Physics

James S. Walker

Chapter 13

Oscillations About Equilibrium - all with Video Answers

Educators

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Chapter Questions

Problem 1

A person in a rocking chair completes 12 cycles in 21 $\mathrm{s}$ . What are
the period and frequency of the rocking?

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

While fishing for catfish, a fisherman suddenly notices that the
bobber (a floating device) attached to his line is bobbing up and
down with a frequency of 1.8 $\mathrm{Hz}$ . What is the period of the bobber's motion?

Averell Hause

Carnegie Mellon University

Problem 3

If you dribble a basketball with a frequency of 1.9 $\mathrm{Hz}$ , how much
time does it take for you to complete 12 dribbles?

Averell Hause

Carnegie Mellon University

Problem 4

You take your pulse and observe 74 heartbeats in a minute. What
are the period and frequency of your heartbeat?

Averell Hause

Carnegie Mellon University

Problem 5

BIO Slow-Motion Dragonfly A frame-by-frame analysis of a slow-motion video shows that a hovering dragonfly takes 7 frames to
complete one wing beat. If the video is shot at 240 frames per second, (a) what is the period of the wing beat, and (b) how many
wing beats does the dragonfly perform per second?

Averell Hause

Carnegie Mellon University

Problem 6

Predict/Calculate (a) Your heart beats with a frequency of
1.45 $\mathrm{Hz}$ . How many beats occur in a minute? (b) If the frequency
of your heartbeat increases, will the number of beats in a minute
increase, decrease, or stay the same? (c) How many beats occur in
a minute if the frequency increases to 1.55 Hz?

Averell Hause

Carnegie Mellon University

Problem 7

You rev your car's engine to 3300 $\mathrm{rpm}(\mathrm{rev} / \mathrm{min}) .$ (a) What are
the period and frequency of the engine? (b) If you change the period of the engine to 0.033 s, how many rpms is it doing?

Averell Hause

Carnegie Mellon University

Problem 8

CE A mass moves back and forth in simple harmonic motion
with amplitude $A$ and period $T$ . (a) In terms of $A$ , through what
distance does the mass move in the time $T ?$ (b) Through what distance does it move in the time 5$T / 2 ?$

Averell Hause

Carnegie Mellon University

Problem 9

CE A mass moves back and forth in simple harmonic motion
with amplitude $A$ and period $T$ . (a) In terms of $T,$ how much time
does it take for the mass to move through a total distance of 2$A ?$
(b) How much time does it take for the mass to move through a
total distance of 3$A$ ?

Averell Hause

Carnegie Mellon University

Problem 10

The position of a mass oscillating on a spring is given by
$x=(3.9 \mathrm{cm}) \cos [2 \pi t /(0.38 \mathrm{s})] .($ a) What is the period of this motion? (b) What is the first time the mass is at the position $x=0 ?$

Averell Hause

Carnegie Mellon University

Problem 11

The position of a mass oscillating on a spring is given by
$x=(7.8 \mathrm{cm}) \cos [2 \pi t /(0.68 \mathrm{s})] .$ (a) What is the frequency of this
motion? (b) When is the mass first at the position $x=-7.8 \mathrm{cm} ?$

Averell Hause

Carnegie Mellon University

Problem 12

CE A position-versus-time plot for an object undergoing simple
harmonic motion is given in FIGURE $13-30 .$ Rank the six points indicated in the figure in order of increasing (a) speed, (b) velocity,
and (c) acceleration. Indicate ties where necessary.

Averell Hause

Carnegie Mellon University

Problem 13

CE A mass on a spring oscillates with simple harmonic motion
of amplitude $A$ about the equilibrium position $x=0 .$ Its maximum speed is $v_{\text { max }}$ and its maximum acceleration is $a_{\max }$ (a) What
is the speed of the mass at $x=0 ?$ (b) What is the acceleration of
the mass at $x=0 ?$ (c) What is the speed of the mass at $x=A ?$
(d) What is the acceleration of the mass at $x=A ?$

Averell Hause

Carnegie Mellon University

Problem 14

A mass oscillates on a spring with a period of 0.63 s and an amplitude of 4.4 $\mathrm{cm} .$ Write an equation giving $x$ as a function of time,
assuming the mass starts at $x=A$ at time $t=0$ .

Averell Hause

Carnegie Mellon University

Problem 15

Predict/Calculate Molecular Oscillations An atom in a molecule oscillates about its equilibrium position with a frequency
of $2.00 \times 10^{14} \mathrm{Hz}$ and a maximum displacement of 3.50 $\mathrm{nm}$ .
(a) Write an expression giving $x$ as a function of time for this atom,
assuming that $x=A$ at $t=0 .$ (b) If, instead, we assume that $x=0$
at $t=0,$ would your expression for position versus time use a sine
function or a cosine function? Explain.

Averell Hause

Carnegie Mellon University

Problem 16

A mass oscillates on a spring with a period $T$ and an amplitude
0.48 $\mathrm{cm} .$ The mass is at the equilibrium position $x=0$ at $t=0$ ,
and is moving in the positive direction. Where is the mass at the
times (a) $t=T / 8,$ (b) $t=T / 4,$ (c) $t=T / 2,$ and $(\mathrm{d}) t=3 T / 4$ ?
(c) Plot your results for parts (a) through (d) with the vertical axis
representing position and the horizontal axis representing time.

Averell Hause

Carnegie Mellon University

Problem 17

The position of a mass on a spring is given by
$x=(3.8 \mathrm{cm}) \cos [2 \pi t /(0.88 \mathrm{s})]$ . (a) What is the period, $T,$ of this
motion? (b) Where is the mass at $t=0.55$ s? (c) Show that the
mass is at the same location at $0.55 \mathrm{s}+T$ seconds as it is at 0.55 $\mathrm{s}$ .

Averell Hause

Carnegie Mellon University

Problem 18

Predict/Calculate A mass attached to a spring oscillates with
a period of 3.35 $\mathrm{s}$ . (a) If the mass starts from rest at $x=0.0440 \mathrm{m}$
and time $t=0,$ where is it at time $t=6.37 \mathrm{s} ?$ (b) Is the mass moving in the positive or negative $x$ direction at $t=6.37 \mathrm{s} ?$ Explain.

Averell Hause

Carnegie Mellon University

Problem 19

A lawn sprinkler oscillates with simple harmonic motion of
period $T=52.0 \mathrm{s},$ and sprays water with an angle to the vertical
$(\theta=0)$ that varies from $\theta=-45^{\circ}$ to $\theta=45^{\circ},$ as shown in FlGURE
$13-31$ . Water from the sprinkler lands in a nearby garden patch
when the angle the sprinkler makes with the vertical is greater
than $\theta=36^{\circ} .$ For what amount of time is the sprinkler watering
the garden during one complete cycle of its oscillation?

Averell Hause

Carnegie Mellon University

Problem 20

A ball rolls on a circular track of radius 0.62 $\mathrm{m}$ with a constant
angular speed of 1.3 $\mathrm{rad} / \mathrm{s}$ in the counterclockwise direction. If the
angular position of the ball at $t=0$ is $\theta=0,$ find the $x$ component of the ball's position at the times $2.5 \mathrm{s}, 5.0 \mathrm{s},$ and 7.5 $\mathrm{s}$ . Let
$\theta=0$ correspond to the positive $x$ direction.

Averell Hause

Carnegie Mellon University

Problem 21

An object executing simple harmonic motion has a maximum
speed of 4.1 $\mathrm{m} / \mathrm{s}$ and a maximum acceleration of 0.85 $\mathrm{m} / \mathrm{s}^{2} .$ Find
(a) the amplitude and (b) the period of this motion.

Mukesh Devi

Numerade Educator

Problem 22

A child rocks back and forth on a porch swing with an amplitude
of 0.204 $\mathrm{mm}$ and a period of 2.80 s. Assuming the motion is approxi-
mately simple harmonic, find the child's maximum speed.

Averell Hause

Carnegie Mellon University

Problem 23

Predict/Calculate $\mathrm{A} 30.0$ -g goldfinch lands on a slender
branch, where it oscillates up and down with simple harmonic
motion of amplitude 0.0335 $\mathrm{m}$ and period 1.65 s. (a) What is the
maximum acceleration of the finch? Express your answer as a fraction of the acceleration of gravity, $($ b) What is the maximum
speed of the goldfinch? (c) At the time when the goldfinch experiences its maximum acceleration, is its speed a maximum or a
minimum? Explain.

Averell Hause

Carnegie Mellon University

Problem 24

BIO Tuning Forks in Neurology Tuning forks are used in the diagnosis of nervous afflictions known as large-fiber polyneuropathies,
which are often manifested in the form of reduced sensitivity to
vibrations. Disorders that can result in this type of pathology include diabetes and nerve damage from exposure to heavy metals.
The tuning fork in FiGURE $13-32$ has a frequency of 165 Hz. If the
tips of the fork move with an amplitude of $1.10 \mathrm{mm},$ find (a) their
maximum speed and (b) their maximum acceleration. Give your
answer to part (b) as a multiple of $g$ .

Averell Hause

Carnegie Mellon University

Problem 25

A vibrating structural beam in a spacecraft can cause problems
if the frequency of vibration is fairly high. Even if the amplitude
of vibration is only a fraction of a millimeter, the acceleration of
the beam can be several times greater than the acceleration due to
gravity. As an example, find the maximum acceleration of a beam
that vibrates with an amplitude of 0.25 $\mathrm{mm}$ at the rate of 110 vibrations per second. Give your answer as a multiple of $g$ .

Averell Hause

Carnegie Mellon University

Problem 26

A peg on a turntable moves with a constant tangential speed
of 0.82 $\mathrm{m} / \mathrm{s}$ in a circle of radius 0.33 $\mathrm{m}$ . The peg casts a shadow
on a wall. Find the following quantities related to the motion of
the shadow: (a) the period, (b) the amplitude, (c) the maximum
speed, and (d) the maximum magnitude of the acceleration.

Averell Hause

Carnegie Mellon University

Problem 27

The pistons in an internal combustion engine undergo a motion
that is approximately simple harmonic motion. If the amplitude of
the motion is $3.5 \mathrm{cm},$ and the engineruns at 1700 $\mathrm{rev} / \mathrm{min}$ , find (a)
the maximum acceleration of the pistons and (b) their maximum
speed.

Averell Hause

Carnegie Mellon University

Problem 28

Vomit Comet NASA trains astronauts to deal with weightlessness
(and its associated nausea) by flying them in the "vomit comet,"
a modified $\mathrm{KC}-135$ airplane that flies in an oscillating path with a
period of 72 seconds and a maximum acceleration of magnitude
9.81 $\mathrm{m} / \mathrm{s}^{2} .$ What is the amplitude of the airplane's oscillation, assuming it undergoes simple harmonic motion?

Averell Hause

Carnegie Mellon University

Problem 29

A $0.84-\mathrm{kg}$ air cart is attached to a spring and allowed to oscillate. If the displacement of the air cart from equilibrium is
$x=(10.0 \mathrm{cm}) \cos \left[\left(2.00 \mathrm{s}^{-1}\right) t+\pi\right],$ find (a) the maximum kinetic
energy of the cart and ( b ) the maximum force exerted on it by the
spring.

Averell Hause

Carnegie Mellon University

Problem 30

Predict/Calculate A person rides on a mechanical bucking
horse (see FiGURE $13-33$ ) that oscillates up and down with simple
harmonic motion. The period of the bucking is 0.74 s and the
amplitude is slowly increasing. At a certain amplitude the rider
must hang on to prevent separating from the mechanical horse.
\begin{equation}
\begin{array}{l}{\text { (a) Give a strategy that will allow you to calculate this amplitude. }} \\ {\text { (b) Carry out your strategy and find the desired amplitude. }}\end{array}
\end{equation}

Averell Hause

Carnegie Mellon University

Problem 31

An object moves with simple harmonic motion of period $T$
and amplitude $A .$ During one complete cycle, for what length of
time is the speed of the object greater than $v_{\max } / 2 ?$

Averell Hause

Carnegie Mellon University

Problem 32

An object executing simple harmonic motion has a maximum
speed $v_{\max }$ and a maximum acceleration $a_{\max }$ . Find (a) the amplitude and (b) the period of this motion. Express your answers in
terms of $v_{\max }$ and $a_{\max }$ .

Averell Hause

Carnegie Mellon University

Problem 33

CE Predict/Explain If a mass $m$ is attached to a given spring, its
period of oscillation is $T .$ If two such springs are connected end to
end and the same mass $m$ is attached, (a) is the resulting period of
oscillation greater than, less than, or equal to $T ?$ (b) Choose the
best explanation from among the following:
\begin{equation}
\begin{array}{l}{\text { I. Connecting two springs together makes the spring stiffer, }} \\ {\text { which means that less time is required for an oscillation. }} \\ {\text { II. The period of oscillation does not depend on the length of a }} \\ {\text { spring, only on its force constant and the mass attached to it. }} \\ {\text { The longer spring stretches more easily, and hence takes longer }} \\ {\text { to complete an oscillation. }}\end{array}
\end{equation}

Averell Hause

Carnegie Mellon University

Problem 34

CE Predict/Explain An old car with worn-out shock absorbers
oscillates with a given frequency when it hits a speed bump. If
the driver adds a couple of passengers to the car and hits another
speed bump, (a) is the car's frequency of oscillation greater than,
less than, or equal to what it was before? (b) Choose the best expla-
nation from among the following:
\begin{equation}
\begin{array}{l}{\text { I Increasing the mass on a spring increases its period, and }} \\ {\text { hence decreases its frequency. }} \\ {\text { I. The frequency depends on the force constant of the spring }} \\ {\text { but is independent of the mass. }} \\ {\text { III. Adding mass makes the spring oscillate more rapidly, which }} \\ {\text { increases the frequency. }}\end{array}
\end{equation}

Averell Hause

Carnegie Mellon University

Problem 35

CE Predict/Explain The two blocks in FIGURE $13-34$ have the same
mass, $m$ . All the springs have the same force constant, $k,$ and are at
their equilibrium length. When the blocks are set into oscillation,
(a) is the period of block 1 greater than, less than, or equal to the
period of block 2$?$ (b) Choose the best explanation from among the
following:
\begin{equation}
\begin{array}{l}{\text { I. Springs in parallel are stiffer than springs in series; therefore }} \\ {\text { the period of block } 1 \text { is smaller than the period of block } 2 \text { . }}\\{\text { II. The two blocks experience the same restoring for a given }} \\ {\text { displacement from equilibrium, and hence they have equal }} \\ {\text { periods of oscillation. }} \\ {\text { III.The force of the two springs on block } 2 \text { partially cancel one }} \\ {\text { another, leading to a longer period of oscillation. }}\end{array}
\end{equation}

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 36

A 0.49 -kg mass attached to a spring undergoes simple harmonic
motion with a period of 0.67 $\mathrm{s}$ . What is the force constant of the
spring?

Supratim Pal

Numerade Educator

Problem 37

A freshly caught catfish is placed on a spring scale, and it oscillates up and down with a period of 0.204 s. If the spring constant
of the scale is $2160 \mathrm{N} / \mathrm{m},$ what is the mass of the cattish?

Averell Hause

Carnegie Mellon University

Problem 38

CE System A consists of a mass $m$ attached to a spring with a
force constant $k ;$ system $B$ has a mass 2$m$ attached to a spring with
a force constant $k ;$ system $C$ has a mass 3$m$ attached to a spring
with a force constant 6$k$ and system $D$ has a mass $m$ attached to
a spring with a force constant 4$k .$ Rank these systems in order of
increasing period of oscillation.

Averell Hause

Carnegie Mellon University

Problem 39

Find the periods of block 1 and block 2 in Figure $13-34,$ given
that $k=49.2 \mathrm{N} / \mathrm{m}$ and $\mathrm{m}=1.25 \mathrm{kg} .$

Averell Hause

Carnegie Mellon University

Problem 40

When a 0.62 -kg mass is attached to a vertical spring, the spring
stretches by 12 $\mathrm{cm} .$ How much mass must be attached to the spring
to result in a 0.67 -s period of oscillation?

Averell Hause

Carnegie Mellon University

Problem 41

A spring with a force constant of 82 $\mathrm{N} / \mathrm{m}$ is attached to a $0.47-\mathrm{kg}$
mass. Assuming that the amplitude of motion is $3.4 \mathrm{cm},$ determine the following quantities for this system: (a) $\omega,(b) v_{\max },(c) T.$

Averell Hause

Carnegie Mellon University

Problem 42

A bunch of grapes is placed in a spring scale at a supermarket.
The grapes oscillate up and down with a period of 0.48 s, and the
spring in the scale has a force constant of 650 $\mathrm{N} / \mathrm{m} .$ What are
(a) the mass and (b) the weight of the grapes?

Averell Hause

Carnegie Mellon University

Problem 43

Two people with a combined mass of 125 $\mathrm{kg}$ hop into an old car
with worn-out shock absorbers. This causes the springs to compress by 8.00 $\mathrm{cm} .$ When the car hits a bump in the road, it oscillates up and down with a period of 1.65 s. Find (a) the total load
supported by the springs and (b) the mass of the car.

Averell Hause

Carnegie Mellon University

Problem 44

A 0.95 -kg mass attached to a vertical spring of force constant
130 $\mathrm{N} / \mathrm{m}$ oscillates with a maximum speed of 0.45 $\mathrm{m} / \mathrm{s}$ . Find the following quantities related to the motion of the mass: (a) the period,
(b) the amplitude, (c) the maximum magnitude of the acceleration.

Averell Hause

Carnegie Mellon University

Problem 45

When a 0.184 -kg mass is attached to a vertical spring, it causes
the spring to stretch a distance $d .$ If the mass is now displaced
slightly from equilibrium, it is found to make 25.0 oscillations in
12.6 s. Find the stretch distance, $d$ .

Averell Hause

Carnegie Mellon University

Problem 46

Predict/ Calculate The springs of a $511-\mathrm{kg}$ motorcycle have an
effective force constant of 9130 $\mathrm{N} / \mathrm{m}$ . (a) If a person sits on the motorcycle, does its period of oscillation increase, decrease, or stay the
same? (b) By what percent and in what direction does the period
of oscillation change when a 112 -kg person rides the motorcycle?

Averell Hause

Carnegie Mellon University

Problem 47

Predict/Calculate If a mass $m$ is attached to a given spring,
its period of oscillation is $T$ . If two such springs are connected end
to end, and the same mass $m$ is attached, (a) is its period greater
than, less than, or the same as with a single spring? (b) Verify your
answer to part (a) by calculating the new period, $T,$ in terms of
the old period $T .$

Averell Hause

Carnegie Mellon University

Problem 48

A $0.285-$ kg mass is attached to a spring with a force constant of
53.4 $\mathrm{N} / \mathrm{m} .$ If the mass is displaced 0.180 $\mathrm{m}$ from equilibrium and
released, what is its speed when it is 0.135 $\mathrm{m}$ from equilibrium?

Averell Hause

Carnegie Mellon University

Problem 49

A 1.6 -kg mass attached to a spring oscillates with an amplitude
of 7.3 $\mathrm{cm}$ and a frequency of 2.8 $\mathrm{Hz}$ . What is its energy of motion?

Averell Hause

Carnegie Mellon University

Problem 50

Predict/Calculate $\mathrm{A} 0.40-\mathrm{kg}$ mass is attached to a spring with
a force constant of 26 $\mathrm{N} / \mathrm{m}$ and released from rest a distance of
3.2 $\mathrm{cm}$ from the equilibrium position of the spring. (a) Give a strategy that allows you to find the speed of the mass when it is halfway
to the equilibrium position. (b) Use your strategy to find this speed.

Averell Hause

Carnegie Mellon University

Problem 51

(a) What is the maximum speed of the mass in the previous
problem? (b) How far is the mass from the equilibrium position
when its speed is half the maximum speed?

Averell Hause

Carnegie Mellon University

Problem 52

B1O Astronaut Mass An astronaut uses a Body Mass Measurement
Device to measure her mass. If the force constant of the spring is
$2700 \mathrm{N} / \mathrm{m},$ her mass is 75 $\mathrm{kg}$ , and the amplitude of her oscillation
is $1.9 \mathrm{cm},$ what is her maximum speed during the measurement?

Averell Hause

Carnegie Mellon University

Problem 53

Predict/Calculate $\mathrm{A} 0.505$ -kg block slides on a frictionless horizontal surface with a speed of 1.18 $\mathrm{m} / \mathrm{s}$ . The block encounters an unstretched spring and compresses it 23.2 $\mathrm{cm}$ before coming to rest. (a)
What is the force constant of this spring? (b) For what length of time
is the block in contact with the spring before it comes to rest? (c) If
the force constant of the spring is increased, does the time required to
stop the block increase, decrease, or stay the same? Explain.

Averell Hause

Carnegie Mellon University

Problem 54

A $3.55$ -g bullet embeds itself in a 1.47 -kg block, which is attached
to a spring of force constant 825 $\mathrm{N} / \mathrm{m} .$ If the maximum compression
of the spring is $5.88 \mathrm{cm},$ find (a) the initial speed of the bullet and
(b) the time for the bullet-blocksystem to come to rest.

Averell Hause

Carnegie Mellon University

Problem 55

CE Metronomes, such as the
penguin shown in FIGURE $13-35,$
are useful devices for music students. If it is desired to have the
metronome tick with a greater
frequency, should the penguin's
bow tie be moved upward or
downward?

Averell Hause

Carnegie Mellon University

Problem 56

Predict/Explain A grandfather clock keeps correct time
at sea level. If the clock is taken
to the top of a nearby mountain,
(a) would you expect it to keep
correct time, run slow, or run
fast? (b) Choose the best explanation from among the following:
\begin{equation}
\begin{array}{l}{\text { 1. Gravity is weaker at the top of the mountain, leading to a }} \\ {\text { greater period of oscillation. }} \\ {\text { II. The length of the pendulum is unchanged, and therefore its }} \\ {\text { period remains the same. }} \\ {\text { IIL. The extra gravity from the mountain causes the period to de- }} \\ {\text { crease. }}\end{array}
\end{equation}

Averell Hause

Carnegie Mellon University

Problem 57

An observant fan at a baseball game notices that the radio commentators have lowered a microphone from their booth to just a
few inches above the ground, as shown in FiGURE $13-36 .$ The microphone is used to pick up sound from the field and from the fans.
The fan also notices that the microphone is slowly swinging back
and forth like a simple pendulum. Using her digital watch, she
finds that 10 complete oscillations take 60.0 s. How high above the
field is the radio booth? (Assume the microphone and its cord can
be treated as a simple pendulum.)

Averell Hause

Carnegie Mellon University

Problem 58

A simple pendulum of length 2.3 $\mathrm{m}$ makes 5.0 complete swings
in 37 s. What is the acceleration of gravity at the location of the
pendulum?

Averell Hause

Carnegie Mellon University

Problem 59

United Nations Pendulum A large pendulum with a $200-$ lb gold-
plated bob 12 inches in diameter is on display in the lobby of the
United Nations building. The pendulum has a length of 75 ft. It is
used to show the rotation of the Earth- for this reason it is referred
to as a Foucault pendulum. What is the least amount of time it takes
for the bob to swing from a position of maximum displacement to
the equilibrium position of the pendulum? (Assume that the accel-
eration due to gravity is $g=9.81 \mathrm{m} / \mathrm{s}^{2}$ at the UN building.)

Averell Hause

Carnegie Mellon University

Problem 60

Predict/Calculate If the pendulum in the previous problem
were to be taken to the Moon, where the acceleration of gravity
is $g / 6,$ (a) would its period increase, decrease, or stay the same?
(b) Check your result in part (a) by calculating the period of the
pendulum on the Moon.

Justin Swantek

Numerade Educator

Problem 61

A Hula Hoop hangs from a peg. Find the period of the hoop as
it gently rocks back and forth on the peg. (For a hoop with axis at
the rim $I=2 m R^{2},$ where $R$ is the radius of the hoop.)

Averell Hause

Carnegie Mellon University

Problem 62

A fireman tosses his 0.98 -kg hat onto a peg, where it oscillates as
a physical pendulum (FiGURE $13-37 )$ . If the center of mass of the hat
is 8.4 $\mathrm{cm}$ from the pivot point, and its period of oscillation is 0.73 $\mathrm{s}$
what is the moment of inertia of the hat about the pivot point?

Averell Hause

Carnegie Mellon University

Problem 63

Predict/Calculate Consider a meterstick that oscillates back
and forth about a pivot point at one of its ends. (a) Is the period of
a simple pendulum of length $L=1.00 \mathrm{m}$ greater than, less than,
or the same as the period of the meterstick? Explain. (b) Find the
length $L$ of a simple pendulum that has a period equal to the period of the meterstick.

Averell Hause

Carnegie Mellon University

Problem 64

On the construction site for a new skyscraper, a uniform beam
of steel is suspended from one end. If the beam swings back and
forth with a period of 2.00 s, what is its length?

Averell Hause

Carnegie Mellon University

Problem 65

BIO (a) Find the period of a child's leg as it swings about the
hip joint. Assume the leg is 0.55 m long and can be treated as a
uniform rod. (b) Estimate the child's walking speed.

Averell Hause

Carnegie Mellon University

Problem 66

Suspended from the ceiling of an elevator is a simple pendulum of length $L .$ What is the period of this pendulum if the elevator (a) accelerates upward with an acceleration $a,$ or (b) accelerates
downward with an acceleration whose magnitude is greater than
zero but less than $g$ ? Give your answer in terms of $L, g,$ and $a$ .

Averell Hause

Carnegie Mellon University

Problem 67

CE An object undergoes simple harmonic motion with a period
$T$ . In the time 3$T / 2$ the object moves through a total distance of
12$D .$ In terms of $D,$ what is the object's amplitude of motion?

Averell Hause

Carnegie Mellon University

Problem 68

CE If the amplitude of a simple harmonic oscillator is doubled,
by what multiplicative factor do the following quantities change:
(a) angular frequency, (b) frequency, (c) period, (d) maximum
speed, (e) maximum acceleration, (f) total mechanical energy?

Averell Hause

Carnegie Mellon University

Problem 69

CE A mass $m$ is suspended from the ceiling of an elevator by a
spring of force constant $k$ . When the elevator is at rest, the period
of the mass is $T .$ Does the period increase, decrease, or remain the
same when the elevator (a) moves upward with constant speed or
(b) moves upward with constant acceleration?

Averell Hause

Carnegie Mellon University

Problem 70

CE A pendulum of length $L$ is suspended from the ceiling of an
elevator. When the elevator is at rest, the period of the pendulum
is T. Does the period increase, decrease, or remain the same when
the elevator (a) moves upward with constant speed or (b) moves
upward with constant acceleration?

Averell Hause

Carnegie Mellon University

Problem 71

A 1.3 -kg mass is attached to a spring with a force constant of
52 $\mathrm{N} / \mathrm{m} .$ If the mass is released with a speed of 0.28 $\mathrm{m} / \mathrm{s}$ at a distance of 8.1 $\mathrm{cm}$ from the equilibrium position of the spring, what
is its speed when it is halfway to the equilibrium position?

Averell Hause

Carnegie Mellon University

Problem 72

BIO Measuring an Astronauts Mass An astronaut uses a Body Mass
Measurement Device (BMMD) to determine her mass. What is the
astronaut's mass, given that the force constant of the BMMD is
2600 $\mathrm{N} / \mathrm{m}$ and the period of oscillation is 0.85 $\mathrm{s} ?$ (See the discussion on page 431 for more details on the BMMD.)

Averell Hause

Carnegie Mellon University

Problem 73

Sunspot observations Sunspots vary in number as a function of
time, exhibiting an approximately 11 -year cycle. Galileo made the
first European observations of sunspots in $1610,$ and daily observations were begun in Zurich in 1749 . At the present time we are well
into the 24 th observed cycle. What is the frequency of the sunspot
cycle? Give your answer in Hz.

Averell Hause

Carnegie Mellon University

Problem 74

BIO Weighing a Bacterium Scientists are using tiny, nanoscale
cantilevers 4 micrometers long and 500 nanometers wide-
essentially miniature diving boards - as a sensitive way to measure mass. An example is shown in FIGURE $13-38 .$ The cantilevers
oscillate up and down with a frequency that depends on the mass
placed near the tip, and a laser beam is used to measure the frequency. A single $E .$coli bacterium was measured to have a mass of
665 femtograms $=6.65 \times 10^{-16} \mathrm{kg}$ with this device, as the cantilever oscillated with a frequency of 14.5 $\mathrm{MHz}$ . Treating the cantilever as an ideal, massless spring, find its effective force constant.

Averell Hause

Carnegie Mellon University

Problem 75

CE An object undergoing simple harmonic motion with a
period $T$ is at the position $x=0$ at the time $t=0 .$ At the time
$t=0.25 T$ the position of the object is positive. State whether $x$
is positive, negative, or zero at the following times: (a) $t=1.5 T$ .
(b) $t=2 T,(\mathrm{c}) t=2.25 T,$ and $(\mathrm{d}) t=6.75 T$ .

Averell Hause

Carnegie Mellon University

Problem 76

The maximum speed of a $4.1-\mathrm{kg}$ mass attached to a spring is
$0.78 \mathrm{m} / \mathrm{s},$ and the maximum force exerted on the mass is 13 $\mathrm{N}$ .
(a) What is the amplitude of motion for this mass? (b) What is the
force constant of the spring? (c) What is the frequency of this system?

Averell Hause

Carnegie Mellon University

Problem 77

The acceleration of a block attached to a spring is given by
$a=-\left(0.302 \mathrm{m} / \mathrm{s}^{2}\right) \cos ([2.41 \mathrm{rad} / \mathrm{s}]$ t). (a) What is the frequency of
the block's motion? (b) What is the maximum speed of the block?
(c) What is the amplitude of the block's motion?

Averell Hause

Carnegie Mellon University

Problem 78

Helioseismology In $1962,$ physicists at Cal Tech discovered that
the surface of the Sun vibrates due to the violent nuclear reactions
that roil within its core. This has led to a new field of solar science
known as helioseismology. A typical vibration of the Sun is shown
in FlGURE $13-39 ;$ it has a period of 5.7 minutes. The blue patches
in Figure $13-39$ are moving outward; the red patches are moving
inward. (a) Find the angular frequency of this vibration. (b) The
maximum speed at which a patch of the surface moves during a
vibration is 4.5 $\mathrm{m} / \mathrm{s}$ . What is the amplitude of the vibration, aSsuming it to be simple harmonic motion?

Averell Hause

Carnegie Mellon University

Problem 79

Predict/Calculate $\mathrm{A} 9.50$ -g bullet, moving horizontally with
an initial speed $v_{0 .}$ embeds itself in a 1.45 -kg pendulum bob that is
initially at rest. The length of the pendulum is $L=0.745 \mathrm{m}$ . After
the collision, the pendulum swings to one side and comes to rest
when it has gained a vertical height of 12.4 $\mathrm{cm} .$ (a) Is the kinetic
energy of the bullet-bob system immediately after the collision
greater than, less than, or the same as the kinetic energy of the
system just before the collision? Explain. (b) Find the initial speed
of the bullet. (c) How much time does it take for the bullet-bob
system to come to rest for the first time?

Averell Hause

Carnegie Mellon University

Problem 80

BIO Spiderweb Oscillations A 1.44 -g spider oscillates on its web,
which has a damping constant of $3.30 \times 10^{-5} \mathrm{kg} / \mathrm{s} .$ How much
time does it take for the spider's amplitude of oscillation to decrease by 10.0 percent?

Averell Hause

Carnegie Mellon University

Problem 81

A service dog tag (FIGURE $13-40$ ) is a circular disk of radius 1.9 $\mathrm{cm}$
and mass 0.013 $\mathrm{kg}$ that can pivot about a small hole near its rim.
Refer to Table $10-1$ to determine the moment of inertia about the
perpendicular axis that passes through the pivot hole, and find
the period of oscillation of the dog tag.

Averell Hause

Carnegie Mellon University

Problem 82

Calculate the ratio of the kinetic energy to the potential energy
of a simple harmonic oscillator when its displacement is half its
amplitude.

Averell Hause

Carnegie Mellon University

Problem 83

A 0.340 -kg mass slides on a frictionless floor with a speed of
1.34 $\mathrm{m} / \mathrm{s} .$ The mass strikes and compresses a spring with a force
constant of 53.4 $\mathrm{N} / \mathrm{m} .$ (a) How far does the mass travel after contacting the spring before it comes to rest? (b) How much time does
it take for the spring to stop the mass?

Averell Hause

Carnegie Mellon University

Problem 84

A shock absorber is designed to quickly damp out the oscillations that a car would otherwise make because it is suspended
on springs. (a) Find the period of oscillation of a 1610 -kg car that
is suspended by springs that make an effective force constant of
$5.75 \times 10^{4} \mathrm{N} / \mathrm{m} .$ (b) Find the damping constant $b$ that will reduce
the amplitude of oscillations of this car by a factor of 5.00 within a
time equal to half the period of oscillation.

Averell Hause

Carnegie Mellon University

Problem 85

Predict/Calculate FIGURE $13-41$ shows a displacement-versus-time
graph of the periodic motion of a 3.8 -kg mass on a spring.
(a) Referring to the figure, do you expect the maximum speed
of the mass to be greater than, less than, or equal to 0.50 $\mathrm{m} / \mathrm{s} ?$
Explain. (b) Calculate the maximum speed of the mass. (c) How
much energy is stored in this system?

Averell Hause

Carnegie Mellon University

Problem 86

Predict/Calculate A 3.2 -kg mass on a spring oscillates as
shown in the displacement-versus-time graph in Figure $13-41.$
\begin{equation}\begin{array}{l}{\text { (a) Referring to thegraph, atwhat timesbetween } t=0 \text { and } t=6.0 \mathrm{s}} \\ {\text { does the mass experience a force of maximum magnitude? Explain. }}\end{array}\end{equation}
\begin{equation}
\begin{array}{l}{\text { (b) Calculate the magnitude of the maximum force exerted on the }} \\ {\text { mass. (c) At what times shown in the graph does the mass expe- }} \\ {\text { rience zero force? Explain. (d) How much force is exerted on the }} \\ {\text { mass at the time } t=0.50 \text { s? }}\end{array}
\end{equation}

Averell Hause

Carnegie Mellon University

Problem 87

A 0.45 -kg crow lands on a slender branch and bobs up and
down with a period of 1.5 s. An eagle flies up to the same branch,
scaring the crow away, and lands. The eagle now bobs up and
down with a period of 4.8 s. Treating the branch as an ideal spring,
find (a) the effective force constant of the branch and (b) the mass
of the eagle.

Averell Hause

Carnegie Mellon University

Problem 88

A mass $m$ is connected to
the bottom of a vertical spring
whose force constant is $k$ . Attached to the bottom of the mass
is a string that is connected to a
second mass $m,$ as shown in FIGURE $13-42 .$ Both masses are undergoing simple harmonic vertical
motion of amplitude $A .$ At the
instant when the acceleration
of the masses is a maximum in
the upward direction the string
breaks, allowing the lower mass
to drop to the floor. Find the resulting amplitude of motion of
the remaining mass.

Averell Hause

Carnegie Mellon University

Problem 89

Predict/Calculate Consider the pendulum shown in FIGURE 13-
43. Note that the pendulum's string is stopped by a peg when the
bob swings to the left, but moves freely when the bobswings to the
right. (a) Is the period of this pendulum greater than, less than, or
the same as the period of the same pendulum without the peg? (b)
Calculate the period of this pendulum in terms of $L$ and $\ell .(\mathrm{c})$ Evaluate your result for $L=1.0 \mathrm{m}$ and $\ell=0.25 \mathrm{m} .$

Averell Hause

Carnegie Mellon University

Problem 90

An object undergoes simple harmonic motion of amplitude $A$
and angular frequency $\omega$ about the equilibrium point $x=0 .$ Use
energy conservation to show that the speed of the object at the
general position $x$ is given by the following expression:
$$v=\omega \sqrt{A^{2}-x^{2}}$$

Averell Hause

Carnegie Mellon University

Problem 91

A physical pendulum consists of a light rod of length $L$ suspended in the middle. A large mass $m_{1}$ is attached to one end of
the rod, and a lighter mass $m_{2}$ is attached to the other end, as illustrated in FIGURE $13-44 .$ Find the period of oscillation for this pendulum.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 92

Predict/Calculate A vertical hollow tube is connected to a
speaker, which vibrates vertically with simple harmonic motion
(FIGURE $13-45$ ). The speaker operates with constant amplitude, $A$
but variable frequency, $f .$ A slender pencil is placed inside the
tube. (a) At low frequencies the pencil stays in contact with the
speaker at all times; at higher frequencies the pencil begins to rattle. Explain the reason for this behavior. (b) Find an expression for
the frequency at which rattling begins.

Averell Hause

Carnegie Mellon University

Problem 94

What is the temperature in degrees Fahrenheit if a cricket is observed to give 35 chirps in 13 s?
\begin{equation}
\text { A. }13^{\circ} \mathrm{F} \quad \text { B. } 35^{\circ} \mathrm{F} \quad \text { C. } 74^{\circ} \mathrm{F} \quad \text { D. } 90^{\circ} \mathrm{F}
\end{equation}

Averell Hause

Carnegie Mellon University

Problem 95

What is the frequency of the cricket's chirping (in Hz) when the
temperature is $68^{\circ} \mathrm{F}$ ?
\begin{equation}
\text { A. }0.45 \mathrm{Hz} \quad \text { B. } 2.2 \mathrm{Hz} \quad \text { C. } 5.2 \mathrm{Hz} \quad \text { D. } 29 \mathrm{Hz}
\end{equation}

Averell Hause

Carnegie Mellon University

Problem 96

Suppose the temperature decreases uniformly from $75^{\circ} \mathrm{F}$ to
63 'in 12 minutes. How many chirps does the cricket produce
during this time?
\begin{equation}
\begin{array}{lllll}{\text { A. } 28} & {\text { B. } 1700} & {\text { C. } 3800} & {\text { D. } 22,000}\end{array}
\end{equation}

Averell Hause

Carnegie Mellon University

Problem 97

Predict/Calculate REFERRING TO EXAMPLE 13.5 Suppose we can
change the plane's period of oscillation, while keeping its amplitude of motion equal to 30.0 $\mathrm{m}$ (a) If we want to reduce the maximum acceleration of the plane, should we increase or decrease the
period? Explain. (b) Find the period that results in a maximum
acceleration of 1.0 $\mathrm{g}$ .

Averell Hause

Carnegie Mellon University

Problem 98

Predict/Calculate REFERRING TO EXAMPLE $13-12$ Suppose the force
constant of the spring is doubled, but the mass and speed of the
block are still 0.980 $\mathrm{kg}$ and 1.32 $\mathrm{m} / \mathrm{s}$ , respectively. (a) By what multiplicative factor do you expect the maximum compression of the
spring to change? Explain. (b) Find the new maximum compression of the spring. (c) Find the time required for the mass to come
to rest after contacting the spring.

Averell Hause

Carnegie Mellon University

Problem 99

Predict/Calculate REFERRING TO EXAMPLE $13-12$ (a) If the block's
initial speed is increased, does the total time the block is in contact with the spring increase, decrease, or stay the same? (b) Find
the total time of contact for $v_{0}=1.65 \mathrm{m} / \mathrm{s}, m=0.980 \mathrm{kg},$ and
$k=245 \mathrm{N} / \mathrm{m} .$

Averell Hause

Carnegie Mellon University