Physics

John D. Cutnell, Kenneth W. Johnson

Chapter 10

Simple Harmonic Motion and Elasticity - all with Video Answers

Educators

Chapter Questions

Problem 1

A hand exerciser utilizes a coiled spring. A force of $89.0 \mathrm{~N}$ is required to compress the spring by $0.0191 \mathrm{~m}$. Determine the force needed to compress the spring by $0.0508 \mathrm{~m}$.

Zachary Warner

Zachary Warner

Numerade Educator

Problem 2

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

Zachary Warner

Numerade Educator

Problem 3

A $0.70-\mathrm{kg}$ 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?

Zachary Warner

Numerade Educator

Problem 4

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

Shoukat Ali

Other Schools

Problem 5

A car is hauling a 92-kg trailer, to which it is connected by a spring. The spring constant is $2300 \mathrm{~N} / \mathrm{m}$. The car accelerates with an acceleration of $0.30 \mathrm{~m} / \mathrm{s}^{2} .$ By how much does the spring stretch?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 6

In a room that is $2.44 \mathrm{~m}$ high, a spring (unstrained length $=0.30 \mathrm{~m}$ ) hangs from the ceiling. A board whose length is $1.98 \mathrm{~m}$ is attached to the free end of the spring. The board hangs straight down, so that its $1.98-\mathrm{m}$ 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?

Zachary Warner

Numerade Educator

Problem 7

In $0.750 \mathrm{~s},$ a $7.00-\mathrm{kg}$ block is pulled through a distance of $4.00 \mathrm{~m}$ 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 $415 \mathrm{~N} / \mathrm{m}$. By how much does the spring stretch?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 8

A $10.1-\mathrm{kg}$ uniform board is wedged into a corner and held by a spring at a $50.0^{\circ}$ angle, as the drawing shows. The spring has a spring constant of $176 \mathrm{~N} / \mathrm{m}$ and is parallel to the floor. Find the amount by which the spring is stretched from its unstrained length.

Supratim Pal

Numerade Educator

Problem 9

Interactive Solution $\underline{10.9}$ at discusses a method used to solve this problem. To measure the static friction coefficient between a $1.6-\mathrm{kg}$ block and a vertical wall, the setup shown in the drawing is used. A spring (spring constant $=510 \mathrm{~N} / \mathrm{m}$ ) 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 $0.039 \mathrm{~m}$ what is the coefficient of static friction?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 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 $0.160 \mathrm{~m}$. 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?

Zachary Warner

Numerade Educator

Problem 11

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

Zachary Warner

Numerade Educator

Problem 12

A $30.0-\mathrm{kg}$ 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 $325 \mathrm{~N} / \mathrm{m}$. 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 $\overrightarrow{\mathbf{F}}$ 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 $\overrightarrow{\mathbf{F}}$.

Zachary Warner

Numerade Educator

Problem 13

A $15.0-\mathrm{kg}$ 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 $5.00 \mathrm{~m} / \mathrm{s}$ in $0.500 \mathrm{~s}$. In the process, the spring is stretched by $0.200 \mathrm{~m}$. The block is then pulled at a constant speed of $5.00 \mathrm{~m} / \mathrm{s}$, during which time the spring is stretched by only $0.0500 \mathrm{~m}$. Find (a) the spring constant of the spring and (b) the coefficient of kinetic friction between the block and the table

Zachary Warner

Numerade Educator

Problem 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 $7.54 \times 10^{4} \mathrm{rad} / \mathrm{s} .$ How many times does the diaphragm move back and forth?

Zachary Warner

Numerade Educator

Problem 15

Atoms in a solid are not stationary, but vibrate about their equilibrium positions. Typically, the frequency of vibration is about $f=2.0 \times 10^{12} \mathrm{~Hz},$ and the amplitude is about $1.1 \times 10^{-11} \mathrm{~m}$. For a typical atom, what is its (a) maximum speed and (b) maximum acceleration?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 16

In Concept Simulation 10.3 at you can explore the concepts that are important in this problem. A block of mass $m=0.750 \mathrm{~kg}$ is fastened to an unstrained horizontal spring whose spring constant is $k=82.0 \mathrm{~N} / \mathrm{m} .$ The block is given a displacement of $+0.120 \mathrm{~m}$ 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 $\omega$ 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.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 17

Concept Simulation 10.3 at illustrates the concepts pertinent to this problem. An $0.80-$ $\mathrm{kg}$ 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 $\omega,$ (c) the spring constant $k,$ (d) the speed of the object at $t=1.0 \mathrm{~s},$ and $(\mathrm{e})$ the magnitude of the object's acceleration at $t=1.0 \mathrm{~s}$

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 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 $45.0 \mathrm{~cm} .$ Determine (a) the amplitude and (b) the angular frequency of the motion. (c) What is the maximum speed attained by the person?

Zachary Warner

Numerade Educator

Problem 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 $174 \mathrm{~N} / \mathrm{m}$. 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

Zachary Warner

Numerade Educator

Problem 20

Multiple-Concept Example 6 reviews the principles that play a role in this problem. A bungee jumper, whose mass is $82 \mathrm{~kg}$, 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 $9.6 \mathrm{~s}$. Ignoring air resistance and assuming that the bungee cord is an ideal spring, determine its spring constant.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 21

Interactive Solution $\underline{10.21}$ at presents a model for solving this problem. A spring (spring constant $=112 \mathrm{~N} / \mathrm{m}$ ) 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 $\mathrm{Hz}$ ) of the motion. (b) Determine the amplitude at which the block will lose contact with the spring.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 22

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

Supratim Pal

Numerade Educator

Problem 23

The drawing shows a top view of a frictionless horizontal surface, where there are two springs with particles of mass $m_{1}$ and $m_{2}$ attached to them. Each spring has a spring constant of $120 \mathrm{~N} / \mathrm{m}$. 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 $x=0 \mathrm{~m}$ if $(\mathrm{a}) \mathrm{m}_{1}=m_{2}=3.0 \mathrm{~kg}$ and $(\mathrm{b}) m_{1}=3.0 \mathrm{~kg}$ and
$m_{2}=27 \mathrm{~kg} ?$

Zachary Warner

Numerade Educator

Problem 24

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

Supratim Pal

Numerade Educator

Problem 25

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

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 26

A rifle fires a $2.10 \times 10^{-2}$ kg 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 $9.10 \times 10^{-2} \mathrm{~m}$ from its unstrained length. The pellet rises to a maximum height of $6.10 \mathrm{~m}$ above its position on the compressed spring. Ignoring air resistance, determine the spring constant.

Zachary Warner

Numerade Educator

Problem 27

Concept Simulation 10.1 at allows you to explore the concepts to which this problem relates. A $2.00-\mathrm{kg}$ object is hanging from the end of a vertical spring. The spring constant is $50.0 \mathrm{~N} / \mathrm{m}$. The object is pulled $0.200 \mathrm{~m}$ 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 $h=0 \mathrm{~m}$.

Manish Jain

Numerade Educator

Problem 28

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

Shoukat Ali

Other Schools

Problem 29

Refer to Interactive Solution $\underline{10.29}$ at for help in solving this problem. A heavy-duty stapling gun uses a $0.140-\mathrm{kg}$ metal rod that rams against the staple to eject it. The rod is pushed by a stiff spring called a "ram spring" $(k=32000 \mathrm{~N} / \mathrm{m})$. The mass of this spring may be ignored. Squeezing the handle of the gun first compresses the ram spring by $3.0 \times 10^{-2} \mathrm{~m}$ from its unstrained length and then releases it. Assuming that the ram spring is oriented vertically and is still compressed by $0.8 \times 10^{-2} \mathrm{~m}$ when the downwardmoving ram hits the staple, find the speed of the ram at the instant of contact.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 30

A $3.2-\mathrm{kg}$ 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 $1.8 \mathrm{~J}$. What is the elastic potential energy of the system when the $3.2-\mathrm{kg}$ block is replaced by a $5.0-\mathrm{kg}$ block?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 31

A $1.00 \times 10^{-2} \mathrm{~kg}$ block is resting on a horizontal frictionless surface and is attached to a horizontal spring whose spring constant is $124 \mathrm{~N} / \mathrm{m}$. The block is shoved parallel to the spring axis and is given an initial speed of $8.00 \mathrm{~m} / \mathrm{s}$, while the spring is initially unstrained. What is the amplitude of the resulting simple harmonic motion?

Zachary Warner

Numerade Educator

Problem 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 $0.065 \mathrm{~m}$, released from rest, and subsequently oscillate back and forth with an angular frequency of $11.3 \mathrm{rad} / \mathrm{s}$. What is the speed of the object at the instant when the spring is stretched by $0.048 \mathrm{~m}$ relative to its unstrained length?

Zachary Warner

Numerade Educator

Problem 33

A $1.1-\mathrm{kg}$ object is suspended from a vertical spring whose spring constant is $120 \mathrm{~N} /$ $\mathrm{m}$
(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 $0.20 \mathrm{~m}$ and released from rest. Find the speed with which the object passes through its original position on the way
up.

Shoukat Ali

Other Schools

Problem 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 $7.0 \mathrm{rad} / \mathrm{s}$. The drawing indicates the position of the block when the spring is unstrained. This position is labeled " $x=0 \mathrm{~m}$." The drawing also shows a small bottle located $0.080 \mathrm{~m}$ to the right of this position. The block is pulled to the right, stretching the spring by $0.050 \mathrm{~m}$, 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 $v_{0}$. Ignoring the width of the block, find $v_{0}$

William Dunkerton

William Dunkerton

Numerade Educator

Problem 35

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

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 36

A $1.00 \times 10^{-2}$ kg bullet is fired horizontally into a $2.50-\mathrm{kg}$ wooden block attached to one end of a massless, horizontal spring $(k=845 \mathrm{~N} / \mathrm{m})$. 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 $0.200 \mathrm{~m}$. What is the speed of the bullet?

Zachary Warner

Numerade Educator

Problem 37

A 70.0 -kg circus performer is fired from a cannon that is elevated at an angle of $40.0^{\circ}$ 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 $3.00 \mathrm{~m}$ 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 $26.8 \mathrm{~m}$ between this point and the net. Ignore friction and air resistance and determine the effective spring constant of the firing mechanism.

Shoukat Ali

Other Schools

Problem 38

A $0.200-\mathrm{m}$ uniform bar has a mass of $0.750 \mathrm{~kg}$ and is released from rest in the vertical position, as the drawing indicates. The spring is initially unstrained and has a spring constant of $k=25.0 \mathrm{~N} / \mathrm{m}$. Find the tangential speed with which end A strikes the horizontal surface.

Zachary Warner

Numerade Educator

Problem 39

If the period of a simple pendulum is to be $2.0 \mathrm{~s},$ what should be its length?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 40

A simple pendulum is made from a $0.65-\mathrm{m}$ -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?

Zachary Warner

Numerade Educator

Problem 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 $9.2 \mathrm{~s}$. What is the height of the tower?

Supratim Pal

Numerade Educator

Problem 42

The length of a simple pendulum is $0.79 \mathrm{~m}$ and the mass of the particle (the "bob") at the end of the cable is $0.24 \mathrm{~kg}$. The pendulum is pulled away from its equilibrium position by an angle of $8.50^{\circ}$ 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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 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 $T_{\mathrm{A}} / T_{\mathrm{B}}$ of their periods for small-angle oscillations.

Zachary Warner

Numerade Educator

Problem 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 $=0.66 \mathrm{~s}$ ), 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?

Zachary Warner

Numerade Educator

Problem 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 $R$.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 46

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

Zachary Warner

Numerade Educator

Problem 47

A student's CD player is mounted on four cylindrical rubber blocks. Each cylinder has a height of $0.030 \mathrm{~m}$ and a cross-sectional area of $1.2 \times 10^{-3} \mathrm{~m}^{2},$ and the shear modulus for rubber is $2.6 \times 10^{6} \mathrm{~N} / \mathrm{m}^{2}$. If a horizontal force of magnitude $32 \mathrm{~N}$ 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.

Supratim Pal

Numerade Educator

Problem 48

The femur is a bone in the leg whose minimum crosssectional area is about $4.0 \times 10^{-4} \mathrm{~m}^{2}$. A compressional force in excess of $6.8 \times 10^{4} \mathrm{~N}$ 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?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 49

A piece of aluminum is surrounded by air at a pressure of $1.01 \times 10^{3} \mathrm{~Pa}$. The aluminum is placed in a vacuum chamber where the pressure is reduced to zero. Determine the fractional change $\Delta V / V_{0}$ in the volume of the aluminum.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 50

Multiple-Concept Example 13 presents a model for solving this type of problem. A $59-$ $\mathrm{kg}$ water skier is being pulled by a nylon tow rope that is attached to a boat. The unstretched length of the rope is $12 \mathrm{~m}$ and its cross-sectional area is $2.0 \times 10^{-5} \mathrm{~m}^{2}$. As the skier moves, a resistive force (due to the water) of magnitude $130 \mathrm{~N}$ 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 $0.85 \mathrm{~m} / \mathrm{s}^{2} ?$

Zachary Warner

Numerade Educator

Problem 51

The drawing shows a $160-\mathrm{kg}$ crate hanging from the end of a steel bar. The length of the bar is $0.10 \mathrm{~m},$ and its crosssectional area is $3.2 \times 10^{-4} \mathrm{~m}^{2}$. Neglect the weight of the bar itself and determine (a) the shear stress on the bar and (b) the vertical deflection $\Delta Y$ of the right end of the bar.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 52

A copper cube, $0.30 \mathrm{~m}$ on a side, is subjected to two shearing forces, each of which has a magnitude $F=6.0 \times 10^{6} \mathrm{~N}$ (see the drawing). Find the angle $\theta$ (in degrees), which is one measure of how the shape of the block has been altered by shear deformation.

Zachary Warner

Numerade Educator

Problem 53

Two metal beams are joined together by four rivets, as the drawing indicates. Each rivet has a radius of $5.0 \times 10^{-3} \mathrm{~m}$ and is to be exposed to a shearing stress of no more than $5.0 \times 10^{8} \mathrm{~Pa}$. What is the maximum tension $\overrightarrow{\mathrm{T}}$ that can be applied to each beam, assuming that each rivet carries one-fourth of the total load?

Zachary Warner

Numerade Educator

Problem 54

When subjected to a force of compression, the length of a bone decreases by $2.7 \times 10^{-5} \mathrm{~m} .$ When this same bone is subjected to a tensile force of the same magnitude, by how much does it stretch?

Zachary Warner

Numerade Educator

Problem 55

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

Zachary Warner

Numerade Educator

Problem 56

Between each pair of vertebrae in the spinal column is a cylindrical disc of cartilage. Typically, this disc has a radius of about $3.0 \times 10^{-2} \mathrm{~m}$ and a thickness of about $7.0 \times 10^{-3} \mathrm{~m}$. The shear modulus of cartilage is $1.2 \times 10^{7} \mathrm{~N} / \mathrm{m}^{2}$. Suppose a shearing force of magnitude $11 \mathrm{~N}$ 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?

Zachary Warner

Numerade Educator

Problem 57

A block of copper is securely fastened to the floor. A force of $1800 \mathrm{~N}$ 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.

Manish Jain

Numerade Educator

Problem 58

A piece of mohair taken from an Angora goat has a radius of $31 \times 10^{-6} \mathrm{~m}$. What is the least number of identical pieces of mohair that should be used to suspend a $75-\mathrm{kg}$ person, so the strain $\Delta L / L_{0}$ experienced by each piece is less than $0.010 ?$ Assume that the tension is the same in all the pieces.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 59

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 $Y_{\text {Composite }}$ of Young's modulus should be used in Equation 10.17 when applied to the composite rod? Note that the change $\Delta L_{\text {Composite }}$ in the length of the composite rod is the sum of the changes in length of the aluminum and tungsten rods.

Manish Jain

Numerade Educator

Problem 60

A die is designed to punch holes with a radius of $1.00 \times 10^{-2} \mathrm{~m}$ in a metal sheet that is $3.0 \times 10^{-3} \mathrm{~m}$ thick, as the drawing illustrates. To punch through the sheet, the die must exert a shearing stress of $3.5 \times 10^{8} \mathrm{~Pa}$. What force $\overrightarrow{\mathbf{F}}$ must be applied to the die?

Zachary Warner

Numerade Educator

Problem 61

A die is designed to punch holes with a radius of $1.00 \times 10^{-2} \mathrm{~m}$ in a metal sheet that is $3.0 \times 10^{-3} \mathrm{~m}$ thick, as the drawing illustrates. To punch through the sheet, the die must exert a shearing stress of $3.5 \times 10^{8} \mathrm{~Pa}$. What force $\overrightarrow{\mathbf{F}}$ must be applied to the die?

Zachary Warner

Numerade Educator

Problem 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 $12^{\circ}$ slope by a steel cable. The cable has a cross-sectional area of $7.8 \times 10^{-5} \mathrm{~m}^{2}$. The cable applies a force to the skier, and, in doing so, the cable stretches by $2.0 \times 10^{-4} \mathrm{~m}$. A frictional force of magnitude $68 \mathrm{~N}$ 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 $1.1 \mathrm{~m} / \mathrm{s}^{2},$ what is the original (unstretched) length of the cable?

Zachary Warner

Numerade Educator

Problem 63

An $8.0-\mathrm{kg}$ stone at the end of a steel wire is being whirled in a circle at a constant tangential speed of $12 \mathrm{~m} / \mathrm{s}$. The stone is moving on the surface of a frictionless horizontal table. The wire is $4.0 \mathrm{~m}$ long and has a radius of $1.0 \times 10^{-3} \mathrm{~m}$. Find the strain in the wire.

Zachary Warner

Numerade Educator

Problem 64

A square plate is $1.0 \times 10^{-2} \mathrm{~m}$ thick, measures $3.0 \times 10^{-2} \mathrm{~m}$ on a side, and has a mass of $7.2 \times 10^{-2} \mathrm{~kg}$. The shear modulus of the material is $2.0 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}$. 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 $10-32 a$. 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 $\Delta X$ (see Figure $10-32 b$ ) that can be created by the applied force just before the plate begins to move.

Manish Jain

Numerade Educator

Problem 65

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 $26^{\circ}$ with respect to the horizontal. What is the radius of the wire?

Manish Jain

Numerade Educator

Problem 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 $1.5 \mathrm{~m},$ and its cross-sectional area is $1.3 \times 10^{-5} \mathrm{~m}^{2}$. 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.

Zachary Warner

Numerade Educator

Problem 67

The shock absorbers in the suspension system of a car are in such bad shape that they have no effect on the behavior of the springs attached to the axles. Each of the identical springs attached to the front axle supports $320 \mathrm{~kg}$. A person pushes down on the middle of the front end of the car and notices that it vibrates through five cycles in $3.0 \mathrm{~s}$. Find the spring constant of either spring.

Supratim Pal

Numerade Educator

Problem 68

Refer to Conceptual Example 2 as an aid in solving this problem. A 100 -coil spring has a spring constant of $420 \mathrm{~N} / \mathrm{m}$. It is cut into four shorter springs, each of which has 25 coils. One end of a 25 -coil spring is attached to a wall. An object of mass $46 \mathrm{~kg}$ is attached to the other end of the spring, and the system is set into horizontal oscillation. What is the angular frequency of the motion?

Zachary Warner

Numerade Educator

Problem 69

Concept Simulation 10.2 at allows you to explore the effect of the acceleration due to gravity on pendulum motion, which is the focus of this problem. Astronauts on a distant planet set up a simple pendulum of length $1.2 \mathrm{~m}$. The pendulum executes simple harmonic motion and makes 100 complete vibrations in $280 \mathrm{~s}$. What is the acceleration due to gravity?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 70

Multiple-Concept Example 6 presents a model for solving this problem. As far as vertical oscillations are concerned, a certain automobile can be considered to be mounted on four identical springs, each having a spring constant of $1.30 \times 10^{5} \mathrm{~N} / \mathrm{m}$. Four identical passengers sit down inside the car, and it is set into a vertical oscillation that has a period of $0.370 \mathrm{~s}$. If the mass of the empty car is $1560 \mathrm{~kg}$, determine the mass of each passenger. Assume that the mass of the car and its passengers is distributed evenly over the springs.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 71

A $3500-\mathrm{kg}$ statue is placed on top of a cylindrical concrete $\left(Y=2.3 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}\right)$ stand. The stand has a crosssectional area of $7.3 \times 10^{-2} \mathrm{~m}^{2}$ and a height of $1.8 \mathrm{~m}$. By how much does the statue compress the stand?

Supratim Pal

Numerade Educator

Problem 72

The graph shows the force $F$, that an archer applies to the string of a long bow versus the string's displacement $x$. Drawing back this bow is analogous to stretching a spring. From the data in the graph determine the effective spring constant of the bow.

Manish Jain

Numerade Educator

Problem 73

The pressure increases by $1.0 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}$ for every meter of depth beneath the surface of the ocean. At what depth does the volume of a Pyrex glass cube, $1.0 \times 10^{-2} \mathrm{~m}$ on an edge at the ocean's surface, decrease by $1.0 \times 10^{-10} \mathrm{~m}^{3} ?$

Zachary Warner

Numerade Educator

Problem 74

Review Conceptual Example 8 before starting this problem. A block is attached to a horizontal spring and oscillates back and forth on a frictionless horizontal surface at a frequency of $3.00 \mathrm{~Hz}$. The amplitude of the motion is $5.08 \times 10^{-2} \mathrm{~m}$. At the point where the block has its maximum speed, it suddenly splits into two identical parts, only one part remaining attached to the spring.
(a) What is the amplitude and the frequency of the simple harmonic motion that exists after the block splits? (b) Repeat part (a), assuming that the block splits when it is at one of its extreme positions.

Zachary Warner

Numerade Educator

Problem 75

Interactive LearningWare 10.1 at reviews the concepts involved in this problem. A spring stretches by $0.018 \mathrm{~m}$ when a $2.8-\mathrm{kg}$ object is suspended from its end. How much mass should be attached to this spring so that its frequency of vibration is $f=3.0$ $\mathrm{Hz} ?$

Rashmi Sinha

Numerade Educator

Problem 76

An 86.0 -kg climber is scaling the vertical wall of a mountain. His safety rope is made of nylon that, when stretched, behaves like a spring with a spring constant of $1.20 \times 10^{3} \mathrm{~N} / \mathrm{m} .$ He accidentally slips and falls freely for $0.750 \mathrm{~m}$ before the rope runs out of slack. How much is the rope stretched when it breaks his fall and momentarily brings him to rest?

Zachary Warner

Numerade Educator

Problem 77

Refer to Interactive Solution $\underline{10.77}$ at to review a method by which this problem can be solved. An $11.2-\mathrm{kg}$ block and a $21.7-\mathrm{kg}$ block are resting on a horizontal frictionless surface. Between the two is squeezed a spring (spring constant $=1330 \mathrm{~N} / \mathrm{m}$ ). The spring is compressed by $0.141 \mathrm{~m}$ from its unstrained length and is not attached permanently to either block. With what speed does each block move away after the mechanism keeping the spring squeezed is released and the spring falls away?

Manish Jain

Numerade Educator

Problem 78

A gymnast does a one-arm handstand. The humerus, which is the upper arm bone between the elbow and the shoulder joint, may be approximated as a 0.30 -m-long cylinder with an outer radius of $1.00 \times 10^{-2} \mathrm{~m}$ and a hollow inner core with a radius of $4.0 \times 10^{-3} \mathrm{~m} .$ Excluding the arm, the mass of the gymnast is $63 \mathrm{~kg}$. (a) What is the compressional strain of the humerus? (b) By how much is the humerus compressed?

Zachary Warner

Numerade Educator

Problem 79

The front spring of a car's suspension system has a spring constant of $1.50 \times 10^{6} \mathrm{~N} / \mathrm{m}$ and supports a mass of $215 \mathrm{~kg}$. The wheel has a radius of $0.400 \mathrm{~m}$. The car is traveling on a bumpy road, on which the distance between the bumps is equal to the circumference of the wheel. Due to resonance, the wheel starts to vibrate strongly when the car is traveling at a certain minimum linear speed. What is this speed?

Supratim Pal

Numerade Educator

Problem 80

Interactive LearningWare 10.2 at explores the approach taken in problems such as this one. A spring is mounted vertically on the floor. The mass of the spring is negligible. A certain object is placed on the spring to compress it. When the object is pushed further down by just a bit and then released, one up/down oscillation cycle occurs in $0.250 \mathrm{~s}$. However, when the object is pushed down by $5.00 \times 10^{-2} \mathrm{~m}$ to point $P$ and then released, the object flies entirely off the spring. To what height above point $P$ does the object rise in the absence of air resistance?

Manish Jain

Numerade Educator

Problem 81

A solid brass sphere is subjected to a pressure of $1.0 \times 10^{5} \mathrm{~Pa}$ due to the earth's atmosphere. On Venus the pressure due to the atmosphere is $9.0 \times 10^{6} \mathrm{~Pa}$. By what fraction $\Delta r / r_{0}$ (including the algebraic sign) does the radius of the sphere change when it is exposed to the Venusian atmosphere? Assume that the change in radius is very small relative to the initial radius.

Zachary Warner

Numerade Educator

Problem 82

A tray is moved horizontally back and forth in simple harmonic motion at a frequency of $f=2.00 \mathrm{~Hz}$. On this tray is an empty cup. Obtain the coefficient of static friction between the tray and the cup, given that the cup begins slipping when the amplitude of the motion is $5.00 \times 10^{-2} \mathrm{~m}$.

Zachary Warner

Numerade Educator

Problem 83

A tray is moved horizontally back and forth in simple harmonic motion at a frequency of $f=2.00 \mathrm{~Hz}$. On this tray is an empty cup. Obtain the coefficient of static friction between the tray and the cup, given that the cup begins slipping when the amplitude of the motion is $5.00 \times 10^{-2} \mathrm{~m}$.

Zachary Warner

Numerade Educator

Problem 84

A cylindrically shaped piece of collagen (a substance found in the body in connective tissue) is being stretched by a force that increases from 0 to $3.0 \times 10^{-2} \mathrm{~N}$. The length and radius of the collagen are, respectively, 2.5 and $0.091 \mathrm{~cm},$ and Young's modulus is $3.1 \times 10^{6} \mathrm{~N} / \mathrm{m}^{2}$. (a) If the stretching obeys Hooke's law, what is the spring constant $k$ for collagen? (b) How much work is done by the variable force that stretches the collagen? (See 6.9 for a discussion of the work done by a variable force.)

Manish Jain

Numerade Educator