Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 15

Oscillations - all with Video Answers

Educators

Chapter Questions

Problem 1

An object undergoing simple harmonic motion takes $0.25 \mathrm{~s}$ to travel from one point of zero velocity to the next such point. The distance between those points is $36 \mathrm{~cm}$. Calculate the (a) period,
(b) frequency, and (c) amplitude of the motion.

Joanna Josey

Joanna Josey

Numerade Educator

Problem 2

A $0.12 \mathrm{~kg}$ body undergoes simple harmonic motion of amplitude $8.5 \mathrm{~cm}$ and period $0.20 \mathrm{~s}$. (a) What is the magnitude of the maximum force acting on it? (b) If the oscillations are produced by a spring, what is the spring constant?

Vishal Gupta

Numerade Educator

Problem 3

What is the maximum acceleration of a platform that oscillates at amplitude $2.20 \mathrm{~cm}$ and frequency $6.60 \mathrm{~Hz}$ ?

Joanna Josey

Numerade Educator

Problem 4

An automobile can be considered to be mounted on four identical springs as far as vertical oscillations are concerned. The springs of a certain car are adjusted so that the oscillations have a frequency of $3.00$ $\mathrm{H} z$ (a) What is the spring constant of each spring if the mass of the car is $1450 \mathrm{~kg}$ and the mass is evenly distributed over the springs? (b) What will be the oscillation frequency if five passengers, averaging $73.0 \mathrm{~kg}$ each, ride in the car with an even distribution of mass?

Supratim Pal

Numerade Educator

Problem 5

In an electric shaver, the blade moves back and forth over a distance of $2.0 \mathrm{~mm}$ in simple harmonic motion, with frequency $120 \mathrm{~Hz}$. Find (a) the amplitude, (b) the maximum blade speed, and
(c) the magnitude of the maximum blade acceleration.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 6

A particle with a mass of $1.00 \times 10^{-20} \mathrm{~kg}$ is oscillating with simple harmonic motion with a period of $1.00 \times 10^{-5} \mathrm{~s}$ and a maximum speed of $1.00 \times 10^{3} \mathrm{~m} / \mathrm{s}$. Calculate (a) the angular frequency and (b) the maximum displacement of the particle.

Willis James

Numerade Educator

Problem 7

A loudspeaker produces a musical sound by means of the oscillation of a diaphragm whose amplitude is limited to $1.00 \mu \mathrm{m} .$ (a) At what frequency is the magnitude $a$ of the diaphragm's acceleration equal to $g$ ? (b) For greater frequencies, is $a$ greater than or less than $g$ ?

Joanna Josey

Numerade Educator

Problem 8

What is the phase constant for the harmonic oscillator with the position function $x(t)$ given in Fig. 15 30 if the position function has the form $x=x_{m} \cos (\omega t+\phi) ?$ The ver-
tical axis scale is set by $x_{s}=6.0 \mathrm{~cm}$.

Supratim Pal

Numerade Educator

Problem 9

$=9$ The position function $x=$ $(6.0 \mathrm{~m}) \cos [(3 \pi \mathrm{rad} / \mathrm{s}) t+\pi / 3 \mathrm{rad}]$
gives the simple harmonic motion of a body. At $t=2.0 \mathrm{~s}$, what are the
(a) displacement,
(b) velocity, (c)
acceleration, and (d) phase of the motion? Also, what are the (e) frequency and (f) period of the motion?

Supratim Pal

Numerade Educator

Problem 10

An oscillating block-spring system takes $0.75 \mathrm{~s}$ to begin re-
peating its motion. Find (a) the period, (b) the frequency in hertz, and
(c) the angular frequency in radians per second.

Supratim Pal

Numerade Educator

Problem 11

In Fig. $15-31$, two identical springs of spring constant $7580 \mathrm{~N} / \mathrm{m}$
are attached to a block of mass $0.245 \mathrm{~kg}$. What is the frequency of oscillation on the frictionless floor?

Joanna Josey

Numerade Educator

Problem 12

What is the phase constant for the harmonic oscillator with the velocity function $v(t)$ given in Fig. $15-32$ if the position function $x(t)$ has the form $x=x_{m} \cos (\omega t+\phi) ?$ The vertical axis
scale is set by $v_{s}=4.0 \mathrm{~cm} / \mathrm{s}$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 13

An oscillator consists of a block of mass $0.500 \mathrm{~kg}$ connected to
a spring. When set into oscillation with amplitude $35.0 \mathrm{~cm}$, the oscillator repeats its motion every $0.500 \mathrm{~s}$. Find the (a) period, (b) frequency, (c) angular frequency, (d) spring constant, (e) maximum speed, and (f) magnitude of the maximum force on the block from the spring.

Joanna Josey

Numerade Educator

Problem 14

A simple harmonic oscillator consists of a block of mass $2.00 \mathrm{~kg}$ attached to a spring of spring constant $100 \mathrm{~N} / \mathrm{m} .$ When $t=1.00 \mathrm{~s}$, the position and velocity of the block are $x=0.129$ $\mathrm{m}$ and $v=3.415 \mathrm{~m} / \mathrm{s}$. (a) What is the amplitude of the oscillations? What were the (b) position and (c) velocity of the block at $t=0 \mathrm{~s}$ ?

Averell Hause

Carnegie Mellon University

Problem 15

Two particles oscillate in simple harmonic motion along a common straight-line segment of length $A$. Each particle has a period of $1.5 \mathrm{~s}$, but they differ in phase by $\pi / 6 \mathrm{rad}$. (a) How far apart are they (in terms of $A$ ) $0.50 \mathrm{~s}$ after the lagging particle leaves one end of the path? (b) Are they then moving in the same direction, toward each other, or away from each other?

Joanna Josey

Numerade Educator

Problem 16

Two particles execute simple harmonic motion of the same amplitude and frequency along close parallel lines. They pass each other moving in opposite directions each time their displacement is half their amplitude. What is their phase difference?

Averell Hause

Carnegie Mellon University

Problem 17

An oscillator consists of a block attached to a spring ( $k=$ $400 \mathrm{~N} / \mathrm{m}) .$ At some time $t$, the position (measured from the system's equilibrium location), velocity, and acceleration of the block are $x=0.100 \mathrm{~m}, v=-13.6 \mathrm{~m} / \mathrm{s}$, and $a=-123 \mathrm{~m} / \mathrm{s}^{2} .$ Calculate (a) the
frequency of oscillation, (b) the mass of the block, and (c) the amplitude of the motion.

Joanna Josey

Numerade Educator

Problem 18

At a certain harbor, the tides cause the ocean surface to rise and fall a distance $d$ (from highest level to lowest level) in simple harmonic motion, with a period of $12.5 \mathrm{~h}$. How long does it take for the water to fall a distance $0.250 d$ from its highest level?

Averell Hause

Carnegie Mellon University

Problem 19

A block rides on a piston (a squat cylindrical piece) that is moving vertically with simple harmonic motion. (a) If the SHM has period $1.0 \mathrm{~s}$, at what amplitude of motion will the block and piston separate? (b) If the piston has an amplitude of $5.0 \mathrm{~cm}$, what is the maximum frequency for which the block and piston will be in contact continuously?

Joanna Josey

Numerade Educator

Problem 20

Figure $15-33 a$ is a partial graph of the position function
$x(t)$ for a simple harmonic oscillator with an angular frequency of
$1.20 \mathrm{rad} / \mathrm{s}$; Fig. $15-33 b$ is a partial graph of the corresponding velocity function $v(t) .$ The vertical axis scales are set by $x_{s}=$ $5.0 \mathrm{~cm}$ and $v_{s}=5.0 \mathrm{~cm} / \mathrm{s}$. What
is the phase constant of the SHM if the position function $x(t)$ is in the general form $x=$ $x_{m} \cos (\omega t+\phi) ?$

Averell Hause

Carnegie Mellon University

Problem 21

In Fig. $15-31$, two springs are attached to a block that can oscillate over a frictionless floor. If the left spring is removed, the block oscillates at a frequency of $30 \mathrm{~Hz}$. If, instead, the spring on the right is removed, the block oscillates at a frequency of 45 $\mathrm{Hz}$. At what frequency does the block oscillate with both springs attached?

Joanna Josey

Numerade Educator

Problem 22

Figure $15-34$ shows block 1 of mass $0.200 \mathrm{~kg}$ sliding to the right over a frictionless elevated surface at a speed of $8.00 \mathrm{~m} / \mathrm{s}$. The block undergoes an elastic collision with stationary block 2, which is attached to a spring of spring constant $1208.5 \mathrm{~N} / \mathrm{m} .$ (Assume
that the spring does not affect the collision.) After the collision, block 2 oscillates in SHM with a period of $0.140 \mathrm{~s}$, and block 1 slides off the opposite end of the elevated surface, landing a distance $d$ from the base of that surface after falling height $h=4.90$ $\mathrm{m}$. What is the value of $d ?$

Averell Hause

Carnegie Mellon University

Problem 23

A block is on a horizontal surface (a shake table) that is moving back and forth horizontally with simple harmonic motion of frequency $2.0 \mathrm{~Hz}$. The coefficient of static friction between block and surface is $0.50 .$ How great can the amplitude of the SHM be if the block is not to slip along the surface?

Joanna Josey

Numerade Educator

Problem 24

In Fig. $15-35$, two springs are joined and connected to a block of mass $0.245 \mathrm{~kg}$ that is set oscillating over a frictionless floor. The springs each have spring constant $k=$ $6430 \mathrm{~N} / \mathrm{m}$. What is the frequency of the oscillations?

Averell Hause

Carnegie Mellon University

Problem 25

In Fig. 15-36, a block weighing $14.0 \mathrm{~N}$, which can slide without friction on an incline at angle $\theta=40.0^{\circ}$, is connected to the top of the incline by a massless spring of unstretched length $0.450$ $\mathrm{m}$ and spring constant $120 \mathrm{~N} / \mathrm{m} .$ (a) How far from the top of the incline is the block's equilibrium point? (b)
If the block is pulled slightly down the incline and released, what is the period of the resulting oscillations?

Joanna Josey

Numerade Educator

Problem 26

In Fig. 15-37, two blocks $(m=1.8 \mathrm{~kg}$ and $M=10 \mathrm{~kg}$ ) and
a spring $(k=200 \mathrm{~N} / \mathrm{m})$ are arranged on a horizontal, frictionless surface. The coefficient of static friction between the two blocks is $0.40$. What amplitude of simple harmonic motion of the spring-blocks system puts the smaller block on the verge of slipping over the larger blocK?

Averell Hause

Carnegie Mellon University

Problem 27

When the displacement in SHM is one-half the amplitude $x_{m}$, what fraction of the total energy is (a) kinetic energy and
(b) potential energy? (c) At what displacement, in terms of the amplitude, is the energy of the system half kinetic energy and half potential energy?

Joanna Josey

Numerade Educator

Problem 28

Figure $15-38$ gives the onedimensional potential energy well for a $2.0 \mathrm{~kg}$ particle (the function $U(x)$ has the form $b x^{2}$ and the vertical axis scale is set by $U_{s}=2.0 \mathrm{~J}$ ).
(a) If the particle passes through the equilibrium position with a velocity of $85 \mathrm{~cm} / \mathrm{s}$, will it be turned back before it reaches $x=15 \mathrm{~cm} ?$
(b) If yes, at what position, and if no, what is the speed of the particle at $x=15 \mathrm{~cm}$ ?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 29

Find the mechanical energy of a block-spring system with a spring constant of $1.3 \mathrm{~N} / \mathrm{cm}$ and an amplitude of $2.4 \mathrm{~cm}$.

Joanna Josey

Numerade Educator

Problem 30

An oscillating block-spring system has a mechanical energy of 1.00 J, an amplitude of $10.0 \mathrm{~cm}$, and a maximum speed of $1.20 \mathrm{~m} / \mathrm{s}$. Find (a) the spring constant, (b) the mass of the block, and (c) the frequency of oscillation.

Averell Hause

Carnegie Mellon University

Problem 31

A $5.00 \mathrm{~kg}$ object on a horizontal frictionless surface is attached to a spring with $k=1000 \mathrm{~N} / \mathrm{m}$. The object is displaced from equilibrium $50.0 \mathrm{~cm}$ horizontally and given an initial velocity of $10.0$ $\mathrm{m} / \mathrm{s}$ back toward the equilibrium position. What are (a) the motion's frequency, (b) the initial potential energy of the block-spring system, (c) the initial kinetic energy, and
(d) the motion's amplitude?

Joanna Josey

Numerade Educator

Problem 32

Figure $15-39$ shows the $\mathrm{ki}$ netic energy $K$ of a simple harmonic oscillator versus its position $x$. The vertical axis scale is set by $K_{\mathrm{s}}=4.0 \mathrm{~J}$. What is the spring constant?

Supratim Pal

Numerade Educator

Problem 33

A block of mass $M=5.4$ $\mathrm{kg}$, at rest on a horizontal frictionless table, is attached to a rigid support by a spring of constant $k=6000 \mathrm{~N} / \mathrm{m}$. A bullet of mass $m=9.5 \mathrm{~g}$ and velocity $\vec{v}$ of magnitude $630 \mathrm{~m} / \mathrm{s}$ strikes and is embedded in the block (Fig. 15.
40). Assuming the compression of the spring is negligible until the bullet is embedded, determine (a) the speed of the block immediately after the collision and (b)
the amplitude of the resulting simple harmonic motion.

Supratim Pal

Numerade Educator

Problem 34

In Fig. $15-41$, block 2 of mass $2.0 \mathrm{~kg}$ oscillates on the end of a spring in SHM with a period of 20 $\mathrm{ms}$. The block's position is given by $x=(1.0 \mathrm{~cm}) \cos (\omega t+\pi / 2) . \underline{\text { Block }}$
of mass $4.0 \mathrm{~kg}$ slides toward block 2
with a velocity of magnitude $6.0 \mathrm{~m} / \mathrm{s}$, directed along the spring's length. The two blocks undergo a completely inelastic collision at time $t=5.0 \mathrm{~ms}$. (The duration of the collision is much less than the period of motion.) What is the amplitude of the SHM after the collision?

Supratim Pal

Numerade Educator

Problem 35

A $10 \mathrm{~g}$ particle undergoes SHM with an amplitude of $2.0 \mathrm{~mm}$, a maximum acceleration of magnitude $8.0 \times 10^{3} \mathrm{~m} / \mathrm{s}^{2}$, and an unknown phase constant $\phi$. What are (a) the period of the motion,
(b) the maximum speed of the particle, and (c) the total mechanical energy of the oscillator? What is the magnitude of the force on the particle when the particle is at (d) its maximum displacement and (e) half its maximum displacement?

Joanna Josey

Numerade Educator

Problem 36

If the phase angle for a block-spring system in SHM is $\pi / 6$ rad and the block's position is given by $x=x_{m} \cos (\omega t+\phi)$, what is the ratio of the kinetic energy to the potential energy at time $t=0 ?$

Averell Hause

Carnegie Mellon University

Problem 37

A massless spring hangs from the ceiling with a small object attached to its lower end. The object is initially held at rest in a position $y_{i}$ such that the spring is at its rest length. The object is then released from $y_{i}$ and oscillates up and down, with its lowest position being $10 \mathrm{~cm}$ below $y_{i}$. (a) What is the frequency of the oscillation? (b) What is the speed of the object when it is $8.0 \mathrm{~cm}$ below the initial position? (c) An object of mass $300 \mathrm{~g}$ is attached to the first object, after which the system oscillates with half the original frequency. What is the mass of the first object? (d) How far below $y_{i}$ is the new equilibrium (rest) position with both objects attached to the spring?

Joanna Josey

Numerade Educator

Problem 38

A $95 \mathrm{~kg}$ solid sphere with a $15 \mathrm{~cm}$ radius is suspended by a vertical wire. A torque of $0.20 \mathrm{~N} \cdot \mathrm{m}$ is required to rotate the sphere through an angle of $0.85$ rad and then maintain that orientation. What is the period of the oscillations that result when the sphere is then released?

Averell Hause

Carnegie Mellon University

Problem 39

The balance wheel of an old-fashioned watch oscillates with angular amplitude $\pi$ rad and period $0.500 \mathrm{~s}$. Find
(a) the maximum angular speed of the wheel, (b) the angular speed at displacement $\pi / 2 \mathrm{rad}$, and $(\mathrm{c})$ the magnitude of the angular acceleration at displacement $\pi / 4$ rad.

Joanna Josey

Numerade Educator

Problem 40

A physical pendulum consists of a meter stick that is piv-
oted at a small hole drilled through the stick a distance $d$ from the 50
$\mathrm{cm}$ mark. The period of oscillation is $2.5 \mathrm{~s}$. Find $d$.

Supratim Pal

Numerade Educator

Problem 41

In Fig. $15-42$, the pendulum consists of a uniform disk with radius $r=10.0 \mathrm{~cm}$ and mass $500 \mathrm{~g}$ attached to a uniform rod with length $L=500 \mathrm{~mm}$ and mass 270
g. (a) Calculate the rotational inertia of the pendulum about the pivot point. (b) What is the distance between the pivot point and
the center of mass of the pendulum? (c) Calculate the period of oscillation.

Joanna Josey

Numerade Educator

Problem 42

Suppose that a simple pendulum consists of a small $60.0 \mathrm{~g}$ bob at the end of a cord of negligible mass. If the angle $\theta$ between the cord and the vertical is given by
$$
\theta=(0.0800 \mathrm{rad}) \cos [(4.43 \mathrm{rad} / \mathrm{s}) t+\phi] \text { , }
$$
what are (a) the pendulum's length and (b) its maximum kinetic energy?

Averell Hause

Carnegie Mellon University

Problem 43

If the physical pendulum of Fig. $15-13$ and the associated sample problem is inverted and suspended at point $P$, what is its period of oscillation? (b) Is the period now greater than, less than, or equal to its previous value?

Joanna Josey

Numerade Educator

Problem 44

A physical pendulum consists of two meter-long sticks joined together as shown in Fig. $15-43 .$ What is the pendulum's period of oscillation about a pin inserted through point $A$ at the center of the horizontal stick?

Averell Hause

Carnegie Mellon University

Problem 45

A performer seated on a trapeze is swinging back and forth with a period of $8.85 \mathrm{~s}$. If she stands up, thus raising the center of mass of
the trapeze $+$ performer system by $35.0 \mathrm{~cm}$, what will be the new period of the system? Treat trapeze $+$ performer as a simple pendulum.

Joanna Josey

Numerade Educator

Problem 46

A physical pendulum has a center of oscillation at distance $2 L / 3$ from its point of suspension. Show that the distance between the point of suspension and the center of oscillation for a physical pendulum of any form is $I / m h$, where $I$ and $h$ have the meanings in Eq. $15-29$ and $m$ is the mass of the pendulum.

Supratim Pal

Numerade Educator

Problem 47

In Fig. $15-44$, a physical pendulum consists of a uniform solid disk (of radius $R=2.35 \mathrm{~cm}$ ) supported in a vertical plane by a pivot located a distance $d=1.75 \mathrm{~cm}$ from the center of the disk. The disk is displaced by a small angle and released. What is the period of the resulting simple harmonic motion?

Joanna Josey

Numerade Educator

Problem 48

A rectangular block, with face lengths $a=35 \mathrm{~cm}$ and $b=45 \mathrm{~cm}$, is to be
suspended on a thin horizontal rod running through a narrow hole in the block. The block is then to be set swinging about the rod like a pendulum, through small angles so that it is in SHM. Figure $15-45$ shows one possible position of the hole, at distance $r$ from the block's center, along a line connecting the center with a corner.
(a) Plot the
period versus distance $r$ along that line such that the minimum in the curve is apparent. (b) For what value of $r$ does that minimum occur? There is a line of points around the block's center for which the period of swinging has the same minimum value. (c) What shape does that line make?

Averell Hause

Carnegie Mellon University

Problem 49

The angle of the pendulum of Fig. $15-11 b$ is given by $\theta=$ $\theta_{m} \cos [(4.44 \mathrm{rad} / \mathrm{s}) t+\phi] .$ If at $t=0$
$\theta=0.040 \mathrm{rad}$ and $d \theta^{\prime} d t=-0.200$
$\mathrm{rad} / \mathrm{s}$, what are (a) the phase con-
stant $\phi$ and (b) the maximum angle $\theta_{m} ?$ (Hint: Don't confuse the rate $d \theta^{\prime} d t$ at which $\theta$ changes with the $\omega$ of the SHM.)

Supratim Pal

Numerade Educator

Problem 50

A thin uniform rod $($ mass $=0.50 \mathrm{~kg}$ ) swings about an axis that passes through one end of the rod and is perpendicu-
lar to the plane of the swing. The rod swings with a period of $1.5 \mathrm{~s}$ and an angular amplitude of $10^{\circ}$.
(a) What is the length of the rod?
(b) What is the maximum kinetic energy of the rod as it swings?

Supratim Pal

Numerade Educator

Problem 51

In Fig. $15-46$, a stick of length $L=1.85 \mathrm{~m}$ oscillates as $\mathrm{a}$ physical pendulum. (a) What value of distance $x$ between the stick's center of mass and its pivot point $O$ gives the least period? (b) What is that least period?

Joanna Josey

Numerade Educator

Problem 52

The $3.00 \mathrm{~kg}$ cube in Fig. $15-47$ has edge lengths $d=6.00 \mathrm{~cm}$ and is mounted on an axle through its center. A spring $(k=1200 \mathrm{~N} / \mathrm{m})$ connects the cube's upper corner to a rigid wall. Initially the spring is at its rest length. If the cube is rotated $3^{\circ}$ and released, what is the period of the resulting $\mathrm{SHM} ?$

Averell Hause

Carnegie Mellon University

Problem 53

In the overhead view of Fig. 15 48 , a long uniform rod of mass $0.600 \mathrm{~kg}$ is free to
rotate in a horizontal plane about a vertical axis through its center. A spring with force constant $k=1850$ $\mathrm{N} / \mathrm{m}$ is connected horizontally between one end of the rod and a fixed wall. When the rod is in equilibrium, it is parallel to the wall. What is the period of the small os-
cillations that result when the rod is rotated slightly and released?

Supratim Pal

Numerade Educator

Problem 54

In Fig. $15-49 a$, a metal plate is mounted on an axle through its center of mass. A spring with $k=2000 \mathrm{~N} / \mathrm{m}$ connects a wall with a point on the rim a distance $r=2.5 \mathrm{~cm}$ from the center of mass. Initially the spring is at its rest length. If the plate is rotated by $7^{\circ}$ and released, it rotates about the axle in SHM, with its angular position given by Fig. $15-49 b$. The horizontal axis scale is set by $t_{s}=20 \mathrm{~ms}$ What is the rotational inertia of the plate about its center of mass?

Averell Hause

Carnegie Mellon University

Problem 55

A pendulum is formed by pivoting a long thin rod about a point on the rod. In a series of experiments, the period is measured as a function of the distance $x$ between the pivot point and the rod's center. (a) If the rod's length is $L=2.20 \mathrm{~m}$ and its mass is $m=22.1 \mathrm{~g}$, what is the minimum period? (b) If $x$ is cho-
sen to minimize the period and then $L$ is increased, does the period increase, decrease, or remain the same? (c) If, instead, $m$ is increased without $L$ increasing, does the period increase, decrease, or remain the same?

Supratim Pal

Numerade Educator

Problem 56

In Fig. $15-50$, a $2.50 \mathrm{~kg}$ disk of diameter $D=42.0 \mathrm{~cm}$ is supported by a rod of length $L=76.0$ $\mathrm{cm}$ and negligible mass that is pivoted at its end. (a) With the massless torsion spring unconnected, what is the period of oscillation? (b) With the torsion spring connected, the rod is vertical at equilibrium. What is the torsion constant of the spring if the period of oscillation has been decreased by $0.500 \mathrm{~s}$ ?

Averell Hause

Carnegie Mellon University

Problem 57

The amplitude of a lightly damped oscillator decreases by $3.0 \%$ during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?

Joanna Josey

Numerade Educator

Problem 58

For the damped oscillator system shown in Fig. $15-16$, with $m=250 \mathrm{~g}, k=85 \mathrm{~N} / \mathrm{m}$, and $b=70 \mathrm{~g} / \mathrm{s}$, what is the ratio of the oscil-
lation amplitude at the end of 20 cycles to the initial oscillation amplitude?

Supratim Pal

Numerade Educator

Problem 59

For the damped oscillator system shown in Fig. 15-16, the block has a mass of $1.50 \mathrm{~kg}$ and the spring constant is $8.00 \mathrm{~N} / \mathrm{m} .$ The damping force is given by $-b(d x / d t)$, where $b=230$ $\mathrm{g} / \mathrm{s}$. The block is pulled down $12.0 \mathrm{~cm}$ and released. (a) Calculate the time required for the amplitude of the resulting oscillations to fall to one-third of its initial value. (b) How many oscillations are made by the block in this time?

Joanna Josey

Numerade Educator

Problem 60

The suspension system of a $2000 \mathrm{~kg}$ automobile "sags" $10 \mathrm{~cm}$ when the chassis is placed on it. Also, the oscillation amplitude decreases by $50 \%$ each cycle. Estimate the values of (a) the spring constant $k$ and (b) the damping constant $b$ for the spring and shock absorber system of one wheel, assuming each wheel supports $500 \mathrm{~kg}$.

Averell Hause

Carnegie Mellon University

Problem 61

For Eq. $15-45$, suppose the amplitude $x_{m}$ is given by
$$
x_{m}=\frac{F_{m}}{\left[m^{2}\left(\omega_{d}^{2}-\omega^{2}\right)^{2}+b^{2} \omega_{d}^{2}\right]^{1 / 2}},
$$
where $F_{m}$ is the (constant) amplitude of the external oscillating force exerted on the spring by the rigid support in Fig. $15-16 .$ At resonance, what are the (a) amplitude and (b) velocity amplitude of the oscillating object?

Joanna Josey

Numerade Educator

Problem 62

Hanging from a horizontal beam are nine simple pendulums of the following lengths: (a) $0.10$, (b) $0.30,(\mathrm{c}) 0.40,($ d) $0.80$, (e) $1.2$,
(f) $2.8,(\mathrm{~g}) 3.5,(\mathrm{~h}) 5.0$, and (i) $6.2 \mathrm{~m}$. Suppose the beam undergoes horizontal oscillations with angular frequencies in the range from $2.00 \mathrm{rad} / \mathrm{s}$ to $4.00 \mathrm{rad} / \mathrm{s}$. Which of the pendulums will be (strongly) set in motion?

Averell Hause

Carnegie Mellon University

Problem 63

A $1000 \mathrm{~kg}$ car carrying four $82 \mathrm{~kg}$ people travels over a "washboard" dirt road with corrugations $4.0 \mathrm{~m}$ apart. The car bounces with maximum amplitude when its speed is $16 \mathrm{~km} / \mathrm{h}$. When the car stops, and the people get out, by how much does the car body rise on its suspension?

Joanna Josey

Numerade Educator

Problem 64

Although California is known for earthquakes, it has large regions dotted with precariously balanced rocks that would be easily toppled by even a mild earthquake. Apparently no major earthquakes have occurred in those regions. If an earthquake were to put such a rock into sinusoidal oscillation (parallel to the ground) with a frequency of $2.2 \mathrm{~Hz}$, an oscillation amplitude of $1.0$ $\mathrm{cm}$ would cause the rock to topple. What would be the magnitude of the maximum acceleration of the oscillation, in terms of $g$ ?

Supratim Pal

Numerade Educator

Problem 65

A loudspeaker diaphragm is oscillating in simple harmonic motion with a frequency of $440 \mathrm{~Hz}$ and a maximum displacement of $0.75 \mathrm{~mm}$. What are the (a) angular frequency, (b) maximum speed, and (c) magnitude of the maximum acceleration?

Joanna Josey

Numerade Educator

Problem 66

A uniform spring with $k=8600 \mathrm{~N} / \mathrm{m}$ is cut into pieces 1 and 2 of unstretched lengths $L_{1}=7.0 \mathrm{~cm}$ and $L_{2}=10 \mathrm{~cm} .$ What are
(a) $k_{1}$ and (b) $k_{2}$ ? A block attached to the original spring as in Fig. $15-7$ oscillates at $200 \mathrm{~Hz}$. What is the oscillation frequency of the block attached to (c) piece 1 and (d) piece $2 ?$

Averell Hause

Carnegie Mellon University

Problem 67

In Fig. $15-51$, three $10000 \mathrm{~kg}$ ore cars are held at rest on a mine railway using a cable that is parallel to the rails, which are inclined at angle $\theta=30^{\circ}$. The cable stretches 15 $\mathrm{cm}$ just before the coupling between the two lower cars breaks, detaching the lowest car. Assuming that the cable obeys Hooke's law, find the (a) frequency and (b) amplitude of the resulting oscillations of the remaining two cars.

Joanna Josey

Numerade Educator

Problem 68

A $2.00 \mathrm{~kg}$ block hangs from a spring. A $300 \mathrm{~g}$ body hung below the block stretches the spring $2.00 \mathrm{~cm}$ farther. (a) What is the spring constant? (b) If the $300 \mathrm{~g}$ body is removed and the block is set into oscillation, find the period of the motion.

Averell Hause

Carnegie Mellon University

Problem 69

In the engine of a locomotive, a cylindrical piece known as a piston oscillates in SHM in a cylinder head (cylindrical chamber) with an angular frequency of 180 rev/min. Its stroke (twice the amplitude) is $0.76 \mathrm{~m}$. What is its maximum speed?

Joanna Josey

Numerade Educator

Problem 70

A wheel is free to rotate about its fixed axle. A spring is attached to one of its spokes a distance $r$ from the axle, as shown in Fig. $15-52 .$ (a) Assuming that the wheel is a hoop of mass $m$ and radius $R$,
what is the angular frequency $\omega$ of small oscillations of this system in terms of $m, R, r$, and the spring constant $k ?$ What is $\omega$ if (b) $r=R$ and (c) $r=0 ?$

Averell Hause

Carnegie Mellon University

Problem 71

A $50.0 \mathrm{~g}$ stone is attached to the bottom of a vertical spring and set vibrating. If the maximum speed of the stone is $15.0 \mathrm{~cm} / \mathrm{s}$ and the period is $0.500 \mathrm{~s}$, find the (a) spring
constant of the spring, (b) amplitude of the motion, and (c) frequency of oscillation.

Joanna Josey

Numerade Educator

Problem 72

A uniform circular disk whose radius $R$ is $12.6 \mathrm{~cm}$ is suspended as a physical pendulum from a point on its rim. (a) What is its period? (b) At what radial distance $r<R$ is there a pivot point that gives the same period?

Averell Hause

Carnegie Mellon University

Problem 73

A vertical spring stretches $9.6 \mathrm{~cm}$ when a $1.3 \mathrm{~kg}$ block
is hung from its end. (a) Calculate the spring constant. This block is then displaced an additional $5.0 \mathrm{~cm}$ downward and released from rest. Find the (b) period, (c) frequency, (d) amplitude, and
(e) maximum speed of the resulting SHM.

Joanna Josey

Numerade Educator

Problem 74

A massless spring with spring constant $19 \mathrm{~N} / \mathrm{m}$ hangs vertically. A body of mass $0.20 \mathrm{~kg}$ is attached to its free end and then released. Assume that the spring was unstretched before the body was released. Find (a) how far below the initial position the body descends, and the (b) frequency and (c) amplitude of the resulting SHM.

Averell Hause

Carnegie Mellon University

Problem 75

A $4.00 \mathrm{~kg}$ block is suspended from a spring with $k=500 \mathrm{~N} / \mathrm{m}$. A $50.0 \mathrm{~g}$ bullet is fired into the block from directly below with a speed of $150 \mathrm{~m} / \mathrm{s}$ and becomes embedded in the block. (a) Find the amplitude of the resulting SHM. (b) What percentage of the original kinetic energy of the bullet is transferred to mechanical energy of the oscillator?

Joanna Josey

Numerade Educator

Problem 76

A $55.0 \mathrm{~g}$ block oscillates in SHM on the end of a spring with $k=1500 \mathrm{~N} / \mathrm{m}$ according to $x=x_{m} \cos (\omega t+\phi) .$ How long does
the block take to move from position $+0.800 x_{m}$ to (a) position $+0.600 x_{m}$ and (b) position $-0.800 x_{m} ?$

Averell Hause

Carnegie Mellon University

Problem 77

Figure $15-53$ gives the position of a $20 \mathrm{~g}$ block oscillating in SHM on the end of a spring. The horizontal axis scale is set by $t_{s}=40.0 \mathrm{~ms}$. What are (a) the maximum kinetic energy of the block and (b) the number of times per second that maximum is reached? (Hint: Measuring a slope will probably not be very accurate. Find another approach.)

Joanna Josey

Numerade Educator

Problem 78

Figure $15-53$ gives the position $x(t)$ of a block oscillating in SHM on the end of a spring $\left(t_{s}=40.0 \mathrm{~ms}\right)$. What are (a) the speed and (b) the magnitude of the radial acceleration of a particle in the corresponding uniform circular motion?

Supratim Pal

Numerade Educator

Problem 79

Figure $15-54$ shows the kinetic energy $K$ of a simple pendulum versus its angle $\theta$ from the vertical. The vertical axis scale is set by $K_{s}=10.0 \mathrm{~mJ}$. The pendulum bob has mass $0.200 \mathrm{~kg}$. What is the length of the pendulum?

Joanna Josey

Numerade Educator

Problem 80

A block is in SHM on the end of a spring, with position given by $x=x_{m} \cos (\omega t+\phi)$. If $\phi=\pi / 5 \mathrm{rad}$,
then at $t=0$ what percentage of the
total mechanical energy is potential energy?

Averell Hause

Carnegie Mellon University

Problem 81

A simple harmonic oscillator consists of a $0.50 \mathrm{~kg}$ block attached to a spring. The block slides back and forth along a straight line on a frictionless surface with equilibrium point $x=0$. At $t=0$ the block is at $x=0$ and moving in the positive $x$ direction. A graph of the magnitude of the net force $\vec{F}$ on the block as a function of its
position is shown in Fig. $15-55$. The vertical scale is set by $F_{s}=$ $75.0 \mathrm{~N}$. What are (a) the amplitude and (b) the period of the motion, (c) the magnitude of the maximum acceleration, and (d) the maximum kinetic energy?

Joanna Josey

Numerade Educator

Problem 82

A simple pendulum of length $20 \mathrm{~cm}$ and mass $5.0 \mathrm{~g}$ is
suspended in a race car traveling with constant speed $70 \mathrm{~m} / \mathrm{s}$ around a circle of radius $50 \mathrm{~m}$. If the pendulum undergoes small oscillations in a radial direction about its equilibrium position, what is the frequency of oscillation?

Averell Hause

Carnegie Mellon University

Problem 83

The scale of a spring balance that reads from 0 to $15.0 \mathrm{~kg}$ is $12.0 \mathrm{~cm}$ long. A package suspended from the balance is found to oscillate vertically with a frequency of $2.00 \mathrm{~Hz}$. (a) What is the spring constant? (b) How much does the package weigh?

Joanna Josey

Numerade Educator

Problem 84

A $0.10 \mathrm{~kg}$ block oscillates back and forth along a straight line on a frictionless horizontal surface. Its displacement from the origin is given by
$$
x=(10 \mathrm{~cm}) \cos [(10 \mathrm{rad} / \mathrm{s}) t+\pi / 2 \mathrm{rad}]
$$
(a) What is the oscillation frequency? (b) What is the maximum speed acquired by the block? (c) At what value of $x$ does this occur? (d) What is the magnitude of the maximum acceleration of the block? (e) At what value of $x$ does this occur?
(f) What force, applied to the block by the spring, results in the given oscillation?

Averell Hause

Carnegie Mellon University

Problem 85

The end point of a spring oscillates with a period of $2.0 \mathrm{~s}$ when a block with mass $m$ is attached to it. When this mass is increased by $2.0 \mathrm{~kg}$, the period is found to be $3.0 \mathrm{~s}$. Find $m$.

Joanna Josey

Numerade Educator

Problem 86

The tip of one prong of a tuning fork undergoes SHM of frequency $1000 \mathrm{~Hz}$ and amplitude $0.40 \mathrm{~mm}$. For this tip, what is the magnitude of the (a) maximum acceleration, (b) maximum velocity, (c) acceleration at tip displacement $0.20 \mathrm{~mm}$, and (d) velocity at tip displacement $0.20 \mathrm{~mm}$ ?

Averell Hause

Carnegie Mellon University

Problem 87

A flat uniform circular disk has a mass of $3.00 \mathrm{~kg}$ and a radius of $70.0 \mathrm{~cm}$. It is suspended in a horizontal plane by a vertical wire attached to its center. If the disk is rotated $2.50 \mathrm{rad}$ about the wire, a torque of $0.0600 \mathrm{~N} \cdot \mathrm{m}$ is required to maintain that orientation. Calculate (a) the rotational inertia of the disk about the wire, (b) the torsion constant, and (c) the angular frequency of this torsion pendulum when it is set oscillating.

Joanna Josey

Numerade Educator

Problem 88

A block weighing $20 \mathrm{~N}$ oscillates at one end of a vertical spring for which $k=100 \mathrm{~N} / \mathrm{m} ;$ the other end of the spring is attached to a ceiling. At a certain instant the spring is stretched $0.30 \mathrm{~m}$ beyond its relaxed length (the length when no object is attached) and the block has zero velocity. (a) What is the net force on the block at this instant? What are the (b) amplitude and (c) period of the resulting simple harmonic motion? (d) What is the maximum kinetic energy of the block as it oscillates?

Averell Hause

Carnegie Mellon University

Problem 89

A $3.0 \mathrm{~kg}$ particle is in simple harmonic motion in one dimension and moves according to the equation
$$
x=(5.0 \mathrm{~m}) \cos [(\pi / 3 \mathrm{rad} / \mathrm{s}) t-\pi / 4 \mathrm{rad}]
$$
with $t$ in seconds. (a) At what value of $x$ is the potential energy of the particle equal to half the total energy? (b) How long does the particle take to move to this position $x$ from the equilibrium position?

Joanna Josey

Numerade Educator

Problem 90

A particle executes linear SHM with frequency $0.25 \mathrm{~Hz}$ about the point $x=0$. At $t=0$, it has displacement $x=0.37 \mathrm{~cm}$ and zero velocity. For the motion, determine the (a) period, (b) angular frequency, (c) amplitude, (d) displacement $x(t),(\mathrm{e})$ velocity $v(t)$,
(f) maximum speed, (g) magnitude of the maximum acceleration,
(h) displacement at $t=3.0 \mathrm{~s}$, and
(i) speed at $t=3.0 \mathrm{~s}$

Averell Hause

Carnegie Mellon University

Problem 91

What is the frequency of a simple pendulum $2.0 \mathrm{~m}$ long
(a) in a room, (b) in an elevator accelerating upward at a rate of $2.0 \mathrm{~m} / \mathrm{s}^{2}$, and $(\mathrm{c})$ in free fall?

Joanna Josey

Numerade Educator

Problem 92

A grandfather clock has a pendulum that consists of a thin brass disk of radius $r=15.00 \mathrm{~cm}$ and mass $1.000$ kg that is attached to a long thin rod of negligible mass. The pendulum swings freely about an axis perpendicular to the rod and through the end of the rod opposite the disk, as shown in Fig. $15.56 .$ If the pendulum is to have a period of $2.000 \mathrm{~s}$ for small oscillations at a place where $g=9.800 \mathrm{~m} / \mathrm{s}^{2}$, what must be the rod length $L$ to the nearest tenth of a millimeter?

Averell Hause

Carnegie Mellon University

Problem 93

A $4.00 \mathrm{~kg}$ block hangs from a spring, extending it $16.0 \mathrm{~cm}$ from its
unstretched position. (a) What is the spring constant? (b) The block is removed, and a $0.500 \mathrm{~kg}$ body is hung from the same spring. If the spring is then stretched and released, what is its period of oscillation?

Joanna Josey

Numerade Educator

Problem 94

What is the phase constant for SMH with $a(t)$ given in Fig. $15-57$ if the position function $x(t)$ has the form $x=x_{m} \cos (\omega t+\phi)$ and $a_{s}=4.0 \mathrm{~m} / \mathrm{s}^{2} ?$

Averell Hause

Carnegie Mellon University

Problem 95

An engineer has an odd-shaped $10 \mathrm{~kg}$ object and needs to find its rotational inertia about an axis through its center of mass. The object is supported on a wire stretched along the desired axis. The wire has a torsion constant
$\kappa=0.50 \mathrm{~N} \cdot \mathrm{m}$. If this torsion pendulum oscillates through 20 cycles in $50 \mathrm{~s}$. what is the rotational inertia of the object?

Joanna Josey

Numerade Educator

Problem 96

A spider can tell when its web has captured, say, a fly because the fly's thrashing causes the web threads to oscillate. A spider can even determine the size of the fly by the frequency of
the oscillations. Assume that a fly oscillates on the capture thread on which it is caught like a block on a spring. What is the ratio of oscillation frequency for a fly with mass $m$ to a fly with mass $2.5 m$ ?

Averell Hause

Carnegie Mellon University

Problem 97

A torsion pendulum }\end{array}$
consists of a metal disk with a wire running through its center and soldered in place. The wire is mounted vertically on clamps and pulled taut. Figure $15-58 a$ gives the magnitude $\tau$ of the torque
needed to rotate the disk about its center (and thus twist the wire) versus the rotation angle $\theta .$ The vertical axis scale is set by $\tau_{\mathrm{s}}=4.0 \times 10^{-3} \mathrm{~N} \cdot \mathrm{m} .$ The disk is rotated to $\theta=0.200 \mathrm{rad}$ and then
released. Figure $15-58 b$ shows the resulting oscillation in terms of angular position $\theta$ versus time $t$. The horizontal axis scale is set by $t_{s}=0.40 \mathrm{~s} .$ (a) What is the rotational inertia of the disk about its center? (b) What is the maximum angular speed $d \theta / d t$ of the disk? (Caution: Do not confuse the (constant) angular frequency of the SHM with the (varying) angular speed of the rotating disk, even though they usually have the same symbol $\omega .$ Hint: The potential energy $U$ of a torsion pendulum is equal to $\frac{1}{2} \kappa \theta^{2}$, analogous to $U=\frac{1}{2} k x^{2}$ for a spring.

Supratim Pal

Numerade Educator

Problem 98

When a $20 \mathrm{~N}$ can is hung from the bottom of a vertical spring, it causes the spring to stretch $20 \mathrm{~cm} .$ (a) What is the spring constant? (b) This spring is now placed horizontally on a frictionless table. One end of it is held fixed, and the other end is attached to a $5.0 \mathrm{~N}$ can. The can is then moved (stretching the spring) and released from rest. What is the period of the resulting oscillation?

Averell Hause

Carnegie Mellon University

Problem 99

For a simple pendulum, find the angular amplitude $\theta_{m}$ at which the restoring torque required for simple harmonic motion deviates from the actual restoring torque by $1.0 \%$. (See "Trigonometric Expansions" in Appendix E.)

Joanna Josey

Numerade Educator

Problem 100

In Fig. $15-59$, a solid cylinder attached to a horizontal spring $(k=$ $3.00 \mathrm{~N} / \mathrm{m}$ ) rolls without slipping along a horizontal surface. If the system is released from rest when the spring is stretched by $0.250 \mathrm{~m}$, find
(a) the translational kinetic energy
and (b) the rotational kinetic energy of the cylinder as it passes through the equilibrium position. (c) Show that under these conditions the cylinder's center of mass executes simple harmonic motion with period
$$
T=2 \pi \sqrt{\frac{3 M}{2 k}}
$$
where $M$ is the cylinder mass. (Hint: Find the time derivative of the total mechanical energy.)

Averell Hause

Carnegie Mellon University

Problem 101

A $1.2 \mathrm{~kg}$ block sliding on a horizontal frictionless surface is attached to a horizontal spring with $k=480 \mathrm{~N} / \mathrm{m}$. Let $x$ be the displacement of the block from the position at which the spring is unstretched. At $t=0$ the block passes through $x=0$ with a speed of $5.2 \mathrm{~m} / \mathrm{s}$ in the positive $x$ direction. What are the (a) frequency and (b) amplitude of the block's motion? (c) Write an expression for $x$ as a function of time.

Joanna Josey

Numerade Educator

Problem 102

A simple harmonic oscillator consists of an $0.80 \mathrm{~kg}$ block attached to a spring $(k=200 \mathrm{~N} / \mathrm{m}) .$ The block slides on a horizontal frictionless surface about the equilibrium point $x=0$ with a total mechanical energy of $4.0 \mathrm{~J} .$ (a) What is the amplitude of the oscillation? (b) How many oscillations does the block complete in $10 \mathrm{~s}$ ?
(c) What is the maximum kinetic energy attained by the block? (d) What is the speed of the block at $x=0.15 \mathrm{~m} ?$

Samuel Smith

Numerade Educator

Problem 103

A block sliding on a horizontal frictionless surface is attached to a horizontal spring with a spring constant of $600 \mathrm{~N} / \mathrm{m}$. The block executes SHM about its equilibrium position with a period of $0.40 \mathrm{~s}$ and an amplitude of $0.20 \mathrm{~m}$. As the block slides through its equilibrium position, a $0.50 \mathrm{~kg}$ putty wad is dropped
vertically onto the block. If the putty wad sticks to the block, determine (a) the new period of the motion and (b) the new amplitude of the motion.

Joanna Josey

Numerade Educator

Problem 104

A damped harmonic oscillator consists of a block $(m=$ $2.00 \mathrm{~kg}$ ), a spring $(k=10.0 \mathrm{~N} / \mathrm{m})$, and a damping force $(F=-b v)$. Initially, it oscillates with an amplitude of $25.0 \mathrm{~cm}$; because of the damping, the amplitude falls to three-fourths of this initial value at the completion of four oscillations. (a) What is the value of $b ?$ (b) How much energy has been "lost" during these four oscillations?

Averell Hause

Carnegie Mellon University

Problem 105

A block weighing $10.0 \mathrm{~N}$ is attached to the lower end of a vertical spring $(k=200.0 \mathrm{~N} / \mathrm{m})$, the other end of which is attached to a ceiling. The block oscillates vertically and has a kinetic energy of $2.00 \mathrm{~J}$ as it passes through the point at which the spring is unstretched. (a) What is the period of the oscillation? (b) Use the law of conservation of energy to determine the maximum distance the block moves both above and below the point at which the spring is unstretched. (These are not necessarily the same.)
(c) What is the amplitude of the oscillation? (d) What is the maximum kinetic energy of the block as it oscillates?

Joanna Josey

Numerade Educator

Problem 106

A simple harmonic oscillator consists of a block attached to a spring with $k=200 \mathrm{~N} / \mathrm{m} .$ The block slides on a frictionless surface, with equilibrium point $x=0$ and amplitude $0.20 \mathrm{~m}$. A graph of the block's velocity $v$ as a function of time $t$ is shown in Fig. $15-60 .$ The horizontal scale is set by $t_{s}=0.20 \mathrm{~s}$. What are (a) the period of the SHM, (b) the block's mass, (c) its displacement at $t=0,(\mathrm{~d})$ its acceleration at $t=0.10 \mathrm{~s}$, and $(\mathrm{e})$ its maximum kinetic energy?

Averell Hause

Carnegie Mellon University

Problem 107

The vibration frequencies of atoms in solids at normal temperatures are of the order of $10^{13} \mathrm{~Hz}$. Imagine the atoms to be connected to one another by springs. Suppose that a single silver atom in a solid vibrates with this frequency and that all the other atoms are at rest. Compute the effective spring constant. One mole of silver $(6.02 \times$ $10^{23}$ atoms) has a mass of $108 \mathrm{~g}$.

Joanna Josey

Numerade Educator

Problem 108

Figure $15-61$ shows that if we hang a block on the end of a spring with spring constant $k$, the spring is stretched by distance $h=2.0 \mathrm{~cm}$. If we pull down on the block a short distance and then release it, it oscillates vertically with a certain frequency. What length must a simple pendulum have to swing with that frequency?

Averell Hause

Carnegie Mellon University

Problem 109

The physical pendulum in Fig. $15-62$ has two possible pivot points $A$ and $B$. Point $A$ has a fixed position but $B$ is adjustable along the length of the pendulum as indicated by the scaling. When suspended from $A$, the pendulum has a period of $T=1.80 \mathrm{~s}$ The pendulum is then suspended from $B$, which is moved until the pendulum again has that period. What is the distance $L$ between $A$ and $B ?$

Joanna Josey

Numerade Educator

Problem 110

A common device for entertaining a toddler is a jump seat that hangs from the horizontal portion of a doorframe via elastic cords (Fig. $15-63$ ). Assume that only one cord is on each side in spite of the more realistic arrangement shown. When a child is
placed in the seat, they both descend by a distance $d_{s}$ as the cords stretch (treat them as springs). Then the seat is pulled down an extra distance $d_{m}$ and released, so that the child oscillates vertically, like a block on the end of a spring. Suppose you are the safety engineer for the manufacturer of the seat. You do not want the magnitude of the child's acceleration to exceed $0.20 g$ for fear of hurting the child's neck. If $d_{m}=10 \mathrm{~cm}$, what value of $d_{s}$ corresponds to that acceleration magnitude?

Supratim Pal

Numerade Educator

Problem 111

A $2.0 \mathrm{~kg}$ block executes SHM while attached to a horizontal spring of spring constant $200 \mathrm{~N} / \mathrm{m}$. The maximum speed of the block as it slides on a horizontal frictionless surface is $3.0 \mathrm{~m} / \mathrm{s}$. What are (a) the amplitude of the block's motion, (b) the magnitude of its maximum acceleration, and (c) the magnitude of its minimum acceleration? (d) How long does the block take to complete $7.0 \mathrm{cy}$ cles of its motion?

Supratim Pal

Numerade Educator

Problem 112

In Fig. $15-64, a$ $2500 \mathrm{~kg}$ demolition ball swings from the end of a crane. The
length of the swinging segment of cable is $17 \mathrm{~m}$. (a) Find the period of the swinging, assuming that the system can be treated as a simple pendulum.
(b) Does the period depend on the ball's mass?

Supratim Pal

Numerade Educator

Problem 113

The center of oscillation of a physical $\quad$ pendulum
has this interesting property: If an impulse (assumed horizontal and in the plane of oscillation) acts at the center of oscillation, no oscillations are felt at the point of support. Baseball players (and players of many other sports) know that unless the ball hits the bat at this point (called the "sweet spot" by athletes), the oscillations due to the impact will sting their hands. To prove this property, let the stick in Fig. $15-13 a$ simulate a baseball bat. Suppose that a horizontal force $\vec{F}$ (due to impact with the ball) acts toward the right at $P$, the center of oscillation. The batter is assumed to hold the bat at $O$, the pivot point of the stick. (a) What acceleration does the point $O$ undergo as a result of $\vec{F} ?$ (b) What angular acceleration is produced by $\vec{F}$ about the center of mass of the stick? (c) As a result of the angular acceleration in (b), what linear acceleration does point $O$ undergo? (d) Considering the magnitudes and directions of the accelerations in (a) and (c), convince yourself that $P$ is indeed the "sweet spot."

Joanna Josey

Numerade Educator

Problem 114

A (hypothetical) large slingshot is stretched $2.30 \mathrm{~m}$ to launch a $170 \mathrm{~g}$ projectile with speed sufficient to escape from Earth (11.2 $\mathrm{km} / \mathrm{s}$ ). Assume the elastic bands of the slingshot obey Hooke's law. (a) What is the spring constant of the device if all the elastic potential energy is converted to kinetic energy? (b) Assume that an average person can exert a force of $490 \mathrm{~N}$. How many people are required to stretch the elastic bands?

Averell Hause

Carnegie Mellon University

Problem 115

What is the length of a simple pendulum whose full swing from left to right and then back again takes $3.2 \mathrm{~s}$ ?

Joanna Josey

Numerade Educator

Problem 116

A $2.0 \mathrm{~kg}$ block is attached to the end of a spring with a spring constant of $350 \mathrm{~N} / \mathrm{m}$ and forced to oscillate by an applied force $F=$ $(15 \mathrm{~N}) \sin \left(\omega_{d} t\right)$, where $\omega_{d}=35 \mathrm{rad} / \mathrm{s}$. The damping constant is $b=$
$15 \mathrm{~kg} / \mathrm{s}$. At $t=0$, the block is at rest with the spring at its rest length.
(a) Use numerical integration to plot the displacement of the block for the first $1.0 \mathrm{~s}$. Use the motion near the end of the $1.0 \mathrm{~s}$ interval to estimate the amplitude, period, and angular frequency. Repeat the calculation for (b) $\omega_{d}=\sqrt{k / m}$ and (c) $\omega_{d}=20 \mathrm{rad} / \mathrm{s}$

Averell Hause

Carnegie Mellon University