Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment 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 format that provides enhanced interactivity. See Preface for additional details.
ssm Solution is in the Student Solutions Manual.
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Section 9.1 The Action of Forces and Torques on Rigid Objects

| 1 | ssm You 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 magnitude F of 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 a vertical 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?

| 3 | Interactive Solution 9.3 at presents a model for solving this problem. The wheel of a car has a radius of 0.350 m. The engine of the car applies 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 frictional force?

| 4 | The drawing shows a lower leg being exercised. It has a 49-N weight attached to the foot and is extended at an angle with respect to the vertical. Consider a rotational axis at the knee. (a) When , 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 | ssm The steering wheel of a car has a radius of 0.19 m, while the steering wheel of a truck 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 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 engine produces a thrust of 62 300 N that is parallel to the ground. The rotational axis in the drawing is perpendicular to the plane of the paper. With respect to this axis, find the magnitude of the torque due to (a) the weight and (b) the thrust.

* | 7 | ssm One end of a meter stick is pinned to a table, so the stick can rotate freely in a plane 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 9.1 at reviews the concepts 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 net torque 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.

* | 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 a torque that does not depend 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 the tire and passes through (a) point A, (b) point B, and (c) point C. Express your answers in terms of the magnitude F of the force and the length L of 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 a and 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 distances a and b that locate the axis.

Section 9.2 Rigid Objects in Equilibrium, Section 9.3 Center of Gravity

| 11 | The drawing shows a person (weight, W = 584 N) doing push-ups. Find the normal force exerted by the floor on each hand and each foot, assuming that the person holds this position.

| 12 | A person is standing on a level floor. His head, upper torso, arms, and hands together weigh 438 N and have a center of gravity that is 1.28 m above the floor. His upper 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 for his entire body.

| 13 | ssm Concept Simulation 9.1 at illustrates how the forces can vary in problems of this type. A hiker, who weighs 985 N, is strolling through the 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?

| 14 | Review Multiple-Concept Example 8 before 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/s without 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 9.15 at illustrates how to model 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.

| 17 | ssm A 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 magnitude and direction of the horizontal component of the force applied to the door by (a) the upper hinge and (b) the lower hinge. Determine the magnitude and direction of the force applied by the door to (c) the upper hinge and (d) the lower hinge.

| 18 | In an isometric exercise a person places a hand on a scale and pushes vertically downward, 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 9.2 at and Conceptual Example 7 as 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 front wheel, and the plane’s 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) each of the two rear wheels.

* | 20 | A 1220-N uniform beam is attached to a vertical wall at one end and is supported by a 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 on the left end of the beam.

* | 21 | ssm www A 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 head reads 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 to the 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 of gravity (cg) that is located 0.28 m from the shoulder joint. Find the magnitude of.

* | 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 and direction 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 her shoulders 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 2L and a weight of 120 N. The horizontal crossbar has a negligible weight. Find the force that the crossbar applies to each leg.

** | 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 uniform and 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. The coefficient 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?

Section 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

| 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 the moment 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 the axis.)

| 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/s2. 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 rotate about 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?

Problem 31

| 32 | A rotating door is made from four rectangular sections, as shown in the drawing. The mass of each section is 85 kg. A person pushes on the outer edge of one section with a force of F = 68 N that is directed perpendicular to the section. Determine the magnitude of the door’s angular acceleration.

| 33 | ssm A 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 of the 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 total moment 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 torque causes this object to rotate about axis 1 with an angular acceleration of 4.6 rad/s2. When the same net external torque is used to cause the object to rotate about axis 2, what is the angular acceleration?

| 35 | Interactive Solution 9.35 at presents a method for modeling this problem. Multiple-Concept Example 10 offers 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 76.0 rad/s, and a moment of inertia of 0.615 kg · m2. 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 the magnitude 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 axis of rotation lies along the 0.20-m side, and in the second it lies along the 0.40-m side. The same 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 | ssm A 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 particle has the same moment of inertia as the rod. This distance is called the radius of gyration of the rod.

* | 38 | Multiple-Concept Example 10 reviews the approach and some of the concepts that 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 the dimensions shown in the drawing and a moment of inertia of 0.065 kg · m2 (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.

* | 39 | ssm A stationary bicycle is raised off the ground, and its front wheel (m = 1.3 kg) is rotating 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 the wheel 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 each brake pad applies to the rim?

* | 40 | Multiple-Concept Example 12 reviews the concepts that play roles in this problem. A block (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. Assume that the radius of the cord around the pulley remains constant at a value of 0.040 m during the block’s descent. Find the angular acceleration of the pulley and the tension in the cord.

* | 41 | The parallel axis theorem provides a useful way to calculate the moment of inertia I about an arbitrary axis. The theorem states that , where Icm is the moment of inertia of the object relative to an axis that passes through the center of mass and is parallel to the axis of interest, M is the total mass of the object, and h is 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 radius R relative to an axis that lies on the surface of the cylinder and is perpendicular to the circular ends.

** | 42 | See Multiple-Concept Example 12 to review some of the concepts that come into play here. The crane shown in the drawing is lifting a 180-kg crate upward with an acceleration of 1.2 m /s2. 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 mass of the cable.

Section 9.5 Rotational Work and Energy

| 43 | ssm A 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 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 × 109 J 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 and that 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 | ssm Three objects lie in the x, y plane. Each rotates about the z axis with an angular speed of 6.00 rad/s. The mass m of each object and its perpendicular distance r from the z 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 Figure 8-12), each of which has a mass of 240 kg and 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 9.47 at offers 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 speed v0, 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?

* | 49 | ssm Review Example 13 before attempting this problem. A solid cylinder and a thin-walled hollow cylinder (see Table 9-1) have the same mass and radius. They are rolling horizontally toward the bottom of an incline. The center of mass of each has the same translational 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 | Review Example 13 before 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-mass speed 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. Treat the ball as a thin-walled spherical shell, and determine the range x.

Section 9.6 Angular Momentum

| 52 | A solid disk rotates in the horizontal plane at an angular velocity of 0.067 rad/s with respect 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 velocity of the disk?

| 53 | ssm www Two disks are rotating about the same axis. Disk A has a moment of inertia of 3.4 kg · m2 and an angular velocity of +7.2 rad/s. Disk B is rotating with an angular velocity of −9.8 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 of −2.4 rad/s. The axis of rotation for this unit is the same as that for the separate disks. What is the moment of inertia of disk B?

| 54 | A flat uniform circular disk () is initially stationary. 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 relative to the ground. Find the resulting angular speed of the disk (in rad/s) and describe the direction of the rotation.

| 55 | Interactive Solution 9.55 at illustrates 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 one of 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 it’s going, what is the angular velocity of the rod?

| 56 | Example 14 provides useful background for this problem. A playground carousel is free to rotate about its center on frictionless bearings, and air resistance is negligible. The carousel itself (without riders) has a moment of inertia of 125 kg · m2. 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 person’s mass?

* | 57 | ssm In outer space two identical space modules are joined together by a massless cable. 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 v0 − 17 m/s. Then they pull together until the distance between them is reduced by a factor of two. Determine the final tangential speed vf for each module.

* | 58 | A thin uniform rod is rotating at an angular velocity of 7.0 rad/s about an axis that is perpendicular 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, the block 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 path of 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 center of 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 can move?

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