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.
Note: Unless otherwise indicated, the values for Young’s modulus Y, the shear modulus S, and the bulk modulus B are given, respectively, in Table 10-1, Table 10-2, and Table 10-3.
ssm Solution is in the Student Solutions Manual.
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Section 10.1 The Ideal Spring and Simple Harmonic Motion

| 1 | ssm A hand exerciser utilizes a coiled spring. A force of 89.0 N is required to compress the spring by 0.0191 m. Determine the force needed to compress the spring by 0.0508 m.

| 2 | An archer, about to shoot an arrow, is applying a force of + 240 N to a drawn bowstring. The bow behaves like an ideal spring whose spring constant is 480 N/m. What is the displacement of the bowstring?

| 3 | A 0.70-kg block is hung from and stretches a spring that is attached to the ceiling. A second block is attached to the first one, and the amount that the spring stretches from its unstrained length triples. What is the mass of the second block?

| 4 | A person who weighs 670 N steps onto a spring scale in the bathroom, and the spring compresses by 0.79 cm. (a) What is the spring constant? (b) What is the weight of another person who compresses the spring by 0.34 cm?

| 5 | ssm A car is hauling a 92-kg trailer, to which it is connected by a spring. The spring constant is 2300 N/m. The car accelerates with an acceleration of 0.30 m/s2. By how much does the spring stretch?

| 6 | In a room that is 2.44 m high, a spring (unstrained length = 0.30 m) hangs from the ceiling. A board whose length is 1.98 m is attached to the free end of the spring. The board hangs straight down, so that its 1.98-m length is perpendicular to the floor. The weight of the board (104 N) stretches the spring so that the lower end of the board just extends to, but does not touch, the floor. What is the spring constant of the spring?

* | 7 | ssm In 0.750 s, a 7.00-kg block is pulled through a distance of 4.00 m on a frictionless horizontal surface, starting from rest. The block has a constant acceleration and is pulled by means of a horizontal spring that is attached to the block. The spring constant of the spring is 415 N/m. By how much does the spring stretch?

* | 8 | A 10.1-kg uniform board is wedged into a corner and held by a spring at a 50.0° angle, as the drawing shows. The spring has a spring constant of 176 N/m and is parallel to the floor. Find the amount by which the spring is stretched from its unstrained length.

* | 9 | Interactive Solution 10.9 at discusses a method used to solve this problem. To measure the static friction coefficient between a 1.6-kg block and a vertical wall, the setup shown in the drawing is used. A spring (spring constant = 510 N/m) is attached to the block. Someone pushes on the end of the spring in a direction perpendicular to the wall until the block does not slip downward. If the spring in such a setup is compressed by 0.039 m, what is the coefficient of static friction?

* | 10 | Review Conceptual Example 2 as an aid in solving this problem. An object is attached to the lower end of a 100-coil spring that is hanging from the ceiling. The spring stretches by 0.160 m. The spring is then cut into two identical springs of 50 coils each. As the drawing shows, each spring is attached between the ceiling and the object. By how much does each spring stretch?

* | 11 | ssm A small ball is attached to one end of a spring that has an unstrained length of 0.200 m. The spring is held by the other end, and the ball is whirled around in a horizontal circle at a speed of 3.00 m/s. The spring remains nearly parallel to the ground during the motion and is observed to stretch by 0.010 m. By how much would the spring stretch if it were attached to the ceiling and the ball allowed to hang straight down, motionless?

** | 12 | A 30.0-kg block is resting on a flat horizontal table. On top of this block is resting a 15.0-kg block, to which a horizontal spring is attached, as the drawing illustrates. The spring constant of the spring is 325 N/m. The coefficient of kinetic friction between the lower block and the table is 0.600, and the coefficient of static friction between the two blocks is 0.900. A horizontal force is applied to the lower block as shown. This force is increasing in such a way as to keep the blocks moving at a constant speed. At the point where the upper block begins to slip on the lower block, determine (a) the amount by which the spring is compressed and (b) the magnitude of the force .

** | 13 | A 15.0-kg block rests on a horizontal table and is attached to one end of a massless, horizontal spring. By pulling horizontally on the other end of the spring, someone causes the block to accelerate uniformly and reach a speed of 5.00 m/s in 0.500 s. In the process, the spring is stretched by 0.200 m. The block is then pulled at a constant speed of 5.00 m/s, during which time the spring is stretched by only 0.0500 m. Find (a) the spring constant of the spring and (b) the coefficient of kinetic friction between the block and the table.
Section 10.2 Simple Harmonic Motion and the Reference Circle

| 14 | A loudspeaker diaphragm is producing a sound for 2.5 s by moving back and forth in simple harmonic motion. The angular frequency of the motion is . How many times does the diaphragm move back and forth?

| 15 | ssm www Atoms in a solid are not stationary, but vibrate about their equilibrium positions. Typically, the frequency of vibration is about , and the amplitude is about . For a typical atom, what is its (a) maximum speed and (b) maximum acceleration?

| 16 | In Concept Simulation 10.3 at you can explore the concepts that are important in this problem. A block of mass m = 0.750 kg is fastened to an unstrained horizontal spring whose spring constant is k = 82.0 N/m. The block is given a displacement of + 0.120 m, where the + sign indicates that the displacement is along the + x axis, and then released from rest. (a) What is the force (magnitude and direction) that the spring exerts on the block just before the block is released? (b) Find the angular frequency of the resulting oscillatory motion. (c) What is the maximum speed of the block? (d) Determine the magnitude of the maximum acceleration of the block.

| 17 | Concept Simulation 10.3 at illustrates the concepts pertinent to this problem. An 0.80-kg object is attached to one end of a spring, as in Figure 10-6, and the system is set into simple harmonic motion. The displacement x of the object as a function of time is shown in the drawing. With the aid of these data, determine (a) the amplitude A of the motion, (b) the angular frequency , (c) the spring constant k, (d) the speed of the object at t = 1.0 s, and (e) the magnitude of the object’s acceleration at t = 1.0 s.

| 18 | A person bounces up and down on a trampoline, while always staying in contact with it. The motion is simple harmonic motion, and it takes 1.90 s to complete one cycle. The height of each bounce above the equilibrium position is 45.0 cm. Determine (a) the amplitude and (b) the angular frequency of the motion. (c) What is the maximum speed attained by the person?

* | 19 | Objects of equal mass are oscillating up and down in simple harmonic motion on two different vertical springs. The spring constant of spring 1 is 174 N/m. The motion of the object on spring 1 has twice the amplitude as the motion of the object on spring 2. The magnitude of the maximum velocity is the same in each case. Find the spring constant of spring 2.

* | 20 | Multiple-Concept Example 6 reviews the principles that play a role in this problem. A bungee jumper, whose mass is 82 kg, jumps from a tall platform. After reaching his lowest point, he continues to oscillate up and down, reaching the low point two more times in 9.6 s. Ignoring air resistance and assuming that the bungee cord is an ideal spring, determine its spring constant.

* | 21 | Interactive Solution 10.21 at presents a model for solving this problem. A spring (spring constant = 112 N/m) is mounted on the floor and is oriented vertically. A 0.400-kg block is placed on top of the spring and pushed down to start it oscillating in simple harmonic motion. The block is not attached to the spring. (a) Obtain the frequency (in Hz) of the motion. (b) Determine the amplitude at which the block will lose contact with the spring.

* | 22 | A 3.0-kg block is between two horizontal springs. Neither spring is strained when the block is at the position labeled x = 0 m in the drawing. The block is then displaced a distance of 0.070 m from the position where x = 0 m and is released from rest. (a) What is the speed of the block when it passes back through the x = 0 m position? (b) Determine the angular frequency of this system.

* | 23 | ssm The drawing shows a top view of a frictionless horizontal surface, where there are two springs with particles of mass m1 and m2 attached to them. Each spring has a spring constant of 120 N/m. The particles are pulled to the right and then released from the positions shown in the drawing. How much time passes before the particles are side by side for the first time at x = 0 m if (a) and (b) m1 = 3.0 kg and m2 = 27 kg?
Section 10.3 Energy and Simple Harmonic Motion

| 24 | An archer pulls the bowstring back for a distance of 0.470 m before releasing the arrow. The bow and string act like a spring whose spring constant is 425 N/m. (a) What is the elastic potential energy of the drawn bow? (b) The arrow has a mass of 0.0300 kg. How fast is it traveling when it leaves the bow?

| 25 | A spring is hung from the ceiling. A 0.450-kg block is then attached to the free end of the spring. When released from rest, the block drops 0.150 m before momentarily coming to rest. (a) What is the spring constant of the spring? (b) Find the angular frequency of the block’s vibrations.

| 26 | A rifle fires a pellet straight upward, because the pellet rests on a compressed spring that is released when the trigger is pulled. The spring has a negligible mass and is compressed by from its unstrained length. The pellet rises to a maximum height of 6.10 m above its position on the compressed spring. Ignoring air resistance, determine the spring constant.

| 27 | ssm Concept Simulation 10.1 at allows you to explore the concepts to which this problem relates. A 2.00-kg object is hanging from the end of a vertical spring. The spring constant is 50.0 N/m. The object is pulled 0.200 m downward and released from rest. Complete the table below by calculating the translational kinetic energy, the gravitational potential energy, the elastic potential energy, and the total mechanical energy E for each of the vertical positions indicated. The vertical positions h indicate distances above the point of release, where h = 0 m.

h |  | PE | PE |
(meters) | KE | (gravity) | (elastic) | E
0 |  |  |  |
0.200 |  |  |  |
0.400 |  |  |  |

| 28 | A vertical spring with a spring constant of 450 N/m is mounted on the floor. From directly above the spring, which is unstrained, a 0.30-kg block is dropped from rest. It collides with and sticks to the spring, which is compressed by 2.5 cm in bringing the block to a momentary halt. Assuming air resistance is negligible, from what height (in cm) above the compressed spring was the block dropped?

| 29 | Refer to Interactive Solution 10.29 at for help in solving this problem. A heavy-duty stapling gun uses a 0.140-kg metal rod that rams against the staple to eject it. The rod is pushed by a stiff spring called a “ram spring” (k = 32 000 N/m). The mass of this spring may be ignored. Squeezing the handle of the gun first compresses the ram spring by from its unstrained length and then releases it. Assuming that the ram spring is oriented vertically and is still compressed by when the downward-moving ram hits the staple, find the speed of the ram at the instant of contact.

| 30 | A 3.2-kg block is hanging stationary from the end of a vertical spring that is attached to the ceiling. The elastic potential energy of this spring/mass system is 1.8 J. What is the elastic potential energy of the system when the 3.2-kg block is replaced by a 5.0-kg block?

| 31 | ssm A block is resting on a horizontal frictionless surface and is attached to a horizontal spring whose spring constant is 124 N/m. The block is shoved parallel to the spring axis and is given an initial speed of 8.00 m/s, while the spring is initially unstrained. What is the amplitude of the resulting simple harmonic motion?

* | 32 | A horizontal spring is lying on a frictionless surface. One end of the spring is attached to a wall while the other end is connected to a movable object. The spring and object are compressed by 0.065 m, released from rest, and subsequently oscillate back and forth with an angular frequency of 11.3 rad/s. What is the speed of the object at the instant when the spring is stretched by 0.048 m relative to its unstrained length?

* | 33 | ssm A 1.1-kg object is suspended from a vertical spring whose spring constant is 120 N/m. (a) Find the amount by which the spring is stretched from its unstrained length. (b) The object is pulled straight down by an additional distance of 0.20 m and released from rest. Find the speed with which the object passes through its original position on the way up.

* | 34 | A block rests on a frictionless horizontal surface and is attached to a spring. When set into simple harmonic motion, the block oscillates back and forth with an angular frequency of 7.0 rad/s. The drawing indicates the position of the block when the spring is unstrained. This position is labeled “x = 0 m.” The drawing also shows a small bottle located 0.080 m to the right of this position. The block is pulled to the right, stretching the spring by 0.050 m, and is then thrown to the left. In order for the block to knock over the bottle, it must be thrown with a speed exceeding v0. Ignoring the width of the block, find v0.

* | 35 | Consult Interactive Solution 10.35 at to explore a model for solving this problem. A spring is compressed by 0.0620 m and is used to launch an object horizontally with a speed of 1.50 m/s. If the object were attached to the spring, at what angular frequency (in rad/s) would it oscillate?

** | 36 | A bullet is fired horizontally into a 2.50-kg wooden block attached to one end of a massless, horizontal spring (k = 845 N/m). The other end of the spring is fixed in place, and the spring is unstrained initially. The block rests on a horizontal, frictionless surface. The bullet strikes the block perpendicularly and quickly comes to a halt within it. As a result of this completely inelastic collision, the spring is compressed along its axis and causes the block/bullet to oscillate with an amplitude of 0.200 m. What is the speed of the bullet?

** | 37 | ssm A 70.0-kg circus performer is fired from a cannon that is elevated at an angle of 40.0° above the horizontal. The cannon uses strong elastic bands to propel the performer, much in the same way that a slingshot fires a stone. Setting up for this stunt involves stretching the bands by 3.00 m from their unstrained length. At the point where the performer flies free of the bands, his height above the floor is the same as that of the net into which he is shot. He takes 2.14 s to travel the horizontal distance of 26.8 m between this point and the net. Ignore friction and air resistance and determine the effective spring constant of the firing mechanism.

** | 38 | A 0.200-m uniform bar has a mass of 0.750 kg and is released from rest in the vertical position, as the drawing indicates. The spring is initially unstrained and has a spring constant of k = 25.0 N/m. Find the tangential speed with which end A strikes the horizontal surface.
Section 10.4 The Pendulum

| 39 | If the period of a simple pendulum is to be 2.0 s, what should be its length?

| 40 | A simple pendulum is made from a 0.65-m-long string and a small ball attached to its free end. The ball is pulled to one side through a small angle and then released from rest. After the ball is released, how much time elapses before it attains its greatest speed?

| 41 | A spiral staircase winds up to the top of a tower in an old castle. To measure the height of the tower, a rope is attached to the top of the tower and hung down the left of the staircase. However, nothing is available with which to measure the length of the rope. Therefore, at the bottom of the rope a small object is attached so as to form a simple pendulum that just clears the floor. The period of the pendulum is measured to be 9.2 s. What is the height of the tower?

| 42 | The length of a simple pendulum is 0.79 m and the mass of the particle (the “bob”) at the end of the cable is 0.24 kg. The pendulum is pulled away from its equilibrium position by an angle of 8.50° and released from rest. Assume that friction can be neglected and that the resulting oscillatory motion is simple harmonic motion. (a) What is the angular frequency of the motion? (b) Using the position of the bob at its lowest point as the reference level, determine the total mechanical energy of the pendulum as it swings back and forth. (c) What is the bob’s speed as it passes through the lowest point of the swing?

* | 43 | Multiple-Concept Example 11 explores the concepts that are important in this problem. Pendulum A is a physical pendulum made from a thin, rigid, and uniform rod whose length is d. One end of this rod is attached to the ceiling by a frictionless hinge, so the rod is free to swing back and forth. Pendulum B is a simple pendulum whose length is also d. Obtain the ratio of their periods for small-angle oscillations.

* | 44 | Multiple-Concept Example 11 provides some pertinent background for this problem. A pendulum is constructed from a thin, rigid, and uniform rod with a small sphere attached to the end opposite the pivot. This arrangement is a good approximation to a simple pendulum (period = 0.66 s), because the mass of the sphere (lead) is much greater than the mass of the rod (aluminum). When the sphere is removed, the pendulum no longer is a simple pendulum, but is then a physical pendulum. What is the period of the physical pendulum?

** | 45 | A point on the surface of a solid sphere (radius = R) is attached directly to a pivot on the ceiling. The sphere swings back and forth as a physical pendulum with a small amplitude. What is the length of a simple pendulum that has the same period as this physical pendulum? Give your answer in terms of R.
Section 10.7 Elastic Deformation,
Section 10.8 Stress, Strain, and Hooke’s Law

| 46 | A tow truck is pulling a car out of a ditch by means of a steel cable that is 9.1 m long and has a radius of 0.50 cm. When the car just begins to move, the tension in the cable is 890 N. How much has the cable stretched?

| 47 | ssm A student’s CD player is mounted on four cylindrical rubber blocks. Each cylinder has a height of 0.030 m and a cross-sectional area of , and the shear modulus for rubber is . If a horizontal force of magnitude 32 N is applied to the CD player, how far will the unit move sideways? Assume that each block is subjected to one-fourth of the force.

| 48 | The femur is a bone in the leg whose minimum crosssectional area is about . A compressional force in excess of will fracture this bone. (a) Find the maximum stress that this bone can withstand. (b) What is the strain that exists under a maximum-stress condition?

| 49 | ssm A piece of aluminum is surrounded by air at a pressure of . The aluminum is placed in a vacuum chamber where the pressure is reduced to zero. Determine the fractional change in the volume of the aluminum.

| 50 | Multiple-Concept Example 13 presents a model for solving this type of problem. A 59-kg water skier is being pulled by a nylon tow rope that is attached to a boat. The unstretched length of the rope is 12 m and its cross-sectional area is . As the skier moves, a resistive force (due to the water) of magnitude 130 N acts on her; this force is directed opposite to her motion. What is the change in the length of the rope when the skier has an acceleration whose magnitude is 0.85 m/s2?

| 51 | The drawing shows a 160-kg crate hanging from the end of a steel bar. The length of the bar is 0.10 m, and its crosssectional area is . Neglect the weight of the bar itself and determine (a) the shear stress on the bar and (b) the vertical deflection of the right end of the bar.

| 52 | A copper cube, 0.30 m on a side, is subjected to two shearing forces, each of which has a magnitude (see the drawing). Find the angle (in degrees), which is one measure of how the shape of the block has been altered by shear deformation.

| 53 | ssm Two metal beams are joined together by four rivets, as the drawing indicates. Each rivet has a radius of and is to be exposed to a shearing stress of no more than . What is the maximum tension that can be applied to each beam, assuming that each rivet carries one-fourth of the total load?

| 54 | When subjected to a force of compression, the length of a bone decreases by . When this same bone is subjected to a tensile force of the same magnitude, by how much does it stretch?

| 55 | A copper cylinder and a brass cylinder are stacked end to end, as in the drawing. Each cylinder has a radius of 0.25 cm. A compressive force of F = 6500 N is applied to the right end of the brass cylinder. Find the amount by which the length of the stack decreases.

| 56 | Between each pair of vertebrae in the spinal column is a cylindrical disc of cartilage. Typically, this disc has a radius of about and a thickness of about . The shear modulus of cartilage is . Suppose a shearing force of magnitude 11 N is applied parallel to the top surface of the disc while the bottom surface remains fixed in place. How far does the top surface move relative to the bottom surface?

| 57 | A block of copper is securely fastened to the floor. A force of 1800 N is applied to the top surface of the block, as the drawing shows. Find (a) the amount by which the height of the block is changed and (b) the shear deformation of the block.

| 58 | A piece of mohair taken from an Angora goat has a radius of . What is the least number of identical pieces of mohair that should be used to suspend a 75-kg person, so the strain experienced by each piece is less than 0.010? Assume that the tension is the same in all the pieces.

| 59 | ssm Two rods are identical in all respects except one: one rod is made from aluminum, and the other from tungsten. The rods are joined end to end, in order to make a single rod that is twice as long as either the aluminum or tungsten rod. What is the effective value of Young’s modulus for this composite rod? That is, what value YComposite of Young’s modulus should be used in Equation 10.17 when applied to the composite rod? Note that the change in the length of the composite rod is the sum of the changes in length of the aluminum and tungsten rods.

* | 60 | A die is designed to punch holes with a radius of in a metal sheet that is thick, as the drawing illustrates. To punch through the sheet, the die must exert a shearing stress of . What force must be applied to the die?

Problem 60

* | 61 | A -kg house spider is hanging vertically by a thread that has a Young’s modulus of and a radius of . Suppose that a 95-kg person is hanging vertically on an aluminum wire. What is the radius of the wire that would exhibit the same strain as the spider’s thread, when the thread is stressed by the full weight of the spider?

* | 62 | Consult Multiple-Concept Example 13 to review a model for solving this type of problem. A 61-kg snow skier is being pulled up a 12° slope by a steel cable. The cable has a cross-sectional area of . The cable applies a force to the skier, and, in doing so, the cable stretches by . A frictional force of magnitude 68 N acts on the skis and is directed opposite to the skier’s motion. If the skier’s acceleration up the slope has a magnitude of 1.1 m/s2, what is the original (unstretched) length of the cable?

* | 63 | ssm www An 8.0-kg stone at the end of a steel wire is being whirled in a circle at a constant tangential speed of 12 m/s. The stone is moving on the surface of a frictionless horizontal table. The wire is 4.0 m long and has a radius of . Find the strain in the wire.

* | 64 | A square plate is thick, measures on a side, and has a mass of . The shear modulus of the material is . One of the square faces rests on a flat horizontal surface, and the coefficient of static friction between the plate and the surface is 0.90. A force is applied to the top of the plate, as in Figure 10-32a. Determine (a) the maximum possible amount of shear stress, (b) the maximum possible amount of shear strain, and (c) the maximum possible amount of shear deformation (see Figure 10-32b) that can be created by the applied force just before the plate begins to move.

** | 65 | ssm A steel wire is strung between two supports attached to a ceiling. Initially, there is no tension in the wire when it is horizontal. A 96-N picture is then hung from the left of the wire, as the drawing illustrates, so the ends of the wire make angles of 26° with respect to the horizontal. What is the radius of the wire?

** | 66 | The drawing shows two crates that are connected by a steel wire that passes over a pulley. The unstretched length of the wire is 1.5 m, and its cross-sectional area is . The pulley is frictionless and massless. When the crates are accelerating, determine the change in length of the wire. Ignore the mass of the wire.

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