mhs physics problems chapter 9

7/24/2019 MHS Physics Problems Chapter 9

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Chapter 9. Rotational Dynamics

Conceptual Questions

ssm Solution is in the Student Solutions Manual

1. Sometimes, even with a wrench, one cannot loosen a nut that is frozen tightly to a

bolt. It is often possible to loosen the nut by slipping a long pipe over the wrench handle.The purpose of the pipe is to extend the length of the handle, so that the applied force can

be located farther away from the nut. Explain why this trick works.

2. Explain (a) how it is possible for a large force to produce only a small, or even zero,

torque, and (b) how it is possible for a small force to produce a large torque.

3. A magnetic tape is being played on a cassette deck. The tension in the tape applies a

torque to the supply reel. Assuming the tension remains constant during playback, discuss

how this torque varies as the reel becomes empty.

4. A flat rectangular sheet of plywood can rotate about an axis perpendicular to the sheet

through one corner. How should a force (acting in the plane of the sheet) be applied to the

plywood so as to create the largest possible torque? Give your reasoning.

5. The drawing shows two tubes of toothpaste. The tubes themselves are the same, and

the cap on each tube is threaded and must be removed by unscrewing it. However, onecap is advertised as being an easy-off cap, the other being a normal cap. Using the

concept of torque, explain why the easy-off cap is easier to unscrew than the normal

cap.


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6. Suppose you are standing on a train, both feet together, facing a window. The front of

the train is to your left. The train starts moving forward. To keep from falling, you slide

your right foot out toward the rear of the train. Explain in terms of torque how this actionkeeps you from falling over.

7. Starting in the spring, fruit begins to grow on the outer end of a branch on a pear tree.

Explain how the center of gravity of the peargrowing branch shifts during the course of

the summer.

8. ssmThe free-body diagram in the drawing shows the forces that act on a thin rod. The

three forces are drawn to scale and lie in the plane of the paper. Are these forcessufficient to keep the rod in equilibrium, or are additional forces necessary? Explain.

9. An A-shaped stepladder is standing on frictionless ground. The ladder consists of two

sections joined at the top and kept from spreading apart by a horizontal crossbar. Draw a


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free-body diagram showing the forces that keep onesection of the ladder in equilibrium.

10. The drawing shows a wine rack for a single bottle of wine that seems to defy

common sense as it balances on a table top. Treat the rack and wine bottle as a rigid bodyand draw the external forces that keep it in equilibrium. In particular, where must the

center of gravity of the rigid body be located? Give your reasoning. (Hint: The physics

here is the same as in Figure9.10c.)

11. A flat triangular sheet of uniform material is shown in the drawing. There are threepossible axes of rotation, each perpendicular to the sheet and passing through one corner,

A, B, or C. For which axis is the greatest net external torque required to bring the triangle

up to an angular speed of 10.0 rad/s in 10.0 s, starting from rest? Explain, assuming that

the net torque is kept constant while it is being applied.

12. An object has an angular velocity. It also has an angular acceleration due to torques

that are present. Therefore, the angular velocity is changing. What happens to the angular

velocity if (a) additional torques are applied so as to make the net torque suddenly equal

to zero and (b) all the torques are suddenly removed?

13. The space probe in the drawing is initially moving with a constant translational

velocity and zero angular velocity. (a) When the two engines are fired, each generating athrust of magnitude T, will the translational velocity increase, decrease, or remain the

same? Why? (b) Explain what will happen to the angular velocity.
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14. ssmSit-ups are more difficult to do with your hands placed behind your head insteadof on your stomach. Why?

15. For purposes of computing the translational kinetic energy of a rigid body, its masscan be considered as concentrated at the center of mass. If one wishes to compute thebodys moment of inertia, can the mass be considered as concentrated at the center of

mass? If not, why not?

16. A thin sheet of plastic is uniform and has the shape of an equilateral triangle.Consider two axes for rotation. Both are perpendicular to the plane of the triangle, axis A

passing through the center of the triangle and axis B passing through one corner. If the

angular speed about each axis is the same, for which axis does the triangle have the

greater rotational kinetic energy? Explain.

17. Bob and Bill have the same weight and wear identical shoes. Keeping both feet flat

on the floor and the body straight, Bob can lean over farther than Bill can before falling.Whose center of gravity is closer to the ground? Account for your answer.

18. A hoop, a solid cylinder, a spherical shell, and a solid sphere are placed at rest at the

top of an incline. All the objects have the same radius. They are then released at the same

time. What is the order in which they reach the bottom? Justify your answer.

19. A woman is sitting on the spinning seat of a piano stool with her arms folded. What

happens to her (a) angular velocity and (b) angular momentum when she extends her

arms outward? Justify your answers.


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20. ReviewExample 14as an aid in answering this question. Suppose the ice cap at the

South Pole melted and the water was distributed uniformly over the earths oceans.

Would the earths angular velocity increase, decrease, or remain the same? Explain.

21. The concept behind this question is discussed inExample 14.Many rivers, like the

Mississippi River, flow from north to south toward the equator. These rivers often carry a

large amount of sediment that they deposit when entering the ocean. What effect doesthis redistribution of the earths soil have on the angular velocity of the earth? Why?

22. Example 14provides background for this question. A cloud of interstellar gas is

rotating. Because the gravitational force pulls the gas particles together, the cloudshrinks, and, under the right conditions, a star may ultimately be formed. Would the

angular velocity of the star be less than, equal to, or greater than, the angular velocity of

the rotating gas? Justify your answer.

23. ssmA person is hanging motionless from a vertical rope over a swimming pool. She

lets go of the rope and drops straight down. After letting go, is it possible for her to curl

into a ball and start spinning? Justify your answer.

24. The photograph shows a workman struggling to keep a stack of boxes balanced on a

dolly. The mans right foot is on the axle of the dolly. Assuming that the boxes are

identical, which one creates the greatest torque with respect to the axle? Why?
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Problems

Note to Instructors: Most of the homework problems in this chapter are available forassignment via an on-line homework management program such as Wiley PLUS or

WebAssign, and those marked with the icon are presented in a guided tutorial formatthat provides enhanced interactivity. See Preface for additional details.

ssmSolution is in the Student Solutions Manual.

wwwSolution is available on the World Wide Web atwww.wiley.com/college/cutnell.

This icon represents a biomedical application.

Section9.1The Action of Forces and Torques on Rigid Objects

1. ssmYou are installing a new spark plug in your car, and the manual specifies that it

be tightened to a torque that has a magnitude of 45 N m. Using the data in the drawing,determine the magnitudeFof the force that you must exert on the wrench.

2. In San Francisco a very simple technique is used to turn around a cable car when it

reaches the end of its route. The car rolls onto a turntable, which can rotate about avertical axis through its center. Then, two people push perpendicularly on the car, one at

each end, as shown in the drawing. The turntable is rotated one-half of a revolution to

turn the car around. If the length of the car is 9.20 m and each person pushes with a 185-

N force, what is the magnitude of the net torque applied to the car?
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3. Interactive Solution 9.3atwww.wiley.com/college/cutnellpresents a model for

solving this problem. The wheel of a car has a radius of 0.350 m. The engine of the carapplies a torque of 295 N m to this wheel, which does not slip against the road surface.

Since the wheel does not slip, the road must be applying a force of static friction to the

wheel that produces a countertorque. Moreover, the car has a constant velocity, so this

countertorque balances the applied torque. What is the magnitude of the static frictionalforce?

4. The drawing shows a lower leg being exercised. It has a 49-N weight attached tothe foot and is extended at an angle with respect to the vertical. Consider a rotational

axis at the knee. (a) When = 90.0 , find the magnitude of the torque that the weight

creates. (b) At what angle does the magnitude of the torque equal 15 N m?

5. ssmThe steering wheel of a car has a radius of 0.19 m, while the steering wheel of atruck has a radius of 0.25 m. The same force is applied in the same direction to each.

What is the ratio of the torque produced by this force in the truck to the torque produced
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in the car?

6. The drawing shows a jet engine suspended beneath the wing of an airplane. The

weight of the engine is 10 200 N and acts as shown in the drawing. In flight the engineproduces a thrust of 62 300 N that is parallel to the ground. The rotational axis in thedrawing is perpendicular to the plane of the paper. With respect to this axis, find themagnitude of the torque due to (a) the weight and (b) the thrust.

7. ssmOne end of a meter stick is pinned to a table, so the stick can rotate freely in aplane parallel to the tabletop. Two forces, both parallel to the tabletop, are applied to the

stick in such a way that the net torque is zero. One force has a magnitude of 2.00 N and is

applied perpendicular to the length of the stick at the free end. The other force has a

magnitude of 6.00 N and acts at a 30.0 angle with respect to the length of the stick.Where along the stick is the 6.00-N force applied? Express this distance with respect to

the end that is pinned.

8. Interactive LearningWare 9.1atwww.wiley.com/college/cutnellreviews theconcepts that are important in this problem. One end of a meter stick is pinned to a table,

so the stick can rotate freely in a plane parallel to the tabletop (see the drawing). Two

forces, both parallel to the tabletop, are applied to the stick in such a way that the nettorque is zero. One force has a magnitude of 4.00 N and is applied perpendicular to the

stick at the free end. The other force has a magnitude of 6.00 N and acts at a 60.0 angle

with respect to the stick. Where along the stick is the 6.00-N force applied? Express this

distance with respect to the axis of rotation.
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9. A pair of forces with equal magnitudes, opposite directions, and different lines of

action is called a couple. When a couple acts on a rigid object, the couple produces atorque that does notdepend on the location of the axis. The drawing shows a couple

acting on a tire wrench, each force being perpendicular to the wrench. Determine an

expression for the torque produced by the couple when the axis is perpendicular to thetire and passes through (a) point A, (b) point B, and (c) point C. Express your answers in

terms of the magnitudeFof the force and the lengthLof the wrench.

10.A rotational axis is directed perpendicular to the plane of a square and is located as

shown in the drawing. Two forces, and , are applied to diagonally opposite corners,

and act along the sides of the square, first as shown in part aand then as shown in part b

of the drawing. In each case the net torque produced by the forces is zero. The square is

one meter on a side, and the magnitude of is three times that of . Find the distancesaand bthat locate the axis.


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Section9.2Rigid Objects in Equilibrium, Section9.3Center of Gravity

11. The drawing shows a person (weight, W= 584 N) doing push-ups. Find the normal

force exerted by the floor on eachhand and eachfoot, assuming that the person holds this

position.

12. A person is standing on a level floor. His head, upper torso, arms, and handstogether weigh 438 N and have a center of gravity that is 1.28 m above the floor. Hisupper legs weigh 144 N and have a center of gravity that is 0.760 m above the floor.

Finally, his lower legs and feet together weigh 87 N and have a center of gravity that is

0.250 m above the floor. Relative to the floor, find the location of the center of gravity forhis entire body.

13. ssmConcept Simulation 9.1atwww.wiley.com/college/cutnellillustrates how the

forces can vary in problems of this type. A hiker, who weighs 985 N, is strolling throughthe woods and crosses a small horizontal bridge. The bridge is uniform, weighs 3610 N,

and rests on two concrete supports, one at each end. He stops one-fifth of the way along

the bridge. What is the magnitude of the force that a concrete support exerts on the bridge(a) at the near end and (b) at the far end?
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14. Review Multiple-ConceptExample 8before beginning this problem. A sport utility

vehicle (SUV) and a sports car travel around the same horizontal curve. The SUV has a

static stability factor of 0.80 and can negotiate the curve at a maximum speed of 18 m/swithout rolling over. The sports car has a static stability factor of 1.4. At what maximum

speed can the sports car negotiate the curve without rolling over?

15. Interactive Solution 9.15atwww.wiley.com/college/cutnellillustrates how tomodel this type of problem. A person exerts a horizontal force of 190 N in the test

apparatus shown in the drawing. Find the horizontal force (magnitude and direction)

that his flexor muscle exerts on his forearm.

16. A lunch tray is being held in one hand, as the drawing illustrates. The mass of the

tray itself is 0.200 kg, and its center of gravity is located at its geometrical center. On the

tray is a 1.00-kg plate of food and a 0.250-kg cup of coffee. Obtain the force exerted by

the thumb and the force exerted by the four fingers. Both forces act perpendicular to the

tray, which is being held parallel to the ground.
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17. ssmA uniform door (0.81 m wide and 2.1 m high) weighs 140 N and is hung on two

hinges that fasten the long left side of the door to a vertical wall. The hinges are 2.1 m

apart. Assume that the lower hinge bears all the weight of the door. Find the magnitudeand direction of the horizontal component of the force applied to the doorby (a) the

upper hinge and (b) the lower hinge. Determine the magnitude and direction of the force

applied by the doorto (c) the upper hinge and (d) the lower hinge.

18. In an isometric exercise a person places a hand on a scale and pushes verticallydownward, keeping the forearm horizontal. This is possible because the triceps muscle

applies an upward force perpendicular to the arm, as the drawing indicates. The

forearm weighs 22.0 N and has a center of gravity as indicated. The scale registers 111 N.Determine the magnitude of

19. Review Concept Stimulation 9.2atwww.wiley.com/college/cutnellandConceptual

Example 7as background material for this problem. A jet transport has a weight of

and is at rest on the runway. The two rear wheels are 15.0 m behind the frontwheel, and the planes center of gravity is 12.6 m behind the front wheel. Determine the

normal force exerted by the ground on (a) the front wheel and on (b) eachof the two rear

wheels.

20. A 1220-N uniform beam is attached to a vertical wall at one end and is supported bya cable at the other end. A 1960-N crate hangs from the far end of the beam. Using the

data shown in the drawing, find (a) the magnitude of the tension in the wire and (b) the

magnitude of the horizontal and vertical components of the force that the wall exerts onthe left end of the beam.
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21. ssm wwwA massless, rigid board is placed across two bathroom scales that are

separated by a distance of 2.00 m. A person lies on the board. The scale under his headreads 425 N, and the scale under his feet reads 315 N. (a) Find the weight of the person.

(b) Locate the center of gravity of the person relative to the scale beneath his head.

22. The drawing shows an outstretched arm (0.61 m in length) that is parallel tothe floor. The arm is pulling downward against the ring attached to the pulley system, in

order to hold the 98-N weight stationary. To pull the arm downward, the latissimus dorsi

muscle applies the force in the drawing, at a point that is 0.069 m from the shoulder

joint and oriented at an angle of 29. The arm has a weight of 47 N and a center ofgravity (cg) that is located 0.28 m from the shoulder joint. Find the magnitude of.


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23. ssm A man holds a 178-N ball in his hand, with the forearm horizontal (see the

drawing). He can support the ball in this position because of the flexor muscle force ,

which is applied perpendicular to the forearm. The forearm weighs 22.0 N and has a

center of gravity as indicated. Find (a) the magnitude of and (b) the magnitude anddirection of the force applied by the upper arm bone to the forearm at the elbow joint.

24. A woman who weighs is leaning against a smooth vertical wall, as the

drawing shows. Find (a) the force (directed perpendicular to the wall) exerted on hershoulders by the wall and the (b) horizontal and (c) vertical components of the force

exerted on her shoes by the ground.

25. The drawing shows an inverted A that is suspended from the ceiling by two

vertical ropes. Each leg of the A has a length of 2Land a weight of 120 N. Thehorizontal crossbar has a negligible weight. Find the force that the crossbar applies to

each leg.


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26. The drawing shows an A-shaped ladder. Both sides of the ladder are equal in length.

This ladder is standing on a frictionless horizontal surface, and only the crossbar (which

has a negligible mass) of the A keeps the ladder from collapsing. The ladder is uniformand has a mass of 20.0 kg. Determine the tension in the crossbar of the ladder.

27. Two vertical walls are separated by a distance of 1.5 m, as the drawing shows. Wall

1 is smooth, while wall 2 is not smooth. A uniform board is propped between them. Thecoefficient of static friction between the board and wall 2 is 0.98. What is the length of

the longest board that can be propped between the walls?

Section9.4Newtons Second Law for Rotational Motion About a Fixed Axis
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28. A solid circular disk has a mass of 1.2 kg and a radius of 0.16 m. Each of three

identical thin rods has a mass of 0.15 kg. The rods are attached perpendicularly to the

plane of the disk at its outer edge to form a threelegged stool (see the drawing). Find themoment of inertia of the stool with respect to an axis that is perpendicular to the plane of

the disk at its center. (Hint: When considering the moment of inertia of each rod, note

that all of the mass of each rod is located at the same perpendicular distance from theaxis.)

29. A ceiling fan is turned on and a net torque of 1.8 N m is applied to the blades. The

blades have a total moment of inertia of 0.22 kg m2. What is the angular acceleration of

the blades?

30. A solid cylindrical disk has a radius of 0.15 m. It is mounted to an axle that is

perpendicular to the circular end of the disk at its center. When a 45-N force is applied

tangentially to the disk, perpendicular to the radius, the disk acquires an angular

acceleration of 120 rad/s

2

. What is the mass of the disk?

31. A uniform solid disk with a mass of 24.3 kg and a radius of 0.314 m is free to rotateabout a frictionless axle. Forces of 90.0 and 125 N are applied to the disk, as the drawing

illustrates. What is (a) the net torque produced by the two forces and (b) the angular

acceleration of the disk?

Problem31
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32. A rotating door is made from four rectangular sections, as shown in the drawing. Themass of each section is 85 kg. A person pushes on the outer edge of one section with a

force ofF= 68 N that is directed perpendicular to the section. Determine the magnitude

of the doors angular acceleration.

33. ssmA bicycle wheel has a radius of 0.330 m and a rim whose mass is 1.20 kg. The

wheel has 50 spokes, each with a mass of 0.010 kg. (a) Calculate the moment of inertia ofthe rim about the axle. (b) Determine the moment of inertia of any one spoke,assuming it

to be a long, thin rod that can rotate about one end. (c) Find the totalmoment of inertia of

the wheel, including the rim and all 50 spokes.

34. A long thin rod is cut into two pieces, one being twice as long as the other. To the

midpoint of piece A (the longer piece), piece B is attached perpendicularly, in order to

form the inverted T shown in the drawing. The application of a net external torquecauses this object to rotate about axis 1 with an angular acceleration of 4.6 rad/s

2. When

the same net external torque is used to cause the object to rotate about axis 2, what is the

angular acceleration?


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35. Interactive Solution 9.35atwww.wiley.com/college/cutnellpresents a method formodeling this problem. Multiple-ConceptExample 10offers useful background for

problems like this. A cylinder is rotating about an axis that passes through the center of

each circular end piece. The cylinder has a radius of 0.0830 m, an angular speed of 76.0rad/s, and a moment of inertia of 0.615 kg m

2. A brake shoe presses against the surface

of the cylinder and applies a tangential frictional force to it. The frictional force reduces

the angular speed of the cylinder by a factor of two during a time of 6.40 s. (a) Find themagnitude of the angular deceleration of the cylinder. (b) Find the magnitude of the force

of friction applied by the brake shoe.

36. Two thin rectangular sheets (0.20 m 0.40 m) are identical. In the first sheet the axisof rotation lies along the 0.20-m side, and in the second it lies along the 0.40-m side. Thesame torque is applied to each sheet. The first sheet, starting from rest, reaches its final

angular velocity in 8.0 s. How long does it take for the second sheet, starting from rest, to

reach the same angular velocity?

37. ssmA thin, rigid, uniform rod has a mass of 2.00 kg and a length of 2.00 m. (a) Find

the moment of inertia of the rod relative to an axis that is perpendicular to the rod at one

end. (b) Suppose all the mass of the rod were located at a single point. Determine the

perpendicular distance of this point from the axis in part (a), such that this point particlehas the same moment of inertia as the rod. This distance is called the radius of gyration

of the rod.

38. Multiple-ConceptExample 10reviews the approach and some of the conceptsthat are pertinent to this problem. The drawing shows a model for the motion of the

human forearm in throwing a dart. Because of the force applied by the triceps muscle,

the forearm can rotate about an axis at the elbow joint. Assume that the forearm has thedimensions shown in the drawing and a moment of inertia of 0.065 kg m

2(including the

effect of the dart) relative to the axis at the elbow. Assume also that the force acts

perpendicular to the forearm. Ignoring the effect of gravity and any frictional forces,

determine the magnitude of the force needed to give the dart a tangential speed of 5.0

m /s in 0.10 s, starting from rest.
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39. ssmA stationary bicycle is raised off the ground, and its front wheel (m= 1.3 kg) isrotating at an angular velocity of 13.1 rad/s (see the drawing). The front brake is then

applied for 3.0 s, and the wheel slows down to 3.7 rad/s. Assume that all the mass of thewheel is concentrated in the rim, the radius of which is 0.33 m. The coefficient of kinetic

friction between each brake pad and the rim is . What is the magnitude of the

normal force that eachbrake pad applies to the rim?

40. Multiple-ConceptExample 12reviews the concepts that play roles in this problem. Ablock (mass = 2.0 kg) is hanging from a massless cord that is wrapped around a pulley

( ), as the drawing shows. Initially the pulley is

prevented from rotating and the block is stationary. Then, the pulley is allowed to rotate

as the block falls. The cord does not slip relative to the pulley as the block falls. Assumethat the radius of the cord around the pulley remains constant at a value of 0.040 m

during the blocks descent. Find the angular acceleration of the pulley and the tension inthe cord.
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41. The parall el axis theoremprovides a useful way to calculate the moment of inertiaI

about an arbitrary axis. The theorem states thatI=Icm+Mh2, whereIcmis the moment of

inertia of the object relative to an axis that passes through the center of mass and isparallel to the axis of interest,Mis the total mass of the object, and his the perpendicular

distance between the two axes. Use this theorem and information to determine an

expression for the moment of inertia of a solid cylinder of radiusRrelative to an axis thatlies on the surface of the cylinder and is perpendicular to the circular ends.

42. See Multiple-ConceptExample 12to review some of the concepts that come into

play here. The crane shown in the drawing is lifting a 180-kg crate upward with anacceleration of 1.2 m /s

2. The cable from the crate passes over a solid cylindrical pulley at

the top of the boom. The pulley has a mass of 130 kg. The cable is then wound onto a

hollow cylindrical drum that is mounted on the deck of the crane. The mass of the drum

is 150 kg, and its radius is 0.76 m. The engine applies a counterclockwise torque to the

drum in order to wind up the cable. What is the magnitude of this torque? Ignore the massof the cable.

Section9.5Rotational Work and Energy

43. ssmA flywheel is a solid disk that rotates about an axis that is perpendicular to the

disk at its center. Rotating flywheels provide a means for storing energy in the form of
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rotational kinetic energy and are being considered as a possible alternative to batteries in

electric cars. The gasoline burned in a 300-mile trip in a typical midsize car produces

about 1.2 109J of energy. How fast would a 13-kg flywheel with a radius of 0.30 m

have to rotate to store this much energy? Give your answer in rev/min.

44. Calculate the kinetic energy that the earth has because of (a) its rotation about its

own axis and (b) its motion around the sun. Assume that the earth is a uniform sphere andthat its path around the sun is circular. For comparison, the total energy used in the

United States in one year is about 9.3 1019

J.

45. ssmThree objects lie in thex,yplane. Each rotates about thezaxis with an angular

speed of 6.00 rad/s. The mass mof each object and its perpendicular distance rfrom thez

axis are as follows: (1) , (2) , (3)

. (a) Find the tangential speed of each object. (b) Determine

the total kinetic energy of this system using the expression(c) Obtain the moment of inertia of the system. (d) Find the rotational kinetic energy of

the system using the relation to verify that the answer is the same as that in (b).

46. A helicopter has two blades (see Figure8.12), each of which has a mass of 240 kgand can be approximated as a thin rod of length 6.7 m. The blades are rotating at an

angular speed of 44 rad/s. (a) What is the total moment of inertia of the two blades about

the axis of rotation? (b) Determine the rotational kinetic energy of the spinning blades.

47. Interactive Solution 9.47atwww.wiley.com/college/cutnelloffers a model for

solving problems of this type. A solid sphere is rolling on a surface. What fraction of its

total kinetic energy is in the form of rotational kinetic energy about the center of mass?

48. One end of a thin rod is attached to a pivot, about which it can rotate without friction.Air resistance is absent. The rod has a length of 0.80 m and is uniform. It is hanging

vertically straight downward. The end of the rod nearest the floor is given a linear speedv0, so that the rod begins to rotate upward about the pivot. What must be the value of v0,such that the rod comes to a momentary halt in a straight-up orientation, exactly opposite

to its initial orientation?
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49. ssmReviewExample 13before attempting this problem. A solid cylinder and a thin-

walled hollow cylinder (see Table9.1)have the same mass and radius. They are rolling

horizontally toward the bottom of an incline. The center of mass of each has the sametranslational speed. The cylinders roll up the incline and reach their highest points.

Calculate the ratio of the distances (ssolid/shollow) along the incline through which each

center of mass moves.

50. ReviewExample 13before attempting this problem. A marble and a cube are placed

at the top of a ramp. Starting from rest at the same height, the marble rolls and the cube

slides (no kinetic friction) down the ramp. Determine the ratio of the center-of-massspeed of the cube to the center-of-mass speed of the marble at the bottom of the ramp.

51. A tennis ball, starting from rest, rolls down the hill in the drawing. At the end of the

hill the ball becomes airborne, leaving at an angle of 35 with respect to the ground. Treatthe ball as a thin-walled spherical shell, and determine the rangex.

Section9.6Angular Momentum

52. A solid disk rotates in the horizontal plane at an angular velocity of 0.067 rad/s withrespect to an axis perpendicular to the disk at its center. The moment of inertia of the disk

is 0.10 kg m2. From above, sand is dropped straight down onto this rotating disk, so that

a thin uniform ring of sand is formed at a distance of 0.40 m from the axis. The sand in

the ring has a mass of 0.50 kg. After all the sand is in place, what is the angular velocityof the disk?

53. ssm wwwTwo disks are rotating about the same axis. Disk A has a moment of

inertia of 3.4 kg m2and an angular velocity of +7.2 rad/s. Disk B is rotating with an

angular velocity of9.8 rad/s. The two disks are then linked together without the aid of

any external torques, so that they rotate as a single unit with an angular velocity of2.4

rad/s. The axis of rotation for this unit is the same as that for the separate disks. What isthe moment of inertia of disk B?
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54. A flat uniform circular disk ( ) is initiallystationary. The disk is free to rotate in the horizontal plane about a frictionless axis

perpendicular to the center of the disk. A 40.0-kg person, standing 1.25 m from the axis,begins to run on the disk in a circular path and has a tangential speed of 2.00 m /s relativeto the ground. Find the resulting angular speed of the disk (in rad/s) and describe the

direction of the rotation.

55. Interactive Solution 9.55atwww.wiley.com/college/cutnellillustrates one way of

solving a problem similar to this one. A thin rod has a length of 0.25 m and rotates in a

circle on a frictionless tabletop. The axis is perpendicular to the length of the rod at oneof its ends. The rod has an angular velocity of 0.32 rad/s and a moment of inertia of

. A bug standing on the axis decides to crawl out to the other end of the

rod. When the bug ( ) gets where its going, what is the angularvelocity of the rod?

56. Example 14provides useful background for this problem. A playground carousel is

free to rotate about its center on frictionless bearings, and air resistance is negligible. Thecarousel itself (without riders) has a moment of inertia of 125 kg m

2. When one person

is standing at a distance of 1.50 m from the center, the carousel has an angular velocity of

0.600 rad/s. However, as this person moves inward to a point located 0.750 m from the

center, the angular velocity increases to 0.800 rad/s. What is the persons mass?

57. ssmIn outer space two identical space modules are joined together by a masslesscable. These modules are rotating about their center of mass, which is at the center of the

cable, because the modules are identical (see the drawing). In each module, the cable is

connected to a motor, so that the modules can pull each other together. The initial

tangential speed of each module is v017 m/s. Then they pull together until the distancebetween them is reduced by a factor of two. Determine the final tangential speed vffor

each module.
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58. A thin uniform rod is rotating at an angular velocity of 7.0 rad/s about an axis that isperpendicular to the rod at its center. As the drawing indicates, the rod is hinged at two

places, one-quarter of the length from each end. Without the aid of external torques, the

rod suddenly assumes a u shape, with the arms of the u parallel to the rotation axis.

What is the angular velocity of the rotating u?

59. A platform is rotating at an angular speed of 2.2 rad/s. A block is resting on this

platform at a distance of 0.30 m from the axis. The coefficient of static friction between

the block and the platform is 0.75. Without any external torque acting on the system, theblock is moved toward the axis. Ignore the moment of inertia of the platform and

determine the smallest distance from the axis at which the block can be relocated and still

remain in place as the platform rotates.

60.

A small 0.500-kg object moves on a frictionless horizontal table in a circular pathof radius 1.00 m. The angular speed is 6.28 rad/s. The object is attached to a

string of negligible mass that passes through a small hole in the table at the centerof the circle. Someone under the table begins to pull the string downward to

make the circle smaller. If the string will tolerate a tension of no more than 105

N, what is the radius of the smallest possible circle on which the object canmove?

Additional Problems

61. ssm wwwA square, 0.40 m on a side, is mounted so that it can rotate about an axisthat passes through the center of the square. The axis is perpendicular to the plane of thesquare. A force of 15 N lies in this plane and is applied to the square. What is the

magnitude of the maximum torque that such a force could produce?


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62. Consult Multiple-ConceptExample 10to review an approach to problems such as

this. A CD has a mass of 17 g and a radius of 6.0 cm. When inserted into a player, the CD

starts from rest and accelerates to an angular velocity of 21 rad/s in 0.80 s. Assuming theCD is a uniform solid disk, determine the net torque acting on it.

63. A baggage carousel at an airport is rotating with an angular speed of 0.20 rad/s when

the baggage begins to be loaded onto it. The moment of inertia of the carousel is 1500kg m

2. Ten pieces of baggage with an average mass of 15 kg each are dropped vertically

onto the carousel and come to rest at a perpendicular distance of 2.0 m from the axis of

rotation. (a) Assuming that no net external torque acts on the system of carousel andbaggage, find the final angular speed. (b) In reality, the angular speed of a baggage

carousel does not change. Therefore, what can you say qualitatively about the net

external torque acting on the system?

64. ReviewConceptual Example 7before starting this problem. A uniform plank oflength 5.0 m and weight 225 N rests horizontally on two supports, with 1.1 m of the

plank hanging over the right support (see the drawing). To what distancexcan a personwho weighs 450 N walk on the overhanging part of the plank before it just begins to tip?

65. ssmA particle is located at each corner of an imaginary cube. Each edge of the cube

is 0.25 m long, and each particle has a mass of 0.12 kg. What is the moment of inertia of

these particles with respect to an axis that lies along one edge of the cube?
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66. The drawing shows a bicycle wheel resting against a small step whose height ish= 0.120 m. The weight and radius of the wheel are W= 25.0 N and r= 0.340 m. A

horizontal force is applied to the axle of the wheel. As the magnitude of increases,

there comes a time when the wheel just begins to rise up and loses contact with the

ground. What is the magnitude of the force when this happens?

67. ssmA uniform board is leaning against a smooth vertical wall. The board is at anangle above the horizontal ground. The coefficient of static friction between the

ground and the lower end of the board is 0.650. Find the smallest value for the angle ,

such that the lower end of the board does not slide along the ground.

68. A wrecking ball (weight = 4800 N) is supported by a boom, which may be assumed

to be uniform and has a weight of 3600 N. As the drawing shows, a support cable runsfrom the top of the boom to the tractor. The angle between the support cable and the

horizontal is 32, and the angle between the boom and the horizontal is 48. Find (a) the

tension in the support cable and (b) the magnitude of the force exerted on the lower endof the boom by the hinge at pointP.

69. A cylindrically shaped space station is rotating about the axis of the cylinder to

create artificial gravity. The radius of the cylinder is 82.5 m. The moment of inertia of thestation without people is . Suppose 500 people, with an average mass

of 70.0 kg each, live on this station. As they move radially from the outer surface of the

cylinder toward the axis, the angular speed of the station changes. What is the maximumpossible percentage change in the stations angular speed due to the radial movement of

the people?


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70. ReviewExample 13before attempting this problem. A bowling ball encounters a0.760-m vertical rise on the way back to the ball rack, as the drawing illustrates. Ignore

frictional losses and assume that the mass of the ball is distributed uniformly. The

translational speed of the ball is 3.50 m/s at the bottom of the rise. Find the translational

speed at the top.

71. ssm wwwAn inverted V is made of uniform boards and weighs356 N. Each side

has the same length and makes a 30.0 angle with the vertical, as the drawing shows.Find the magnitude of the static frictional force that acts on the lower end of each leg ofthe V.

72. Multiple-ConceptExample 12deals with a situation that has similarities to this one

and uses some of the same concepts that are needed here. See Concept Simulation 9.4at
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www.wiley.com/college/cutnellto review the principles involved in this problem. By

means of a rope whose mass is negligible, two blocks are suspended over a pulley, as the

drawing shows. The pulley can be treated as a uniform solid cylindrical disk. Thedownward acceleration of the 44.0-kg block is observed to be exactly one-half the

acceleration due to gravity. Noting that the tension in the rope is not the same on each

side of the pulley, find the mass of the pulley.

Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related

quantitative Problem. The Concept Questions involve little or no mathematics. They

focus on the concepts with which the problems deal. Recognizing the concepts is theessential initial step in any problem-solving technique.

73. Concept QuestionsThe drawing shows a rectangular piece of wood. The forces

applied to corners B and D have the same magnitude and are directed parallel to the long

and short sides of the rectangle. An axis of rotation is shown perpendicular to the plane ofthe rectangle at its center. (a) Relative to this axis, which force produces the torque with

the greater magnitude? (b) A force is to be applied to corner A, directed along the shortside of the rectangle. The net torque produced by the three forces is zero. How is the

force at corner A directedfrom A toward B or away from B? Give your reasoning.

ProblemThe magnitudes of the forces at corners B and D are each 12 N. The long side

of the rectangle is twice as long as the short side. Find the magnitude and direction of the

force applied to corner A.

74. Concept QuestionsThe wheels, axle, and handles of a wheelbarrow weigh 60.0

N. The load chamber and its contents weigh 525 N. The drawing shows these two forcesin two different wheelbarrow designs. To support the wheelbarrow in equilibrium, the

mans hands apply a force to the handles that is directed vertically upward. Consider arotational axis at the point where the tire contacts the ground, directed perpendicular to

the plane of the paper. (a) For which design is there a greater total torque from the 60.0-

and 525-N forces? (b) For which design does the torque from the mans force need to be
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greater? (c) For which design does the mans force need to be greater? Account for your

answers.

ProblemFind the magnitude of the mans force for both designs. Be sure that youranswers are consistent with your answers to the Concept Questions.

75. Concept QuestionsPart aof the drawing shows a uniform horizontal beamattached to a vertical wall by a frictionless hinge and supported from below at an angle

by a brace that is attached to a pin. The forces that keep the beam in equilibrium are its

weight and the forces applied to the beam by the hinge and the brace. (a) Part bof the

drawing shows the weight and the force from the brace, while assuming that the

hinge applies a force that is vertical. Can these three forces keep the beam inequilibrium? (b) Part cof the drawing is like part b, except that it assumes the hinge

applies a force that is horizontal rather than vertical. Can the three forces in part ckeepthe beam in equilibrium? (c) Part dof the drawing assumes that the hinge applies a force

that has both a vertical component and a horizontal component . Can the forces in

part dkeep the beam in equilibrium? Justify your answers.

ProblemThe brace makes an angle = 39 with respect to the beam, which has a


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weight of 340 N. Find the magnitudes of the forces , , and .

76. Concept QuestionsMultiple-ConceptExample 10provides one model for

solving this type of problem. Two wheels have the same mass and radius. One has theshape of a hoop and the other the shape of a solid disk. Each wheel starts from rest and

has a constant angular acceleration with respect to a rotational axis that is perpendicular

to the plane of the wheel at its center. Each makes the same number of revolutions in thesame time. (a) Which wheel, if either, has the greater angular acceleration? (b) Which, if

either, has the greater moment of inertia? (c) To which wheel, if either, is a greater net

external torque applied? Explain your answers.

ProblemThe mass and radius of each wheel are 4.0 kg and 0.35 m, respectively. Each

wheel starts from rest and turns through an angle of 13 rad in 8.0 s. Find the net external

torque that acts on each wheel. Check to see that your answers are consistent with your

answers to the Concept Questions.

77. Concept QuestionsTwo thin rods of lengthLare rotating with the same angular

speed (in rad/s) about axes that pass perpendicularly through one end. Rod A is

massless but has a particle of mass 0.66 kg attached to its free end. Rod B has a mass

0.66 kg, which is distributed uniformly along its length. (a) Which has the greatermoment of inertiarod A with its attached particle or rod B? (b) Which has the greater

rotational kinetic energy? Account for your answers.

ProblemThe length of each rod is 0.75 m, and the angular speed is 4.2 rad/s. Find the

kinetic energies of rod A with its attached particle and of rod B. Make sure your answers

are consistent with your answers to the Concept Questions.

78. Concept QuestionsAs seen from above, a playground carousel is rotatingcounterclockwise about its center on frictionless bearings. A person standing still on the

ground grabs onto one of the bars on the carousel very close to its outer edge and climbsaboard. Thus, this person begins with an angular speed of zero and ends up with a

nonzero angular speed, which means that he underwent a counterclockwise angularacceleration. (a) What applies the force to the person to create the torque causing thisacceleration? What is the direction of this force? (b) According to Newtons action-

reaction law, what can you say about the direction of the force applied to the carousel by

the person and about the nature (clockwise or counterclockwise) of the torque that itcreates? (c) Does the torque identified in part (b) increase or decrease the angular speed

of the carousel?
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ProblemThe carousel has a radius of 1.50 m, an initial angular speed of 3.14 rad/s, and a

moment of inertia of 125 kg m2. The mass of the person is 40.0 kg. Find the final

angular speed of the carousel after the person climbs aboard. Verify that your answer isconsistent with your answers to the Concept Questions.

79. Concept QuestionsThe drawing shows two identical systems of objects; each

consists of three small balls (masses m1, m2, and m3) connected by massless rods. In both

systems the axis is perpendicular to the page, but it is located at a different place, asshown. (a) Do the systems necessarily have the same moments of inertia? If not, why

not? (b) The same force of magnitudeFis applied to the same ball in each system (see

the drawing). Is the magnitude of the torque created by the applied force greater for

system A or for system B? Or is the magnitude the same in the two cases? Explain. (c)The two systems start from rest. Will system A or system B have the greater angular

speed at the same later time? Or will they have the same angular speeds? Justify your

answer.

ProblemThe masses of the balls are , and .The magnitude of the force isF= 424 N. (a) For each of the two systems, determine

the moment of inertia about the given axis of rotation. (b) Calculate the torque(magnitude and direction) acting on each system. (c) Both systems start from rest, and

the direction of the force moves with the system and always points along the 4.00-m

rod. What is the angular velocity of each system

mhs physics problems chapter 9

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