Chapter 11

\(\bullet\) 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.

\(\bullet\) 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 \(\mathrm{cm}^{2} ?\)

\(\bullet\) 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 \(\mathrm{cm}^{2}.\)

\(\bullet\) Stress on a mountaineer's rope. A nylon rope used by mountaineers elongates 1.10 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?

\(\bullet\) 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?

\(\bullet\) 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 kg sprinter exerts a force of 13 times his weight on his Achilles tendon, by how much will it stretch?

\(\bullet\) \(\bullet\) Human hair. According to one set of measurements, the tensile strength of hair is 196 \(\mathrm{MPa}\) , 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?

\(\bullet\) \(\bullet\) 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 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?

\(\bullet\) A solid gold bar is pulled up from the hold of the sunken RMS Titanic. (a) What happens to its volume as it goes from the pressure at the ship to the lower pressure at the ocean's surface? (b) The pressure difference is proportional to the depth. How many times greater would the volume change have been had the ship been twice as deep? (c) The bulk modulus of lead is one-fourth that of gold. Find the ratio of the volume change of a solid lead bar to that of a gold bar of equal volume for the same pressure change.

\(\bullet\) In the Challenger Deep of the Marianas Trench, the depth of seawater is 10.9 \(\mathrm{km}\) and the pressure is \(1.16 \times 10^{8}\) 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 \(1.0 \times 10^{3} \mathrm{Pa}\) , 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 sea-water at the depth of the Challenger Deep?

\(\bullet\) 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 \(1.0 \times 10^{4}\) 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?

\(\bullet\) 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?

\(\bullet\) \(\bullet\) 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 \(1.0 \times 10^{4}\) 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?

\(\bullet\) A steel wire has the following properties: $$\begin{array}{l}{\text { Length }=5.00 \mathrm{m}} \\ {\text { Cross- sectional area }=0.040 \mathrm{cm}^{2}} \\ {\text { Young's modulus = } 2.0 \times 10^{11} \mathrm{Pa}} \\ {\text { Shear modulus = } 0.84 \times 10^{11} \mathrm{Pa}} \\ {\text { Proportional limit }=3.60 \times 10^{8} \mathrm{Pa}} \\\ {\text { Breaking stress }=11.0 \times 10^{8} \mathrm{Pa}}\end{array}$$ 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?

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

\(\bullet\) (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 \(B\) 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 a 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?

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

\(\bullet\) 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.

\(\bullet\) The wings of the Blue-throated Hummingbird (Lampornis clemenciae), 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.

\(\bullet\) A 0.500 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 \(x=-0.015 \mathrm{m},(\mathrm{c})\) the magnitude of the maximum acceleration of the glider, (d) the acceleration of the glider at \(x=-0.015 \mathrm{m},\) and \((\mathrm{e})\) the total mechanical energy of the glider at any point in its motion.

\(\bullet\) A 0.150 \(\mathrm{kg}\) toy is undergoing SHM on the end of a horizontal spring with force constant 300.0 \(\mathrm{N} / \mathrm{m} .\) When the object is 0.0120 \(\mathrm{m}\) from its equilibrium position, it is observed to have a speed of 0.300 \(\mathrm{m} / \mathrm{s}\) . Find (a) the total energy of the object at any point in its motion, (b) the amplitude of the motion, and (c) the maximum speed attained by the object during its motion.

\(\bullet\) A 2.00 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.

\(\bullet\) \(\bullet\) 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?

\(\bullet\) \(\bullet\) A mass is oscillating with amplitude \(A\) at the end of a spring. 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?

\(\bullet\) A proud deep-sea fisherman hangs a 65.0 \(\mathrm{kg}\) fish from an ideal spring having negligible mass. The fish stretches the spring 0.120 \(\mathrm{m} .\) (a) What is the force constant of the spring? (b) What is the period of oscillation of the fish if it is pulled down 3.50 \(\mathrm{cm}\) and released?

\(\bullet\) \(\bullet\) One end of a stretched ideal spring is attached to an airtrack and the other is attached to a glider with a mass of 0.355 \(\mathrm{kg}\) . The glider is released and allowed to oscillate in SHM. If the distance of the glider from the fixed end of the spring varies between 1.80 \(\mathrm{m}\) and \(1.06 \mathrm{m},\) and the period of the oscillation is \(2.15 \mathrm{s},\) find (a) the force constant of the spring, (b) the maximum speed of the glider, and (c) the magnitude of the maximum acceleration of the glider.

\(\bullet\) A mass of 0.20 \(\mathrm{kg}\) on the end of a spring oscillates with a period of 0.45 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, and (c) the equation describing the motion of the mass, assuming that \(x\) was a maximum at time \(t=0\) .

\(\bullet\) \(\bullet\) A harmonic oscillator is made by using a 0.600 kg frictionless block and an ideal spring of unknown force constant. The oscillator is found to have a period of 0.150 s. Find the force constant of the spring.

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.

\(\bullet\) \(\bullet\) 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.

\(\bullet\) \(\bullet\) Weighing a virus. In February \(2004,\) scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just 30 nm long) with a laser, first without the virus and then after the virus had attached itself to the silicon. The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached \(\left(f_{\mathrm{s}+\mathrm{v}}\right)\) to the frequency without the virus \(\left(f_{\mathrm{s}}\right)\) is given by the formula $$\frac{f_{\mathrm{S}+\mathrm{V}}}{f_{\mathrm{S}}}=\frac{1}{\sqrt{1+\frac{m_{\mathrm{v}}}{m_{\mathrm{S}}}}},$$ where \(m_{\mathrm{v}}\) is the mass of the virus and \(m_{\mathrm{s}}\) is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of \(2.10 \times 10^{-16} \mathrm{g}\) and a frequency of \(2.00 \times 10^{15} \mathrm{Hz}\) without the virus and \(2.87 \times 10^{14} \mathrm{Hz}\) with the virus. What is the mass of the virus, in grams and femtograms?

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

\(\bullet\) 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?

\(\bullet\) 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?

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

\(\bullet\) 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.

\(\bullet\) A 2.50 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."

\(\bullet\) \(\bullet\) Inside a NASA test vehicle, a 3.50 -kg ball is pulled along by a horizontal ideal spring fixed to a friction-free table. The force constant of the spring is 225 \(\mathrm{N} / \mathrm{m} .\) The vehicle has a steady acceleration of \(5.00 \mathrm{m} / \mathrm{s}^{2},\) and the ball is not oscillating. Suddenly, when the vehicle's speed has reached \(45.0 \mathrm{m} / \mathrm{s},\) its engines turn off, thus eliminating its acceleration but not its velocity. Find (a) the amplitude and (b) the frequency of the resulting oscillations of the ball. (c) What will be the ball's maximum speed relative to the vehicle?

\(\bullet\) \(\bullet\) Four passengers with a combined mass of 250 \(\mathrm{kg}\) compress the springs of a car with wom-out shock absorbers by 4.00 \(\mathrm{cm}\) when they enter it. Model the car and passengers as a single body on a single ideal spring. If the loaded car has a period of vibration of \(1.08 \mathrm{s},\) what is the period of vibration of the empty car?

\(\bullet\) \(\bullet\) An astronaut notices that a pendulum which took 2.50 s for a complete cycle of swing when the rocket was waiting on the launch pad takes 1.25 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.)

\(\bullet\) \(\bullet\) 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?

\(\bullet\) \(\bullet\) 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)?

\(\bullet\) \(\bullet\) 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 11.41 ). A second block with mass \(m\) rests on top of the first block. The coefficient of static friction between the blocks is \(\mu_{s}\) . Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

\(\bullet\) \(\bullet\) Stress on the shinbone. The compressive strength of our bones is important in everyday life. Young's modulus for bone is approximately 14 GPa. Bone can take only about a 1.0\(\%\) change in its length before fracturing. If Hooke's law were to hold up to fracture: (a) What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 3.0 \(\mathrm{cm}^{2} .\) . This is approximately the cross-sectional area of a tibia, or shinbone, at its narrowest point.) (b) Estimate the maximum height from which a 70 \(\mathrm{kg}\) man can jump and not fracture the tibia. Take the time between when he first touches the floor and when he has stopped to be \(0.030 \mathrm{s},\) and assume that the stress is distributed equally between his legs.

\(\bullet\) \(\bullet \mathrm{A} 100 \mathrm{kg}\) 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 cross-sectional 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.

\(\bullet\) \(\bullet\) Rapunzel, Rapunzel, let down your golden hair. In the Grimms' fairy tale Rapunzel, she lets down her golden hair to a length of 20 yards (we'll use \(20 \mathrm{m},\) which is not much different) so that the prince can climb up to her room. Human hair has a Young's modulus of about 490 MPa, and we can assume that Rapunzel's hair can be squeezed into a rope about 2.0 \(\mathrm{cm}\) in cross-sectional diameter. The prince is described as young and handsome, so we can estimate a mass of 60 \(\mathrm{kg}\) for him. (a) Just after the prince has started to climb at constant speed, while he is still near the bottom of the hair, by how many centimeters does he stretch Rapunzel's hair? (b) What is the mass of the heaviest prince that could climb up, given that the maximum tensile stress hair can support is 196 MPa? (Assume that Hooke's law holds up to the breaking point of the hair, even though that would not actually be the case.)