position.

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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.)

The three principal axes of rotation in

the human body are the longitudinal axis, the transverse axis, and the medial axis.



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12.3 Rotational Inertia and RollingIn Figure 12.11, which will roll down the incline with greater accel-eration, 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 inertiathe hollow or the solid cylinder? The answer is, the one with its mass concentrated farthest from the axis of rotationthe 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 gain-ing speed. The solid cylinder will roll with greater acceleration.

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.

CONCEPTCHECK ...

... What happens when objects of the same shape but different sizes are rolled down an incline?

think!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

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

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.


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12.3 Rotational Inertia and Rolling

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 fasterhoops 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.

Objects of the same shape but different

sizes accelerate equally when rolled down an incline.



Laboratory Manual 41, 42

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12.4 Angular MomentumAnything 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, .

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 vec-tor, have the same direction and lie along the axis of rotation.)

In this book, we wont treat the vector nature of angular momen-tum (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 Earths gravity acts to change the direction of the bicycle wheels angular momentum (which is along the wheels 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.

FIGURE 12.12 The turntable has more angular momentum when it is turning at 45 RPM than at

1333 RPM. It has even more

angular momentum if a load is placed on it so its rota-tional inertia is greater.

FIGURE 12.13 The gyroscope is a remark-able device. a. The opera-tion 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 Earths gravity acts on it.

a b

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12.4 Angular Momentum

Key Termslinear momentum, angular momentum, rotational velocity

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 distancethe object you previously swung overhead.

Common MisconceptionWater drains backward in the Southern Hemisphere due to Earths rotation.

FACT Test this for yourself. Youll find it can flow either way, depending on the sinks 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 energythe 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.


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Figure 12.14 illustrates the case of an object that is small com-pared 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 mag-nitude of its linear momentum, mv, multiplied by the radial distance, r. In equation form,

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. Newtons 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 unbal-anced external torque.

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.

CONCEPTCHECK ...

... How does Newtons first law apply to rotating systems?

FIGURE 12.14 An object of concentrated mass m whirling in a cir-cular path of radius r with a speed v has angular momentum mvr.

FIGURE 12.15 The lightweight wheels on racing bikes have less angu-lar momentum than those on recreational bikes, so it takes less effort to get them turning.


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Teaching Tip For the more general case, angular momentum is simply the product of rotational inertia I and angular velocity v.

Teaching Tip Rotational KE is the energy of an object due to its rotational motion. In keeping with the spirit of information overload, we dont treat rotational kinetic energy in this book. FYI, KErot 5 1/2 lv

2.

Newtons 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.



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12.5 Conservation 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 rota-tional 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 over-all 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 mysteri-ous. But its 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.

FIGURE 12.16When the man pulls his arms and the whirling weights inward, he decreases his rotational inertia, and his rotational speed corre-spondingly increases.

For:

Visit:Web Code:

Links on rotational motion

www.SciLinks.org csn 1205

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12.5 Conservation of Angular Momentum

Key Termlaw of conservation of angular momentum

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 Earths rotation? It tends to slow Earths 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.

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 Earths 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.

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.

DemonstrationDemonstration


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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.

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 momen-tum remains zero. When it is over, the cat is not turning. This maneu-ver 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-angular-momentum rotations about any principal axis to orient their bodies in any preferred direction when floating in space.

CONCEPTCHECK ...

... What happens to angular momentum when no net external torque acts on an object?

FIGURE 12.18 After being dropped upside down, the cat rotates so it can land on its feet.

FIGURE 12.17 Rotational speed is controlled by variations in the bodys rotational inertia as angular momen-tum is conserved during a forward somersault.


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Common MisconceptionA 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.



Problem-Solving Exercises in Physics 7-3

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...Show the operation of a gyroscopeeither 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, youll 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.)

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12.6 Simulated GravityIn Chapter 10, we considered a ladybug in a rotating frame of refer-ence. 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 tires 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 rotat-ing 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.

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.

FIGURE 12.19If the spinning wheel freely falls, the ladybugs inside will experience a centrifugalforce that feels like gravitywhen the wheel spins at the appropriate rate.

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.

Link to ASTRONOMY

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12.6 Simulated Gravity

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.

Teaching Tidbit The law of angular momentum conservation is an everynight fact of life to astronomers.


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Support Force Occupants in todays space vehicles feel weight-less because they lack a support force. Theyre 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.

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 grav-ity, it has no reaction counterpartthere 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 Earths surface is due to gravity. Inside a rotating space-ship the acceleration experienced is the centripetal/centrifugal accel-eration 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.

ba

FIGURE 12.20 The man inside this rotating space habitat experiences simu-lated 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.


224

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.

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.


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Small-diameter structures would have to rotate at high speeds to provide a simulated gravitational acceleration of l g. Sensitive and deli-cate 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 structureone almost 2 km in diameter. This is an immense structure compared with the size of todays 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 liv-ing 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.

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 gwithin 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 liveespecially for those who will be prepared to play a role in these new adventures.12.6

CONCEPTCHECK ...... How is gravity simulated?

FIGURE 12.21This NASA depiction of a rotational space colony may be a glimpse into the future.

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. Clarkes 1973 book Rendezvous with Rama, is credited to the Russian scientist Konstantin Tsiolkovsky in 1920.

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225

From within a rotating frame of

reference, there seems to be an outwardly directed centrifugal force, which can simulate gravity.



Concept-Development Practice Book 12-2

PresentationEXPRESS


CONCEPTCHECK ...

...CONCEPTCHECK ...

...


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1 REVIEWConcept Summary

The greater an objects rotational inertia, the more difficult it is to change the rota-tional 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.

Newtons first law of inertia for rotating systems states that an object or system of objects will maintain its angular momen-tum 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 refer-ence, 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)

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 hoops rotational inertia is greater than a solid

cylinders ( 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 spheres 15 mr

2.

think! Answers

For:Visit:Web Code:

Self-Assessment PHSchool.com csa 1200


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REVIEW


TeacherEXPRESS

Virtual Physics Lab 14

ASSESS

Check Concepts 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.


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CHAPTER 12 ROTATIONAL MOTION 227CHAPTER 12 ROTATIONAL MOTION 227

1 ASSESSCheck 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 12.2 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 rota-tional inertia while spinning?

Section 12.3 9. Which will have the greater acceleration

rolling down an inclinea hoop or a solid disk?

Section 12.4 10. Distinguish between linear momentum and

angular momentum.

11. The text says that angular momentum is I,then says it is mvr. Which is it?

12. Momentum is conserved when there is no net external force. When is angular momen-tum conserved?

Section 12.5 13. What does it mean to say that angular mo-

mentum 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?

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?

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227

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.

12. When there is no net external torque

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 doesnt 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.


228228

1 ASSESSConcept Summary

ASSESS (continued)

228

1Think and 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.

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.

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, Perkys an-gular momenta.

b. Rank from greatest to least, the amounts of friction needed to keep Perky from sliding off.

22. Students Art, Bart, Cis, and Dot sit on a ro-tating turntable at different distances from the center as indicated.

(A) Art, m 60 kg, sits at r.

(B) Bart, m 25 kg, sits at r.

(C) Cis, m 50 kg, sits at r.

(D) Dot, m 20 kg, sits at r.

34

12

14

From greatest to least, rank the angular momenta of the four students.

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228

Think and Rank 19. C, B, A

20. B, A, C

21. a. A, B, C b. A, B, C

22. C, D, B, A


11/19/07 10:58:30 AM




Think and Explain 23. Mei Fan says that a basketball has greater ro-

tational inertia than a solid ball of the same size and mass because most of a basketballs 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 doesnt 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 bal-ancing an empty pole?

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 inclinea bowling ball or a volleyball? Defend your answer.

29. Any rolling object takes more time to roll down an inclined plane than a non-roll-ing 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 doesnt think a sliding object slides down an incline faster than a rolling object. With whom do you agree?

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 rotat-ing 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?

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 Earths inhabitants moved to the equator, how would this affect Earths rotational inertia? How would it affect the length of a day?

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229

Think and Explain 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.

24. The bare stick has less rotational inertia and hits the ground first.

25. The stick with the wad of clay at the top has more rotational inertia.

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, Jims analysis is correct. A can of liquid beats a can of solid for the reason stated.

31. One filled with waterit has more rotational inertia.

32. Decrease, in accordance with the conservation of angular momentum.

33. The soil moves farther away from Earths axis. By conservation of angular momentum, like a skater extending their arms, Earths rotation slows, making the days slightly longer.

34. More mass at the equator means that more mass is concentrated further away from Earths axis and therefore Earths rotational inertia increases. By conservation of angular momentum, Earths rotation would be slower, making the days slightly longer.


230230


ASSESS (continued)

230

1 35. If the worlds populations move to the

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 Earths 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?

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.

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 conser-vation of angular momentum contribute to the galaxys present shape and why it rotates faster now than when it was a larger, spheri-cal cloud.

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?

41. Explain why the faster Earth spins, the less a person weighs, whereas the faster a space station spins, the more a person weighs.


230

35. By conservation of angular momentum, less mass at the equator increases the speed of Earths rotation, making the days slightly shorter.

36. By conservation of angular momentum, more mass at the equator slows Earths 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.

40. The occupant experiences a centripetal force that provides support. By Newtons 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 dont press as hard against a weighing scale. But in a space station, we are thrown against the weighing scale and we weigh more.


11/13/07 4:36:14 PM




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 rota-tion. What is the rotational inertia of the two-pod system about its midpoint? Express your answers in kg.m2.

46. The two-pod system in the previous question rotates 1.2 RPM to provide artificial grav-ity for its occupants. If one of the pods pulls in 100 m of cable (bringing the pods closer together), what will be the systems new rota-tion rate?

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?

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 short-ened to 0.60 m the speed of the puck will be 3.6 m/s.

Activity 50. Gather a selection of canned foods. Pre-

dict 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.

More Problem-Solving PracticeAppendix F

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231

Think and Solve 42. From I 5 mr 2, twice the

mass and half the r give I 5 2m(r/2)2 5 l/2, half the original rotational inertia.

43. From I 5 mr 2, twice the mass and twice the length give I 5 2m(2r)2 5 8l, eight times the original rotational inertia.

44. From I 5 mr 2, half the mass and half the length give I 5 (m/2)(r/2)2 5 l / 8, one-eighth the original rotational inertia.

45. I 5 mr 2 5 (1000 kg)(450 m)2 5 2 3 108 kg?m2 (for each pod); 4 3 108 kg?m2 (for two-pod system)

46. (Iv)bef 5 (Iv)aft so: vaft 5 (Iv)bef/ Iaft 5 (mr 2v)bef/(mr

2)aft 5

(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 pucks 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 50. Liquids roll faster because

the PE becomes mainly translational KE with little rotational KE.


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