Chapter 11

A thin, light wire \(75.0 \mathrm{~cm}\) long having a circular cross section \(0.550 \mathrm{~mm}\) in diameter has a \(25.0 \mathrm{~kg}\) weight attached to it, causing it to stretch by \(1.10 \mathrm{~mm}\). (a) What is the stress in this wire? (b) What is the strain of the wire? (c) Find Young's modulus for the material of the wire.

A petite young woman distributes her \(500 \mathrm{~N}\) weight equally over the heels of her high-heeled shoes. Each heel has an area of \(0.750 \mathrm{~cm}^{2}\). (a) What pressure is exerted on the floor by each heel? (b) With the same pressure, how much weight could be supported by two flat- bottomed sandals, each of area \(200 \mathrm{~cm}^{2} ?\)

Biceps muscle. A relaxed biceps muscle requires a force of \(25.0 \mathrm{~N}\) for an elongation of \(3.0 \mathrm{~cm} ;\) under maximum tension, the same muscle requires a force of \(500 \mathrm{~N}\) for the same elongation. Find Young's modulus for the muscle tissue under each of these conditions if the muscle can be modeled as a uniform cylinder with an initial length of \(0.200 \mathrm{~m}\) and a cross-sectional area of \(50.0 \mathrm{~cm}^{2}\)

Stress on a mountaineer's rope. A nylon rope used by mountaineers elongates \(1.10 \mathrm{~m}\) under the weight of a \(65.0 \mathrm{~kg}\) climber. If the rope is \(45.0 \mathrm{~m}\) in length and \(7.0 \mathrm{~mm}\) in diameter, what is Young's modulus for this nylon?

A steel wire \(2.00 \mathrm{~m}\) long with circular cross section must stretch no more than \(0.25 \mathrm{~cm}\) when a \(400.0 \mathrm{~N}\) weight is hung from one of its ends. What minimum diameter must this wire have?

Achilles tendon. The Achilles tendon, which connects the calf muscles to the heel, is the thickest and strongest tendon in the body. In extreme activities, such as sprinting, it can be subjected to forces as high as 13 times a person's weight. According to one set of experiments, the average area of the Achilles tendon is \(78.1 \mathrm{~mm}^{2}\), its average length is \(25 \mathrm{~cm},\) and its average Young's modulus is 1474 MPa. (a) How much tensile stress is required to stretch this muscle by \(5.0 \%\) of its length? (b) If we model the tendon as a spring, what is its force constant? (c) If a \(75 \mathrm{~kg}\) sprinter exerts a force of 13 times his weight on his Achilles tendon, by how much will it stretch?

Human hair. According to one set of measurements, the tensile strength of hair is 196 MPa, which produces a maximum strain of 0.40 in the hair. The thickness of hair varies considerably, but let's use a diameter of \(50 \mu \mathrm{m}\). (a) What is the magnitude of the force giving this tensile stress? (b) If the length of a strand of the hair is \(12 \mathrm{~cm}\) at its breaking point, what was its unstressed length?

The effect of jogging on the knees. High-impact activities such as jogging can cause considerable damage to the cartilage at the knee joints. Peak loads on each knee can be eight times body weight during jogging. The bones at the knee are separated by cartilage called the medial and lateral meniscus. Although it varies considerably, the force at impact acts over approximately \(10 \mathrm{~cm}^{2}\) of this cartilage. Human cartilage has a Young's modulus of about 24 MPa (although that also varies). (a) By what percent does the peak load impact of jogging compress the knee cartilage of a \(75 \mathrm{~kg}\) person? (b) What would be the percentage for a lower-impact activity, such as power walking, for which the peak load is about four times body weight?

In the Challenger Deep of the Marianas Trench, the depth of seawater is \(10.9 \mathrm{~km}\) and the pressure is \(1.16 \times 10^{8} \mathrm{~Pa}\) (about 1150 atmospheres). (a) If a cubic meter of water is taken to this depth from the surface (where the normal atmospheric pressure is about \(\left.1.0 \times 10^{5} \mathrm{~Pa}\right),\) what is the change in its volume? Assume that the bulk modulus for seawater is the same as for freshwater \(\left(2.2 \times 10^{9} \mathrm{~Pa}\right)\). (b) At the surface, seawater has a density of \(1.03 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). What is the density of seawater at the depth of the Challenger Deep?

Effect of diving on blood. It is reasonable to assume that the bulk modulus of blood is about the same as that of water ( \(2.2 \mathrm{GPa}\) ). As one goes deeper and deeper in the ocean, the pressure increases by \(1.0 \times 10^{4} \mathrm{~Pa}\) for every meter below the surface. (a) If a diver goes down \(33 \mathrm{~m}\) (a bit over \(100 \mathrm{ft}\) ) in the ocean, by how much does each cubic centimeter of her blood change in volume? (b) How deep must a diver go so that each drop of blood compresses to half its volume at the surface? Is the ocean deep enough to have this effect on the diver?

Shear forces are applied to a rectangular solid. The same forces are applied to another rectangular solid of the same material, but with three times each edge length. In each case, the forces are small enough that Hooke's law is obeyed. What is the ratio of the shear strain for the larger object to that of the smaller object?

Compression of human bone. The bulk modulus for bone is 15 GPa. (a) If a diver-in-training is put into a pressurized suit, by how much would the pressure have to be raised (in atmospheres) above atmospheric pressure to compress her bones by \(0.10 \%\) of their original volume? (b) Given that the pressure in the ocean increases by \(1.0 \times 10^{4} \mathrm{~Pa}\) for every meter of depth below the surface, how deep would this diver have to go for her bones to compress by \(0.10 \% ?\) Does it seem that bone compression is a problem she needs to be concerned with when diving?

A steel wire has the following properties: $$ \begin{array}{l} \text { Length }=5.00 \mathrm{~m} \\ \text { Cross-sectional area }=0.040 \mathrm{~cm}^{2} \end{array} $$ Young's modulus \(=2.0 \times 10^{11} \mathrm{~Pa}\) Shear modulus \(=0.84 \times 10^{11} \mathrm{~Pa}\) Proportional limit \(=3.60 \times 10^{8} \mathrm{~Pa}\) Breaking stress \(=11.0 \times 10^{8} \mathrm{~Pa}\) The proportional limit is the maximum stress for which the wire still obeys Hooke's law (see point \(\mathrm{B}\) in Figure 11.12 ). The wire is fastened at its upper end and hangs vertically. (a) How great a weight can be hung from the wire without exceeding the proportional limit? (b) How much does the wire stretch under this load? (c) What is the maximum weight that can be supported?

A steel cable with cross-sectional area of \(3.00 \mathrm{~cm}^{2}\) has an elastic limit of \(2.40 \times 10^{8} \mathrm{~Pa}\). Find the maximum upward acceleration that can be given to a \(1200 \mathrm{~kg}\) elevator supported by the cable if the stress is not to exceed one-third of the elastic limit.

Weight lifting. The legs of a weight lifter must ultimately support the weights he has lifted. A human tibia (shinbone) has a circular cross section of approximately \(3.6 \mathrm{~cm}\) outer diameter and \(2.5 \mathrm{~cm}\) inner diameter. (The hollow portion contains marrow.) If a \(90 \mathrm{~kg}\) lifter stands on both legs, what is the heaviest weight he can lift without breaking his legs, assuming that the breaking stress of the bone is \(200 \mathrm{MPa}\) ?

(a) Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern having the same frequency of the note that is sung. If someone sings the note \(\mathrm{B}\) flat that has a frequency of 466 \(\mathrm{Hz}\), how much time does it take the person's vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords? (b) Hearing. When sound waves strike the eardrum, this membrane vibrates with the same frequency as the sound. The highest pitch that typical humans can hear has a period of \(50.0 \mu \mathrm{s}\). What are the frequency and angular frequency of the vibrating eardrum for this sound? (c) Vision. When light having vibrations with angular frequency ranging from \(2.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) to \(4.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light? (d) Ultrasound. High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as X-rays do. To detect small objects such as tumors, a frequency of around \(5.0 \mathrm{MHz}\) is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

Find the period, frequency, and angular frequency of (a) the second hand and (b) the minute hand of a wall clock.

If an object on a horizontal frictionless surface is attached to a spring, displaced, and then released, it oscillates. Suppose it is displaced \(0.120 \mathrm{~m}\) from its equilibrium position and released with zero initial speed. After \(0.800 \mathrm{~s}\), its displacement is found to be \(0.120 \mathrm{~m}\) on the opposite side and it has passed the equilibrium position once during this interval. Find (a) the amplitude, (b) the period, and (c) the frequency of the motion.

The wings of the blue-throated hummingbird, which inhabits Mexico and the southwestern United States, beat at a rate of up to 900 times per minute. Calculate (a) the period of vibration of the bird's wings, (b) the frequency of the wings' vibration, and (c) the angular frequency of the bird's wingbeats.

A \(0.500 \mathrm{~kg}\) glider on an air track is attached to the end of an ideal spring with force constant \(450 \mathrm{~N} / \mathrm{m} ;\) it undergoes simple harmonic motion with an amplitude of \(0.040 \mathrm{~m}\). Compute (a) the maximum speed of the glider, (b) the speed of the glider when it is at \(x=-0.015 \mathrm{~m},\) (c) the magnitude of the maximum acceleration of the glider, (d) the acceleration of the glider at \(x=-0.015 \mathrm{~m},\) and (e) the total mechanical energy of the glider at any point in its motion.

A toy is undergoing SHM on the end of a horizontal spring with force constant \(300.0 \mathrm{~N} / \mathrm{m} .\) When the toy is \(0.120 \mathrm{~m}\) from its equilibrium position, it is observed to have a speed of \(3 \mathrm{~m} / \mathrm{s}\) and a total energy of \(4.4 \mathrm{~J}\). Find (a) the mass of the toy, (b) the amplitude of the motion, and (c) the maximum speed attained by the toy during its motion.

A \(2.00 \mathrm{~kg}\) frictionless block is attached to an ideal spring with force constant \(315 \mathrm{~N} / \mathrm{m} .\) Initially the spring is neither stretched nor compressed, but the block is moving in the negative direction at \(12.0 \mathrm{~m} / \mathrm{s}\) Find (a) the amplitude of the motion, (b) the maximum acceleration of the block, and (c) the maximum force the spring exerts on the block.

You are watching an object that is moving in SHM. When the object is displaced \(0.600 \mathrm{~m}\) to the right of its equilibrium position, it has a velocity of \(2.20 \mathrm{~m} / \mathrm{s}\) to the right and an acceleration of \(8.40 \mathrm{~m} / \mathrm{s}^{2}\) to the left. How much farther from this point will the object move before it stops momentarily and then starts to move back to the left?

A mass is oscillating with amplitude \(A\) at the end of a spring. (a) How far (in terms of \(A\) ) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy? (b) How far is the mass from the equilibrium position when the kinetic energy is \(\frac{1}{10}\) of the total energy?

(a) If a vibrating system has total energy \(E_{0},\) what will its total energy be (in terms of \(E_{0}\) ) if you double the amplitude of vibration? (b) If you want to triple the total energy of a vibrating system with amplitude \(A_{0}\), what should its new amplitude be (in terms of \(A_{0}\) )?

A concrete block is hung from an ideal spring that has a force constant of \(200 \mathrm{~N} / \mathrm{m} .\) The spring stretches \(0.120 \mathrm{~m} .\) (a) What is the mass of the block? (b) What is the period of oscillation of the block if it is pulled down \(1.0 \mathrm{~cm}\) and released? (c) What would be the period of oscillation if the block and spring were placed on the moon?

A mass of \(0.20 \mathrm{~kg}\) on the end of a spring oscillates with a period of \(0.45 \mathrm{~s}\) and an amplitude of \(0.15 \mathrm{~m} .\) Find (a) the velocity when it passes the equilibrium point, (b) the total energy of the system, (c) the spring constant, and (d) the maximum acceleration of the mass.

A harmonic oscillator is made by using a \(0.600 \mathrm{~kg}\) frictionless block and an ideal spring of unknown force constant. The oscillator is found to have a period of \(0.150 \mathrm{~s}\) and a maximum speed of \(2 \mathrm{~m} / \mathrm{s} .\) Find \((\mathrm{a})\) the force constant of the spring and \((\mathrm{b})\) the amplitude of the oscillation.

Weighing astronauts. In order to study the long-term effects of weightlessness, astronauts in space must be weighed (or at least "massed"). One way in which this is done is to seat them in a chair of known mass attached to a spring of known force constant and measure the period of the oscillations of this system. If the \(35.4 \mathrm{~kg}\) chair alone oscillates with a period of \(1.25 \mathrm{~s}\), and the period with the astronaut sitting in the chair is \(2.23 \mathrm{~s},\) find (a) the force constant of the spring and (b) the mass of the astronaut.

An object of unknown mass is attached to an ideal spring with force constant \(120 \mathrm{~N} / \mathrm{m}\) and is found to vibrate with a frequency of \(6.00 \mathrm{~Hz}\). Find (a) the period, (b) the angular frequency, and (c) the mass of this object.

A science museum has asked you to design a simple pendulum that will make 25.0 complete swings in \(85.0 \mathrm{~s}\). What length should you specify for this pendulum?

A simple pendulum in a science museum entry hall is \(3.50 \mathrm{~m}\) long, has a \(1.25 \mathrm{~kg}\) bob at its lower end, and swings with an amplitude of \(11.0^{\circ} .\) How much time does the pendulum take to swing from its extreme right side to its extreme left side?

You've made a simple pendulum with a length of \(1.55 \mathrm{~m}\), and you also have a (very light) spring with force constant \(2.45 \mathrm{~N} / \mathrm{m} .\) What mass should you add to the spring so that its period will be the same as that of your pendulum?

A pendulum consisting of a \(0.5 \mathrm{~kg}\) mass tied to a \(0.1 \mathrm{~m}\) string is set into oscillation at the same moment that a stone is dropped from a 44.1 -m-tall building. How many cycles of oscillation will the pendulum go through before the stone hits the ground?

A pendulum on Mars. A certain simple pendulum has a period on earth of \(1.60 \mathrm{~s}\). What is its period on the surface of Mars, where the acceleration due to gravity is \(3.71 \mathrm{~m} / \mathrm{s}^{2} ?\)

(a) If a pendulum has period \(T\) and you double its length, what is its new period in terms of \(T ?\) (b) If a pendulum has a length \(L\) and you want to triple its frequency, what should be its length in terms of \(L ?\) (c) Suppose a pendulum has a length \(L\) and period \(T\) on earth. If you take it to a planet where the acceleration of freely falling objects is 10 times what it is on earth, what should you do to the length to keep the period the same as on earth? (d) If you do not change the pendulum's length in part (c), what is its period on that planet in terms of \(T ?\) (e) If a pendulum has a period \(T\) and you triple the mass of its bob, what happens to the period (in terms of \(T\) )?

A \(1.35 \mathrm{~kg}\) object is attached to a horizontal spring of force constant \(2.5 \mathrm{~N} / \mathrm{cm}\) and is started oscillating by pulling it \(6.0 \mathrm{~cm}\) from its equilibrium position and releasing it so that it is free to oscillate on a frictionless horizontal air track. You observe that after eight cycles its maximum displacement from equilibrium is only \(3.5 \mathrm{~cm}\). (a) How much energy has this system lost to damping during these eight cycles? (b) Where did the "lost" energy go? Explain physically how the system could have lost energy.

A \(2.50 \mathrm{~kg}\) rock is attached at the end of a thin, very light rope \(1.45 \mathrm{~m}\) long and is started swinging by releasing it when the rope makes an \(11^{\circ}\) angle with the vertical. You record the observation that it rises only to an angle of \(4.5^{\circ}\) with the vertical after \(10 \frac{1}{2}\) swings. (a) How much energy has this system lost during that time? (b) What happened to the "lost" energy? Explain how it could have been "lost."

An astronaut notices that a pendulum that took \(2.50 \mathrm{~s}\) for a complete cycle of swing when the rocket was waiting on the launch pad takes \(1.25 \mathrm{~s}\) for the same cycle of swing during liftoff. What is the acceleration of the rocket? (Hint: Inside the rocket, it appears that \(g\) has increased.)

An object suspended from a spring vibrates with simple harmonic motion. At an instant when the displacement of the object is equal to one-half the amplitude, what fraction of the total energy of the system is kinetic and what fraction is potential?

An apple weighs \(1.00 \mathrm{~N}\). When you hang it from the end of a long spring of force constant \(1.50 \mathrm{~N} / \mathrm{m}\) and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back- and-forth swings do not cause any appreciable change in the length of the spring.) What is the unstretched length of the spring (with the apple removed)?

A block with mass \(M\) rests on a frictionless surface and is connected to a horizontal spring of force constant \(k\), the other end of which is attached to a wall (Figure 11.37 ). A second block with mass \(m\) rests on top of the first block. The coefficient of static friction between the blocks is \(\mu_{\mathrm{s}}\). Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

A \(100 \mathrm{~kg}\) mass suspended from a wire whose unstretched length is \(4.00 \mathrm{~m}\) is found to stretch the wire by \(6.0 \mathrm{~mm}\). The wire has a uniform crosssectional area of \(0.10 \mathrm{~cm}^{2}\). (a) If the load is pulled down a small additional distance and released, find the frequency at which it vibrates. (b) Compute Young's modulus for the wire.

Crude oil with a bulk modulus of \(2.35 \mathrm{GPa}\) is leaking from a deep- sea well \(2250 \mathrm{~m}\) below the surface of the ocean, where the water pressure is \(2.27 \times 10^{7} \mathrm{~Pa}\). Suppose 35,600 barrels of oil leak from the wellhead; assuming all that oil reaches the surface, how many barrels will it be on the surface?

"Seeing" surfaces at the nanoscale. One technique for making images of surfaces at the nanometer scale, including membranes and biomolecules, is dynamic atomic force microscopy. In this technique, a small tip is attached to a cantilever, which is a flexible, rectangular slab supported at one end, like a diving board (Figure 11.40 ). The cantilever vibrates, so the tip moves up and down in simple harmonic motion. In one mode of operation, the resonant frequency for a cantilever with force constant \(k=1000 \mathrm{~N} / \mathrm{m}\) is \(100 \mathrm{kHz}\). As the oscillating tip is brought within a few nanometers of the surface of a sample (as shown in the figure), it experiences an attractive force from the surface. For an oscillation with a small amplitude (typically \(0.050 \mathrm{nm}),\) the force \(F\) that the sample surface exerts on the tip varies linearly with the displacement \(x\) of the tip, \(|F|=k_{\text {surf }} x,\) where \(k_{\text {surf }}\) is the effective force constant for this force. The net force on the tip is therefore \(\left(k+k_{\text {surf }}\right) x,\) and the frequency of the oscillation changes slightly due to the interaction with the surface. Measurements of the frequency as the tip moves over different parts of the sample's surface can provide information about the sample. If we model the vibrating system as a mass on a spring, what is the mass necessary to achieve the desired resonant frequency when the tip is not interacting with the surface? A. \(25 \mathrm{ng}\) B. \(100 \mathrm{ng}\) C. \(25 \mu \mathrm{g}\) D. \(100 \mu \mathrm{g}\)