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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 13

Vibrations and Waves

Educators

cm

Problem 1

A block of mass $m=0.60 \mathrm{kg}$ attached to a spring with force constant 130 $\mathrm{N} / \mathrm{m}$ is free to move on a friction-less, horizontal surface as in Figure $\mathrm{P} 13.1$ . The block is released from rest after the spring is stretched a distance $A=$ 0.13 $\mathrm{m}$ . At that instant, find (a) the force on the block and (b) its acceleration.

Averell H.
Carnegie Mellon University

Problem 2

A spring oriented vertically is attached to a hard horizontal surface as in Figure P13.2. The
spring has a force constant of 1.46 $\mathrm{kN} / \mathrm{m}$ . How much is the spring compressed when a object of mass $m=2.30 \mathrm{kg}$ is placed on top of the spring and the system is at rest?

cm
Charles M.
Numerade Educator

Problem 3

The force constant of a spring is 137 $\mathrm{N} / \mathrm{m}$ . Find the magnitude of the force required to (a) compress the spring by 4.80 $\mathrm{cm}$ from its unstretched length and (b) stretch the spring by 7.36 $\mathrm{cm}$ from its unstretched length.

Averell H.
Carnegie Mellon University

Problem 4

A spring is hung from a ceiling, and an object attached to its lower end stretches the spring by a distance
$d=5.00 \mathrm{cm}$ from its unstretched position when the system is in equilibrium as in Figure P13.4. If the spring constant is $47.5 \mathrm{N} / \mathrm{m},$ determine the mass of the object.

cm
Charles M.
Numerade Educator

Problem 5

A biologist hangs a sample of mass 0.725 $\mathrm{kg}$ on a pair of identical, vertical springs in parallel and slowly lowers the sample to equilibrium, stretching the springs by 0.200 $\mathrm{m}$ . Calculate the value of the spring constant of one of the springs.

Averell H.
Carnegie Mellon University

Problem 6

An archer must exert a force of 375 $\mathrm{N}$ on the bowstring shown in Figure $P 13.6 a$ such that the string makes an angle of $\theta=35.0^{\circ}$ with the vertical. (a) Determine the tension in the bowstring. (b) If the applied force is replaced by a stretched spring as in Figure $P 13.6 \mathrm{b}$ and the spring is stretched 30.0 $\mathrm{cm}$ from its unstretched length, what is the spring constant?

Averell H.
Carnegie Mellon University

Problem 7

A spring 1.50 $\mathrm{m}$ long with force constant 475 $\mathrm{N} / \mathrm{m}$ is hung from the ceiling of an elevator, and a block of mass 10.0 kg is attached to the bottom of the spring. (a) By how much is the spring stretched when the block is slowly lowered to its equilibrium point? (b) If the elevator subsequently accelerates upward at 2.00 $\mathrm{m} / \mathrm{s}^{2}$ , what is the position of the block, taking the equilibrium position found in part (a) as $y=0$ and upwards as the positive $y$ -direction. (c) If the elevator cable snaps during the acceleration, describe the subsequent motion of the block relative to the freely falling elevator. What is the amplitude of its motion?

Averell H.
Carnegie Mellon University

Problem 8

A block of mass $m=2.00 \mathrm{kg}$ is attached to a spring of force constant $k=5.00 \times 10^{2} \mathrm{N} / \mathrm{m}$ that lies on a horizontal frictionless surface as shown in Figure $\mathrm{P} 13.8$ . The block is pulled to a position $x_{i}=5.00 \mathrm{cm}$ to the right of equilibrium and the right of equilibrium and released from rest. Find (a) the work required to stretch the spring and (b) the speed the block has as it passes through equilibrium.

cm
Charles M.
Numerade Educator

Problem 9

A slingshot consists of a light leather cup containing a stone. The cup is pulled back against two parallel rubber bands. It takes a force of 15.0 $\mathrm{N}$ to stretch either one of these bands 1.00 $\mathrm{cm} .$ (a) What is the potential energy stored in the two bands together when a $50.0-\mathrm{g}$ stone is placed in the cup and pulled back 0.200 $\mathrm{m}$ from the equilibrium position? (b) With what speed does the stone leave the slingshot?

Averell H.
Carnegie Mellon University

Problem 10

An archer pulls her bowstring back 0.400 $\mathrm{m}$ by exerting a force that increases uniformly from zero to 230 $\mathrm{N}$ . (a) What is the equivalent spring constant of the bow? (b) How much work is done in pulling the bow?

cm
Charles M.
Numerade Educator

Problem 11

A student pushes the $1.50-\mathrm{kg}$ block in Figure $\mathrm{P} 13.11$ against a horizontal spring, compressing it by 0.125 $\mathrm{m}$ . When released, the block travels across a horizontal surface and up an incline. Neglecting friction, find the block's maximum height if the spring constant is $k=575 \mathrm{N} / \mathrm{m} .$

Averell H.
Carnegie Mellon University

Problem 12

An automobile having a mass of $1.00 \times 10^{3} \mathrm{kg}$ is driven into a brick wall in a safety test. The bumper behaves like a spring with constant $5.00 \times 10^{6} \mathrm{N} / \mathrm{m}$ and is compressed 3.16 $\mathrm{cm}$ as the car is brought to rest. What was the speed of the car before impact, assuming no energy is lost in the collision with the wall?

cm
Charles M.
Numerade Educator

Problem 13

A 10.0 -g bullet is fired into, and embeds itself in, a $2.00-\mathrm{kg}$ block attached to a spring with a force constant of 19.6 $\mathrm{N} / \mathrm{m}$ and having negligible mass. How far is the spring compressed if the bullet has a speed of $300 . \mathrm{m} / \mathrm{s}$ just before it strikes the block and the block slides on a frictionless surface? Note: You must use conservation of momentum in this problem because of the inelastic collision between the bullet and block.

Averell H.
Carnegie Mellon University

Problem 14

An object-spring system moving with simple harmonicmotion has an amplitude $A .(a)$ What is the total energy of the system in terms of $k$ and $A$ only? (b) Suppose at a certain instant the kinetic energy is twice the elastic potential energy. Write an equation describing this situation, using only the variables for the mass $m,$ velocity $v$ spring constant $k,$ and position $x$ . (c) Using the results of parts (a) and (b) and the conservation of energy equation, find the positions $x$ of the object when its kinetic energy equals twice the potential energy stored in the spring. (The answer should in terms of $A$ only. $)$

Averell H.
Carnegie Mellon University

Problem 15

A horizontal block-spring system with the block on a frictionless surface has total mechanical energy $E=47.0 \mathrm{J}$ and a maximum displacement from equilibrium of 0.240 $\mathrm{m} .$ (a) What is the spring constant? (b) What is the kinetic energy of the system at the equilibrium point? (c) If the maximum speed of the block is $3.45 \mathrm{m} / \mathrm{s},$ what is its mass? (d) What is the speed of the block when its displacement is 0.160 $\mathrm{m} ?(\mathrm{e})$ Find the kinetic energy of the block at $x=0.160 \mathrm{m} .$ (f) Find the potential energy stored in the spring when $x=$ 0.160 $\mathrm{m} .$ (g) Suppose the same system is released from rest at $x=0.240 \mathrm{m}$ on a rough surface so that it loses 14.0 $\mathrm{J}$ by the time it reaches its first turning point (after passing equilibrium at $x=0$ ). What is its position at that instant?

Averell H.
Carnegie Mellon University

Problem 16

A 0.250 -kg block attached to a light spring oscillates on a frictionless, horizontal table. The oscillation amplitude is $A=0.125 \mathrm{m}$ and the block moves at 3.00 $\mathrm{m} / \mathrm{s}$ as it passes through equilibrium at $x=0 .$ (a) Find the spring constant, $k$ (b) Calculate the total energy of the block-spring system. (c) Find the block's speed when $x=A / 2$

cm
Charles M.
Numerade Educator

Problem 17

A block-spring system consists of a spring with constant $k=$ 425 $\mathrm{N} / \mathrm{m}$ attached to a $2.00-\mathrm{kg}$ block on a frictionless surface. The block is pulled 8.00 $\mathrm{cm}$ from equilibrium and released from rest. For the resulting oscillation, find the $(\mathrm{a})$ amplitude, (b) angular frequency, (c) frequency, and (d) period. What is the maximum value of the block's (e) velocity and (f) acceleration?

Averell H.
Carnegie Mellon University

Problem 18

A $0.40-\mathrm{kg}$ object connected to a light spring with a force constant of 19.6 $\mathrm{N} / \mathrm{m}$ oscillates on a frictionless horizontal surface. If the spring is compressed 4.0 $\mathrm{cm}$ and released from rest, determine (a) the maximum speed of the object, (b) the
speed of the object when the spring is compressed $1.5 \mathrm{cm},$ and (c) the speed of the object as it passes the point 1.5 $\mathrm{cm}$ from the equilibrium position. (d) For what value of $x$ does the speed equal one-half the maximum speed?

cm
Charles M.
Numerade Educator

Problem 19

At an outdoor market, a bunch of bananas attached to the bottom of a vertical spring of force constant 16.0 $\mathrm{N} / \mathrm{m}$ is set into oscillatory motion with an amplitude of 20.0 $\mathrm{cm} .$ It is observed that the maximum speed of the bunch of bananas is 40.0 $\mathrm{cm} / \mathrm{s} .$ What is the weight of the bananas in newtons?

Averell H.
Carnegie Mellon University

Problem 20

A student stretches a spring, attaches a $1.00-\mathrm{kg}$ mass to it, and releases the mass from rest on a frictionless surface. The resulting oscillation has a period of 0.500 $\mathrm{s}$ and an amplitude of 25.0 $\mathrm{cm}$ . Determine (a) the oscillation frequency, (b) the spring constant, and (c) the speed of the mass when it is halfway to the equilibrium position.

Averell H.
Carnegie Mellon University

Problem 21

A horizontal spring attached to a wall has a force constant of $k=8.50 \times 10^{2} \mathrm{N} / \mathrm{m}$ . A block of mass $m=1.00 \mathrm{kg}$ is attached to the spring and rests on a frictionless, horizontal surface as in Figure $\mathrm{P} 13.21 .$ (a) The block is pulled to a position $x_{i}=6.00 \mathrm{cm}$ from equilibrium and released. Find the potential energy stored in the spring when the block is 6.00 $\mathrm{cm}$ from equilibrium. (b) Find the speed of the block as it passes through the equilibrium position. (c) What is the speed of the block when it is at a position $x_{i} / 2=3.00 \mathrm{cm}$ ?

Averell H.
Carnegie Mellon University

Problem 22

An object moves uniformly around a circular path of radius $20.0 \mathrm{cm},$ making one complete revolution every 2.00 $\mathrm{s}$ . What are (a) the translational speed of the object, (b) the frequency
of motion in hertz, and (c) the angular speed of the object?

cm
Charles M.
Numerade Educator

Problem 23

The wheel in the simplified engine of Figure $P 13.23$ has radius $A=0.250 \mathrm{m}$ and rotates with angular frequency $\omega=12.0 \mathrm{rad} / \mathrm{s}$ . At $t=0$ , the piston is located at $x=A$ . Calculate the piston's (a) position, (b) velocity, and (c) acceleration at $t=1.15 \mathrm{s}$ .

Guilherme B.
Numerade Educator

Problem 24

The period of motion of an object-spring system is $T=0.528$ s when an object of mass $m=238 \mathrm{g}$ is attached to the spring. Find (a) the frequency of motion in hertz and (b) the force constant of the spring. (c) If the total energy of the oscillating motion is $0.234 \mathrm{J},$ find the amplitude of the oscillations.

cm
Charles M.
Numerade Educator

Problem 25

A vertical spring stretches 3.9 $\mathrm{cm}$ when a $10 .$ -g object is hung from it. The object is replaced with a block of mass 25 $\mathrm{g}$ that oscillates up and down in simple harmonic motion. Calculate the period of motion.

Averell H.
Carnegie Mellon University

Problem 26

When four people with a combined mass of 320 $\mathrm{kg}$ sit down in a $2.0 \times 10^{3}-$ -kg car, they find that their weight compresses the springs an additional 0.80 $\mathrm{cm} .$ (a) What is the effective force constant of the springs? (b) The four people get out of the car and bounce it up and down. What is the frequency of the car's vibration?

Averell H.
Carnegie Mellon University

Problem 27

The position of an object connected to a spring varies with time according to the expression $x=(5.2 \mathrm{cm})$ sin $(8.0 \pi t) .$ Find (a) the period of this motion, (b) the frequency of the motion, (c) the amplitude of the motion, and (d) the first time after $t=0$ that the object reaches the position $x=2.6 \mathrm{cm} .$

Averell H.
Carnegie Mellon University

Problem 28

A harmonic oscillator is described by the function $x(t)=$ $(0.200 \mathrm{m})$ cos $(0.350 t) .$ Find the oscillator's (a) maximum velocity and (b) maximum acceleration. Find the oscillator's (c) position, (d) velocity, and (e) acceleration when $t=2.00 \mathrm{s}$

cm
Charles M.
Numerade Educator

Problem 29

A 326 -g object is attached to a spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is $5.83 \mathrm{J},$ find (a) the maximum speed of the object, (b) the force constant of the spring, and (c) the amplitude of the motion.

Averell H.
Carnegie Mellon University

Problem 30

An object executes simple harmonic motion with an amplitude $A$ . (a) At what values of its position does its speed equal half its maximum speed? (b) At what values of its position does its potential energy equal half the total energy?

Averell H.
Carnegie Mellon University

Problem 31

A 2.00 -kg object on a frictionless horizontal track is attached to the end of a horizontal spring whose force constant is 5.00 $\mathrm{N} / \mathrm{m}$ . The object is displaced 3.00 $\mathrm{m}$ to the right from its equilibrium position and then released, initiating simple harmonic motion. (a) What is the force (magnitude and direction) acting on the object 3.50 $\mathrm{s}$ after it is released? (b) How many times does the object oscillate in 3.50 $\mathrm{s}$ ?

Averell H.
Carnegie Mellon University

Problem 32

A spring of negligible mass stretches 3.00 $\mathrm{cm}$ from its relaxed length when a force of 7.50 $\mathrm{N}$ is applied. A $0.500-\mathrm{kg}$ particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to $x=5.00 \mathrm{cm}$ and released from rest at $t=0$ . (a) What is the force constant of the spring? (b) What are the angular frequency $\omega,$ the frequency, and the period of the motion? (c) What is the total energy of the system? (d) What is the amplitude of the motion? (e) What are the maximum velocity and the maximum acceleration of the particle? (f) Determine the displacement $x$ of the particle from the equilibrium position at $t=0.500$ s. (g) Determine the velocity and acceleration of the particle when $t=0.500$ s.

Sheh Lit C.
University of Washington

Problem 33

Given that $x=A \cos (\omega t)$ is a sinusoidal function of time, show that $v(\text { velocity })$ and $a$ (acceleration) are also sinusoidal functions of time. Hint: Use Equations 13.6 and $13.2 .$

Averell H.
Carnegie Mellon University

Problem 34

A man enters a tall tower, needing to know its height. He notes that a long pendulum extends from the ceiling almost to the floor and that its period is 15.5 $\mathrm{s}$ . (a) How tall is the tower? (b) If this pendulum is taken to the Moon, where the free-fall acceleration is $1.67 \mathrm{m} / \mathrm{s}^{2},$ what is the period there?

Averell H.
Carnegie Mellon University

Problem 35

A simple pendulum has a length of 52.0 $\mathrm{cm}$ and makes 82.0 complete oscillations in 2.00 $\mathrm{min}$ . Find (a) the period of the pendulum and (b) the value of $g$ at the location of the pendulum.

Averell H.
Carnegie Mellon University

Problem 36

A seconds pendulum is one that moves through its equilibrium position once each second. (The period of the pendulum is 2.000 s.) The length of a seconds pendulum is 0.9927 $\mathrm{m}$ at Tokyo and 0.9942 $\mathrm{m}$ at Cambridge, England. What is the ratio of the free-fall accelerations at these two locations?

Averell H.
Carnegie Mellon University

Problem 37

A clock is constructed so that it keeps perfect time when its simple pendulum has a period of 1.000 s at locations where $g=9.800 \mathrm{m} / \mathrm{s}^{2} .$ The pendulum bob has length $L=0.2482 \mathrm{m},$ and instead of keeping perfect time, the clock runs slow by 1.500 minutes per day. (a) What is the free-fall acceleration at the clock's location? (b) What length of pendulum bob is required for the clock to keep perfect time?

Averell H.
Carnegie Mellon University

Problem 38

A coat hanger of mass $m=0.238$ kg oscillates on a peg as a physical pendulum as shown in Figure P13.38. The distance from the pivot to the center of mass of the coat hanger is $d=18.0 \mathrm{cm}$ and the period of the motion is $T=1.25 \mathrm{s}$ . Find the moment of inertia of the coat hanger about the pivot.

Averell H.
Carnegie Mellon University

Problem 39

The free-fall acceleration on Mars is 3.7 $\mathrm{m} / \mathrm{s}^{2}$ (a) What length of pendulum has a period of 1.0 $\mathrm{s}$ on Earth? (b) What length of pendulum would have a $1.0-\mathrm{s}$ period on Mars? An object is suspended from a spring with force constant 10.0 $\mathrm{N} / \mathrm{m} .$ Find the mass suspended from this spring that would result in a period of 1.0 $\mathrm{s}(\mathrm{c})$ on Earth and (d) on Mars.

Averell H.
Carnegie Mellon University

Problem 40

A simple pendulum is 5.00 $\mathrm{m}$ long. (a) What is the period of simple harmonic motion for this pendulum if it is located in an elevator accelerating upward at 5.00 $\mathrm{m} / \mathrm{s}^{2} ?$ (b) What is its period if the elevator is accelerating downward at 5.00 $\mathrm{m} / \mathrm{s}^{2} ?(\mathrm{c})$ What is the period of simple harmonic motion for the pendulum if it is placed in a truck that is accelerating horizontally at 5.00 $\mathrm{m} / \mathrm{s}^{2} ?$

Averell H.
Carnegie Mellon University

Problem 41

The sinusoidal wave shown in Figure $\mathrm{P} 13.41$ is traveling in the positive $x$ -direction and has a frequency of 18.0 $\mathrm{Hz}$ . Find the (a) amplitude, (b) wavelength, (c) period, and (d) speed of the wave.

Averell H.
Carnegie Mellon University

Problem 42

An object attached to a spring vibrates with simple harmonic motion as described by Figure $\mathrm{P} 13.42 .$ For this motion, find (a) the amplitude, (b) the period, (c) the angular frequency, (d) the maximum speed, (e) the maximum acceleration, and (f) an equation for its position $x$ in terms of a sine function.

Averell H.
Carnegie Mellon University

Problem 43

Light waves are electromagnetic waves that travel at $3.00 \times$ $10^{8} \mathrm{m} / \mathrm{s}$ . The eye is most sensitive to light having a wavelength of $5.50 \times 10^{-7} \mathrm{m} .$ Find (a) the frequency of this light wave and (b) its period.

Averell H.
Carnegie Mellon University

Problem 44

The distance between two successive minima of a transverse wave is 2.76 $\mathrm{m}$ . Five crests of the wave pass a given point along the direction of travel every 14.0 $\mathrm{s}$ . Find (a) the frequency of the wave and (b) the wave speed.

Averell H.
Carnegie Mellon University

Problem 45

A harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum travels 425 $\mathrm{cm}$ along the rope in 10.0 s. What is the wavelength?

Averell H.
Carnegie Mellon University

Problem 46

A bat can detect small objects, such as an insect, whose size is approximately equal to one wavelength of the sound the bat makes. If bats emit a chirp at a frequency of $60.0 \times 10^{3} \mathrm{Hz}$
and the speed of sound in air is $343 \mathrm{m} / \mathrm{s},$ what is the smallest insect a bat can detect?

Averell H.
Carnegie Mellon University

Problem 47

Orchestra instruments are commonly tuned to match an A-note played by the principal oboe. The Baltimore Symphony Orchestra tunes to an A-note at 440 $\mathrm{Hz}$ while the Boston Symphony Orchestra tunes to 442 $\mathrm{Hz}$ . If the speed of sound is constant at 343 $\mathrm{m} / \mathrm{s}$ , find the magnitude of difference between the wavelengths of these two different A-notes.

Averell H.
Carnegie Mellon University

Problem 48

Ocean waves are traveling to the east at 4.0 $\mathrm{m} / \mathrm{s}$ with a distance of 20.0 $\mathrm{m}$ between crests. With what frequency do the waves hit the front of a boat (a) when the boat is at anchor and (b) when the boat is moving westward at 1.0 $\mathrm{m} / \mathrm{s}$ ?

Averell H.
Carnegie Mellon University

Problem 49

An ethernet cable is 4.00 $\mathrm{m}$ long and has a mass of 0.200 $\mathrm{kg}$ . A transverse wave pulse is produced by plucking one end of the taut cable. The pulse makes four trips down and back
along the cable in 0.800 s. What is the tension in the cable?

Averell H.
Carnegie Mellon University

Problem 50

Workers attach a $25.0-\mathrm{kg}$ mass to one end of a $20.0-\mathrm{m}$ long cable and secure the other end to the top of a stationary crane, suspending the mass in midair. If the cable has a mass
of $12.0 \mathrm{kg},$ determine the speed of transverse waves at (a) the middle and (b) the bottom end of the cable. (Hint: Don't neglect the cable's mass. Because of it, the tension increases from a minimum value at the bottom of the cable to a maximum value at the top.)

Averell H.
Carnegie Mellon University

Problem 51

A piano string of mass per unit length $5.00 \times 10^{-3} \mathrm{kg} / \mathrm{m}$ is under a tension of 1350 $\mathrm{N}$ . Find the speed with which a wave travels on this string.

Averell H.
Carnegie Mellon University

Problem 52

A student taking a quiz finds on a reference sheet the two equations
$$
f=\frac{1}{T} \quad \text { and } \quad v=\sqrt{\frac{T}{\mu}}
$$
She has forgotten what $T$ represents in each equation. (a) Use dimensional analysis to determine the units required for $T$ in each equation. (b) Explain how you can identify the physical quantity each Trepresents from the units.

Averell H.
Carnegie Mellon University

Problem 53

Transverse waves with a speed of 50.0 $\mathrm{m} / \mathrm{s}$ are to be produced on a stretched string. A $5.00-\mathrm{m}$ length of string with a total mass of 0.0600 $\mathrm{kg}$ is used. (a) What is the required tension in the string? (b) Calculate the wave speed in the string if the tension is 8.00 $\mathrm{N}$ .

Averell H.
Carnegie Mellon University

Problem 54

An astronaut on the Moon wishes to measure the local value of $g$ by timing pulses traveling down a wire that has a large object suspended from it. Assume a wire of mass 4.00 $\mathrm{g}$ is 1.60 $\mathrm{m}$ long and has a $3.00-\mathrm{kg}$ object suspended from it. A pulse requires 36.1 $\mathrm{ms}$ to traverse the length of the wire. Calculate $g_{\text { Moon }}$ from these data. (You may neglect the mass of the wire when calculating the tension in it.)

Averell H.
Carnegie Mellon University

Problem 55

A simple pendulum consists of a ball of mass 5.00 $\mathrm{kg}$ hanging from a uniform string of mass 0.0600 $\mathrm{kg}$ and length $L$ . If the period of oscillation of the pendulum is 2.00 $\mathrm{s}$ s, determine the speed of a transverse wave in the string when the pendulum hangs vertically.

Averell H.
Carnegie Mellon University

Problem 56

A string is 50.0 $\mathrm{cm}$ long and has a mass of 3.00 $\mathrm{g} .$ A wave travels
at 5.00 $\mathrm{m} / \mathrm{s}$ along this string. A second string has the same length, but half the mass of the first. If the two strings are under the same tension, what is the speed of a wave along the second string?

Averell H.
Carnegie Mellon University

Problem 57

Tension is maintained in a string as in Figure $P 13.57$ The observed wave speed is $v=$ 24.0 $\mathrm{m} / \mathrm{s}$ when the suspended mass is $m=3.00 \mathrm{kg}$ . (a) What is the mass per unit length of the string? (b) What is the wave speed when the suspended mass is $m=$ 2.00 $\mathrm{kg?}$

Averell H.
Carnegie Mellon University

Problem 58

The elastic limit of a piece of steel wire is $2.70 \times 10^{9}$ Pa. What is the maximum speed at which transverse wave pulses can propagate along the wire without exceeding its elastic limit? (The density of steel is $7.86 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} . )$

Averell H.
Carnegie Mellon University

Problem 59

A 2.65 -kg power line running between two towers has a length of 38.0 $\mathrm{m}$ and is under a tension of 12.5 N. (a) What is the speed of a transverse pulse set up on the line? (b) If the tension in the line was unknown, describe a procedure a worker on the ground might use to estimate the tension.

Averell H.
Carnegie Mellon University

Problem 60

A taut clothesline has length $L$ and a mass $M .$ A transverse pulse is produced by plucking one end of the clothesline. If the pulse makes $n$ round trips along the clothesline in $t$ seconds, find expressions for (a) the speed of the pulse in terms of $n, L,$ and $t$ and (b) the tension $F$ in the clothesline in terms of the same variables and mass $M .$

Averell H.
Carnegie Mellon University

Problem 61

A wave of amplitude 0.30 $\mathrm{m}$ interferes with a second wave of amplitude 0.20 $\mathrm{m}$ traveling in the same direction. What are (a) the largest and (b) the smallest resultant amplitudes that can occur, and under what conditions will these maxima and minima arise?

Averell H.
Carnegie Mellon University

Problem 62

The position of a $0.30-\mathrm{kg}$ object attached to a spring is described by
$$
x=(0.25 \mathrm{m}) \cos (0.4 \pi t)
$$
Find (a) the amplitude of the motion, (b) the spring constant, (c) the position of the object at $t=0.30 \mathrm{s},$ and (d) the object's speed at $t=0.30 \mathrm{s} .$

cm
Charles M.
Numerade Educator

Problem 63

An object of mass 2.00 $\mathrm{kg}$ is oscillating freely on a vertical spring with a period of 0.600 s. Another object of unknown mass on the same spring oscillates with a period of 1.05 s. Find (a) the spring constant $k$ and (b) the unknown mass.

Averell H.
Carnegie Mellon University

Problem 64

A certain tuning fork vibrates at a frequency of 196 $\mathrm{Hz}$ while each tip of its two prongs has an amplitude of 0.850 $\mathrm{mm}$ . (a) What is the period of this motion? (b) Find the wavelength of the sound produced by the vibrating fork, taking the speed of sound in air to be 343 $\mathrm{m} / \mathrm{s}$ .

Averell H.
Carnegie Mellon University

Problem 65

A simple pendulum has mass 1.20 $\mathrm{kg}$ and length 0.700 $\mathrm{m} .$ (a) What is the period of the pendulum near the surface of Earth? (b) If the same mass is attached to a spring, what spring constant would result in the period of motion found in part (a)?

Averell H.
Carnegie Mellon University

Problem 66

A $0.500-$ kg block is released from rest and slides down a frictionless track that begins 2.00 $\mathrm{m}$ above the horizontal, as shown in Figure $\mathrm{P} 13.66 .$ At the bottom of the track, where the surface is horizontal, the block strikes and sticks to a light spring with a spring constant of 20.0 $\mathrm{N} / \mathrm{m} .$ Find the maximum distance the spring is compressed.

Averell H.
Carnegie Mellon University

Problem 67

A $3.00-\mathrm{kg}$ object is fastened to a light spring, with the intervening cord passing over a pulley (Fig. $\mathrm{P} 13.67 ) .$ The pulley is frictionless, and its inertia may be neglected. The object is
released from rest when the spring is unstretched. If the object drops 10.0 $\mathrm{cm}$ before stopping, find (a) the spring constant of the spring and (b) the speed of the object when it is 5.00 $\mathrm{cm}$ below its starting point.

Averell H.
Carnegie Mellon University

Problem 68

A 5.00 -g bullet moving with an initial speed of $400 . \mathrm{m} / \mathrm{s}$ is fired into and passes through a 1.00 $\mathrm{-kg}$ block, as in Figure $\mathrm{P} 13.68$ . The block, initially at rest on a frictionless horizontal surface, is connected to a spring with a spring constant of $900 . \mathrm{N} / \mathrm{m}$ . If the block moves 5.00 $\mathrm{cm}$ to the right after impact, find (a) the speed at which the bullet emerges from the block and (b) the mechanical energy lost in the collision.

Averell H.
Carnegie Mellon University

Problem 69

A large block $P$ executes horizontal simple harmonic motion as it slides across a frictionless surface with a frequency $f=1.50 \mathrm{Hz}$ . Block $B$ rests on it, as shown in Figure $\mathrm{P} 13.69$ , and the coefficient of static friction between the two is $\mu_{s}=0.600 .$ What maximum amplitude of oscillation can the system have if block $B$ is not to slip?

Averell H.
Carnegie Mellon University

Problem 70

A spring in a toy gun has a spring constant of 9.80 $\mathrm{N} / \mathrm{m}$ and can be compressed 20.0 $\mathrm{cm}$ beyond the equilibrium position. A 1.00 -g pellet resting against the spring is propelled forward when the spring is released. (a) Find the muzzle speed of the pellet. (b) If the pellet is fired horizontally from a height of 1.00 $\mathrm{m}$ above the floor, what is its range?

Averell H.
Carnegie Mellon University

Problem 71

A light balloon filled with helium of density 0.179 $\mathrm{kg} /$ $\mathrm{m}^{3}$ is tied to a light string of length $L=3.00 \mathrm{m} .$ The string is tied to the ground, forming an inverted simple pendulum (Fig. $\mathrm{P} 13.71 \mathrm{a} ) .$ If the balloon is displaced slightly from equilibrium, as in Figure $\mathrm{P} 13.71 \mathrm{b},(\mathrm{a})$ show that the motion is simple harmonic and (b) determine the period of the motion. Take the density of air to be 1.29 $\mathrm{kg} / \mathrm{m}^{3} .$ Hint: Use an analogy with the simple pendulum discussed in the text, and see Topic 9.

Averell H.
Carnegie Mellon University

Problem 72

An object of mass $m$ is connected to two rubber bands of length $L$ , each under tension $F,$ as in Figure $\mathrm{Pl} 3.72$ . The object is displaced vertically by a small distance $y$ . Assuming the tension does not change, show that (a) the restoring force is $-(2 F / L) y$ and (b) the system exhibits simple harmonic motion with an angular frequency $\omega=\sqrt{2 F / m L}$

Averell H.
Carnegie Mellon University

Problem 73

Assume a hole is drilled through the center of the Earth. It can be shown that an object of mass $m$ at a distance $r$ from the center of the Earth is pulled toward the center only by the material in the shaded portion of Figure $\mathrm{P} 13.73$ . Assume Earth has a uniform density $\rho .$ Write down Newton's law of gravitation for an object at a distance $r$ from the center of the Earth and show that
the force on it is of the form of Hooke's law, $F=-k r$ , with an effective force constant of $k=\left(\frac{4}{3}\right) \pi \rho G m,$ where $G$ is the gravitational constant.

Averell H.
Carnegie Mellon University

Problem 74

Figure $P 13.74$ shows a crude model of an insect wing. The mass $m$ represents the entire mass of the wing, which pivots about the fulcrum $F$ . The spring represents the surrounding connective tissue. Motion of the wing corresponds to vibration of the spring. Suppose the mass of the wing is 0.30 $\mathrm{g}$ and the effective spring constant of the tissue is $4.7 \times$ $10^{-4} \mathrm{N} / \mathrm{m}$ . If the mass $m$ moves up and down a distance of 2.0 $\mathrm{mm}$ from its position of equilibrium, what is the maximum speed of the outer tip of the wing?

Averell H.
Carnegie Mellon University

Problem 75

A $2.00-\mathrm{kg}$ block hangs without vibrating at the end of a spring $(k=500 . \mathrm{N} / \mathrm{m})$ that is attached to the ceiling of an elevator car. The car is rising with an upward acceleration of $g / 3$ when the acceleration suddenly ceases (at $t=0 ) .$ (a) What is the angular frequency of oscillation of the block after the acceleration ceases? (b) By what amount is the spring stretched during the time that the elevator car is accelerating?

Averell H.
Carnegie Mellon University

Problem 76

A system consists of a vertical spring with force constant $k=1250 \mathrm{N} / \mathrm{m}$ , length $L=1.50 \mathrm{m},$ and object of mass $m=5.00 \mathrm{kg}$ attached to the end (Fig. $\mathrm{P} 13.76 ) .$ The object is $\mathrm{placed}$ at the level of the point of attachment with the spring unstretched, at position $y_{i}=L,$ and then it is released so that it swings like a pendulum. (a) Write Newton's second law symbolically for the system as the object passes through its lowest point. (Note that at the lowest point, $r=L-y_{f} )$ (b) Write the conservation of energy equation symbolically, equating the total mechanical energies at the initial point and lowest point. (c) Find the coordinate position of the lowest point. (d) Will this pendulum's period be greater or less than the period of a simple pendulum with the same mass $m$ and length $I$ ? Explain.

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