College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 13

Vibrations and Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

A $0.60-\mathrm{kg}$ block attached to a spring with force constant $130 \mathrm{~N} / \mathrm{m}$ is free to move on a frictionless, horizontal surface as in Figure 13.7. The block is released from rest after the spring is stretched $0.13 \mathrm{~m}$. At that instant, find (a) the force on the block and (b) its acceleration.

Lisa Tarman

Lisa Tarman

Numerade Educator

Problem 2

When a $4.25-\mathrm{kg}$ object is placed on top of a vertical spring, the spring compresses a distance of $2.62 \mathrm{~cm}$. What is the force constant of the spring?

Keshav Singh

Numerade Educator

Problem 3

A ball dropped from a height of $4.00 \mathrm{~m}$ makes a perfectly elastic collision with the ground. Assuming no mechanical energy is lost due to air resistance, (a) show that the motion is periodic and (b) determine the period of the motion. (c) Is the motion simple harmonic? Explain.

Prabhu Ramji

Numerade Educator

Problem 4

A load of $50 \mathrm{~N}$ attached to a spring hanging vertically stretches the spring $5.0 \mathrm{~cm}$. The spring is now placed horizontally on a table and stretched $11 \mathrm{~cm} .$ (a) What force is required to stretch the spring by that amount? (b) Plot a graph of force (on the $y$ -axis) versus spring displacement from the equilibrium position along the $x$ -axis.

Prabhu Ramji

Numerade Educator

Problem 5

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

Prabhu Ramji

Numerade Educator

Problem 6

An archer must exert a force of $375 \mathrm{~N}$ on the bowstring shown in Figure $\mathrm{P} 13.6 \mathrm{a}$ 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 $\mathrm{P} 13.6 \mathrm{~b}$ and the spring is stretched $30.0 \mathrm{~cm}$ from its unstretched length, what is the spring constant?

Averell Hause

Carnegie Mellon University

Problem 7

ecp 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 $10.0 \mathrm{~kg}$ 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?

Prabhu Ramji

Numerade Educator

Problem 8

A spring-loaded pellet gun is designed to fire $3.00-\mathrm{g}$ projectiles horizontally at a speed of $45.0 \mathrm{~m} / \mathrm{s}$. (a) If the spring is compressed to its maximum design difference of $8.00 \mathrm{~cm}$, what spring constant is required? (b) What maximum force is required to load the gun?

Prabhu Ramji

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 $15 \mathrm{~N}$ to stretch either one

Prabhu Ramji

Numerade Educator

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 Magnusen

Numerade Educator

Problem 11

A child's toy consists of a piece of plastic attached to a spring (Fig. P13.11). The spring is compressed against the floor a distance of $2.00 \mathrm{~cm}$, and the toy is released. If the toy has a mass of $100 \mathrm{~g}$ and rises to a maximum height of $60.0 \mathrm{~cm}$, estimate the force constant of the spring.

Prabhu Ramji

Numerade Educator

Problem 12

An automobile having a mass of $1000 \mathrm{~kg}$ 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?

Prabhu Ramji

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 whose mass is negligible. 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. Why?

Prabhu Ramji

Numerade Educator

Problem 14

ecp An object-spring system moving with simple harmonic motion 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 cquals twice the potential energy stored in the spring. (The answer should in terms of $A$ only.)

Prabhu Ramji

Numerade Educator

Problem 15

GP 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 $0.160 \mathrm{~m} ?$ (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?

Prabhu Ramji

Numerade Educator

Problem 16

ecp A $0.250-\mathrm{kg}$ block resting on a frictionless, horizontal surface is attached to a spring having force constant $83.8 \mathrm{~N} / \mathrm{m}$ as in Figure $\mathrm{P} 13.16 .$ A horizontal force $\overrightarrow{\mathbf{F}}$ causesthe spring to stretch ? distance of $5.46 \mathrm{~cm}$ from its equilibrium position. (a) Find the value of $F$. (b) What is the total energy FIGURE P13.16
stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block immediately after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, how would your answer to part (d) change? What other information would you need to know to answer?

Prabhu Ramji

Numerade Educator

Problem 17

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 when the spring is stretched $1.5 \mathrm{~cm}$. (d) For what value of $x$ does the speed equal one-half the maximum speed?

Prabhu Ramji

Numerade Educator

Problem 18

An object-spring system oscillates with an amplitude of $3.5 \mathrm{~cm}$. If the spring constant is $250 \mathrm{~N} / \mathrm{m}$ and the object has a mass of $0.50 \mathrm{~kg}$, determine (a) the mechanical energy of the system, (b) the maximum speed of the object, and
(c) the maximum acceleration of the object.

Prabhu Ramji

Numerade Educator

Problem 19

At an outdoor market, a bunch of bananas is set into oscillatory motion with an amplitude of $20.0 \mathrm{~cm}$ on a spring with a force constant of $16.0 \mathrm{~N} / \mathrm{m}$. 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?

Prabhu Ramji

Numerade Educator

Problem 20

A $50.0-\mathrm{g}$ object is attached to a horizontal spring with a force constant of $10.0 \mathrm{~N} / \mathrm{m}$ and released from rest with an amplitude of $25.0 \mathrm{~cm}$. What is the velocity of the object when it is halfway to the equilibrium position if the surface is frictionless?

Supratim Pal

Numerade Educator

Problem 21

While riding behind a car taveling at $3.00 \mathrm{~m} / \mathrm{s}$, you notice that one of the car's tires has a small hemispherical bump on its rim, as in Figure P13.21. (a) Explain why the bump, from your viewpoint behind the car, executes simple harmonic motion. (b) If the radius of the car's tires is $0.30 \mathrm{~m}$, what is the bump's period of oscillation?

Prabhu Ramji

Numerade Educator

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 Magnusen

Numerade Educator

Problem 23

Consider the simplified single-piston engine in Figure P13.23. If the wheel rotates at a constant angular speed $\omega$. explain why the piston rod oscillates in simple harmonic motion.

Prabhu Ramji

Numerade Educator

Problem 24

The period of motion of an object-spring system is $0.223 \mathrm{~s}$ when a $35.4$ -g object is attached to the spring. What is the force constant of the spring?

Prabhu Ramji

Numerade Educator

Problem 25

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

Prabhu Ramji

Numerade Educator

Problem 26

When four people with a combined mass of $320 \mathrm{~kg}$ sit down in a car, they find that the car drops $0.80 \mathrm{~cm}$ lower on its springs. Then they get out of the car and bounce it up and down. What is the frequency of the car's vibration if its mass (when it is empty) is $2.0 \times 10^{\mathrm{s}} \mathrm{kg}$ ?

Prabhu Ramji

Numerade Educator

Problem 27

.] A cart of mass $250 \mathrm{~g}$ is placed on a frictionless horizontal air track. A spring having a spring constant of $9.5 \mathrm{~N} / \mathrm{m}$ is attached between the cart and the left end of the track. When in equilibrium, the cart is located $12 \mathrm{~cm}$ from the left end of the track. If the cart is displaced $4.5 \mathrm{~cm}$ from its equilibrium position, find (a) the period at which it oscillates, (b) its maximum speed, and (c) its speed when it is $14 \mathrm{~cm}$ from the left end of the track.

Prabhu Ramji

Numerade Educator

Problem 28

The position of an object connected to a spring varies with time according to the expression $x=(5.2 \mathrm{~cm}) \mathrm{sin}$ $(8.0 \pi t) .$ 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 Hause

Carnegie Mellon University

Problem 29

A 326 -g object is attached to a spring and executes simple harmonic motion with a period of $0.250 \mathrm{~s}$. 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 Hause

Carnegie Mellon University

Problem 30

ecp 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?

Prabhu Ramji

Numerade Educator

Problem 31

A $2.00-\mathrm{kg}$ 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

Prabhu Ramji

Numerade Educator

Problem 32

GP 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 $t=0.500 \mathrm{~s}$. (g) Determine the velocity and acceleration of the particle when $t=0.500 \mathrm{~s}$.

Prabhu Ramji

Numerade Educator

Problem 33

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

Vishal Gupta

Numerade Educator

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 Hause

Carnegie Mellon University

Problem 35

A simple pendulum makes 120 complete oscillations in $3.00$ min at a location where $g=9.80 \mathrm{~m} / \mathrm{s}^{2}$. Find (a) the period of the pendulum and (b) its length.

Matt Braby

Numerade Educator

Problem 36

A "seconds" pendulum is one that moves through its equilibrium position once each second. (The period of the pendulum is $2.000 \mathrm{~s}$.) 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 Hause

Carnegie Mellon University

Problem 37

IA pendulum clock that works perfectly on the Earth is taken to the Moon. (a) Does it run fast or slow there?
(b) If the clock is started at 12:00 midnight, what will it read after one Earth day $(24.0 \mathrm{~h})$ ? Assume the free-fall acceleration on the Moon is $1.63 \mathrm{~m} / \mathrm{s}^{2}$.

Prabhu Ramji

Numerade Educator

Problem 38

An aluminum clock pendulum having a period of $1.00 \mathrm{~s}$ keeps perfect time at $20.0^{\circ} \mathrm{C}$. (a) When placed in a room at a temperature of $-5.0^{\circ} \mathrm{C}$, will it gain time or lose time?
(b) How much time will it gain or lose every hour? Hint:
See Chapter $10 .$

Prabhu Ramji

Numerade Educator

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 $1 \mathrm{~s}$ on Earth? What length of pendulum would have a 1 -s period on Mars?
(b) An object is suspended from a spring with force constant $10 \mathrm{~N} / \mathrm{m}$. Find the mass suspended from this spring that would result in a period of $1 \mathrm{~s}$ on Earth and on Mars.

Prabhu Ramji

Numerade Educator

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 Hause

Carnegie Mellon University

Problem 41

The sinusoidal wave shown in Figure P13.41 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.

Prabhu Ramji

Numerade Educator

Problem 42

An object attached to a spring vibrates with simple harmonic motion as described by Figure P13.42. 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 Hause

Carnegie Mellon University

Problem 43

A certain FM radio station broadcasts jazz music at a frequency of $101.9 \mathrm{MHz}$. Find (a) the wave's period and
(b) its wavelength. (Radio waves are electromagnetic waves that travel at the speed of light, $3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}$.)

Prabhu Ramji

Numerade Educator

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 Hause

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 $30.0 \mathrm{~s}$. Also, a given maximum travels $425 \mathrm{~cm}$ along the rope in $10.0 \mathrm{~s} .$ What is the wavelength?

Averell Hause

Carnegie Mellon University

Problem 46

8. 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 $60.0 \mathrm{kHz}$ and the speed of sound in air is $340 \mathrm{~m} / \mathrm{s}$, what is the smallest insect a bat can detect?

Prabhu Ramji

Numerade Educator

Problem 47

A cork on the surface of a pond bobs up and down two times per second on ripples having a wavelength of $8.50 \mathrm{~cm}$. If the cork is $10.0 \mathrm{~m}$ from shore, how long does it take a ripple passing the cork to reach the shore?

Prabhu Ramji

Numerade Educator

Problem 48

Ocean waves are traveling to the east at $4.0 \mathrm{~m} / \mathrm{s}$ with a distance of $20 \mathrm{~m}$ 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 Hause

Carnegie Mellon University

Problem 49

A phone cord 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 cord. The pulse makes four trips down and back along the cord in $0.800 \mathrm{~s}$. What is the tension in the cord?

Keshav Singh

Numerade Educator

Problem 50

A circus performer stretches a tightrope between two towers. He strikes one end of the rope and sends a wave along it toward the other tower. He notes that it takes the wave $0.800 \mathrm{~s}$ to reach the opposite tower, $20.0 \mathrm{~m}$ away. If a $1-\mathrm{m}$ length of the rope has a mass of $0.350 \mathrm{~kg}$, find the tension in the tightrope.

Prabhu Ramji

Numerade Educator

Problem 51

A transverse pulse moves along a stretched cord of length $6.30 \mathrm{~m}$ having a mass of $0.150 \mathrm{~kg}$. If the tension in the cord is $12.0 \mathrm{~N}$, find (a) the wave speed and (b) the time it takes the pulse to travel the length of the cord.

Prabhu Ramji

Numerade Educator

Problem 52

A taut clothesline is $12.0 \mathrm{~m}$ long and has a mass of $0.375$ kg. A transverse pulse is produced by plucking one end of the clothesline. If the pulse takes $2.96 \mathrm{~s}$ to make six round trips along the clothesline, find (a) the speed of the pulse and (b) the tension in the clothesline.

Prabhu Ramji

Numerade Educator

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 Hause

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 Hause

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 $2.00 \mathrm{~s}$, determine the speed of a transverse wave in the string when the pendulum hangs vertically.

Averell Hause

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 Hause

Carnegie Mellon University

Problem 57

Tension is maintained in a string as in Figure P13.57. The observed wave speed is $24 \mathrm{~m} / \mathrm{s}$ when the suspended mass is $3.0 \mathrm{~kg}$. (a) What is the mass per unit length of the string?
(b) What is the wave speed when the suspended mass is $2.0 \mathrm{~kg} ?$

Prabhu Ramji

Numerade Educator

Problem 58

The elastic limit of a piece of steel wire is $2.70 \times 10^{9} \mathrm{~Pa}$. 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 Hause

Carnegie Mellon University

Problem 59

ecp A $2.65-\mathrm{kg}$ power line running between two towers has a length of $38.0 \mathrm{~m}$ and is under a tension of $12.5 \mathrm{~N}$.
(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.

Prabhu Ramji

Numerade Educator

Problem 60

ecp 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 $M$.

Prabhu Ramji

Numerade Educator

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 Hause

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 $t=0.30 \mathrm{~s}$.

Rashmi Sinha

Numerade Educator

Problem 63

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

Averell Hause

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 Hause

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

Prabhu Ramji

Numerade Educator

Problem 66

A 500 -g 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.

Prabhu Ramji

Numerade Educator

Problem 67

A $3.00-\mathrm{kg}$ object is fastened to a light spring, with the intervening cord passing over a pulley (Fig. P13.67). 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 Hause

Carnegie Mellon University

Problem 68

A $5.00-\mathrm{g}$ 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 Hause

Carnegie Mellon University

Problem 69

A 25 -kg block is connected to a $30-\mathrm{kg}$ block by a light string that passes over a frictionless pulley. The $30-\mathrm{kg}$ block is connected to a light spring of force constant $200 \mathrm{~N} / \mathrm{m}$, as in Figure $\mathrm{P} 13.69 .$ The spring is unstretched when the sys-

Prabhu Ramji

Numerade Educator

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 Hause

Carnegie Mellon University

Problem 71

ecp A light balloon filled with helium of density $0.180 \mathrm{~kg} / \mathrm{m}^{3}$ 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. P13.71a). If the balloon is displaced slightly from equilibrium, as in Figure $\mathrm{P} 13.7 \mathrm{lb}$, show that the motion is simple harmonic and 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 Chapter $9 .$

Prabhu Ramji

Numerade Educator

Problem 72

ecp An object of mass $m$ is connected to two rubber bands of length $L$, each under tension $F$, as in Figure P13.72. 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 $\omega=\sqrt{2 F / m L}$

Prabhu Ramji

Numerade Educator

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 cen-
ter of the Earth is pulled toward the center only by the material in the shaded portion of Figure P13.73. Assume Earth has a uniform density $\rho$. Write down Newton's law of gravitation for $\quad$ FIGURE P13.73 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.

Prabhu Ramji

Numerade Educator

Problem 74

1. Figure $\mathrm{P} 13.74$ 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?

Prabhu Ramji

Numerade Educator

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? This distance will be the amplitude of the ensuing oscillation of the block.

Prabhu Ramji

Numerade Educator

Problem 76

ecp 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 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, $\left.r=L-y_{f} .\right)$ (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 mass L? Explain.

Jacob Schulze

Numerade Educator

Problem 77

] 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 $\mathrm{P} 13.77$, 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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator