Physics for Scientist and Engineers: A Strategic Approach

Randall Knight

Chapter 14

Oscillations - all with Video Answers

Educators

Chapter Questions

Problem 1

When a guitar string plays the note "A," the string vibrates at $440 \mathrm{Hz}$ What is the period of the vibration?

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

An air-track glider attached to a spring oscillates between the $10 \mathrm{cm}$ mark and the $60 \mathrm{cm}$ mark on the track. The glider completes 10 oscillations in 33 s. What are the (a) period, (b) frequency, (c) angular frequency, (d) amplitude, and (e) maximum speed of the glider?

Averell Hause

Carnegie Mellon University

Problem 3

An air-track glider is attached to a spring. The glider is pulled to the right and released from rest at $t=0$ s. It then oscillates with a period of $2.0 \mathrm{s}$ and a maximum speed of $40 \mathrm{cm} / \mathrm{s}$
a. What is the amplitude of the oscillation?
b. What is the glider's position at $t=0.25 \mathrm{s} ?$

Averell Hause

Carnegie Mellon University

Problem 4

What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation .
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 5

What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation .
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 6

An object in simple harmonic motion has an amplitude of $4.0 \mathrm{cm},$ a frequency of $2.0 \mathrm{Hz}$, and a phase constant of $2 \pi / 3$ rad. Draw a position graph showing two cycles of the motion.

Averell Hause

Carnegie Mellon University

Problem 7

An object in simple harmonic motion has an amplitude of $8.0 \mathrm{cm},$ a frequency of $0.25 \mathrm{Hz},$ and a phase constant of $-\pi / 2$ rad. Draw a position graph showing two cycles of the motion.

Averell Hause

Carnegie Mellon University

Problem 8

An object in simple harmonic motion has amplitude 4.0 $\mathrm{cm}$ and frequency $4.0 \mathrm{Hz},$ and at $t=0 \mathrm{s}$ it passes through the equilibrium point moving to the right. Write the function $x(t)$ that describes the object's position.

Averell Hause

Carnegie Mellon University

Problem 9

An object in simple harmonic motion has amplitude 8.0 $\mathrm{cm}$ and frequency $0.50 \mathrm{Hz}$ At $t=0$ s it has its most negative velocity. Write the function $x(t)$ that describes the object's position.

Averell Hause

Carnegie Mellon University

Problem 10

An air-track glider attached to a spring oscillates with a period of 1.5 s. At $t=0$ s the glider is 5.00 cm left of the cquilibrium position and moving to the right at $36.3 \mathrm{cm} / \mathrm{s}$
a. What is the phase constant?
b. What is the phase at $t=0 \mathrm{s}, 0.5 \mathrm{s}, 1.0 \mathrm{s},$ and $1.5 \mathrm{s} ?$

Averell Hause

Carnegie Mellon University

Problem 11

A block attached to a spring with unknown spring constant oscillates with a period of $2.0 \mathrm{s}$. What is the period if
a. The mass is doubled?
b. The mass is halved?
c. The amplitude is doubled?
d. The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.

Averell Hause

Carnegie Mellon University

Problem 12

A $200 \mathrm{g}$ air-track glider is attached to a spring. The glider is pushed in $10 \mathrm{cm}$ and released. A student with a stopwatch finds that 10 oscillations take $12.0 \mathrm{s}$. What is the spring constant?

Averell Hause

Carnegie Mellon University

Problem 13

A $200 \mathrm{g}$ mass attached to a horizontal spring oscillates at a frequency of $2.0 \mathrm{Hz}$. At $t=0 \mathrm{s}$, the mass is at $x=5.0 \mathrm{cm}$ and has $v_{x}=-30 \mathrm{cm} / \mathrm{s} .$ Determine:
a. The period.
b. The angular frequency.
c. The amplitude.
d. The phase constant.
e. The maximum speed.
f. The maximum acceleration.
8. The total energy.
h. The position at $t=0.40 \mathrm{s}$

Averell Hause

Carnegie Mellon University

Problem 14

The position of a $50 \mathrm{g}$ oscillating mass is given by $x(t)=$ $(2.0 \mathrm{cm}) \cos (10 t-\pi / 4),$ where $t$ is in s. Determine:
a. The amplitude.
b. The period.
c. The spring constant.
d. The phase constant.
c. The initial conditions.
f. The maximum speed.
g. The total energy.
h. The velocity at $t=0.40 \mathrm{s}$

Averell Hause

Carnegie Mellon University

Problem 15

A $1.0 \mathrm{kg}$ block is attached to a spring with spring constant 16 N/m. While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of $40 \mathrm{cm} / \mathrm{s}$ What are
a. The amplitude of the subsequent oscillations?
b. The block's speed at the point where $x=\frac{1}{2} A ?$

Averell Hause

Carnegie Mellon University

Problem 16

A spring is hanging from the ceiling. Attaching a $500 \mathrm{g}$ physics book to the spring causes it to stretch $20 \mathrm{cm}$ in order to come to equilibrium.
a. What is the spring constant?
b. From equilibrium, the book is pulled down $10 \mathrm{cm}$ and released. What is the period of oscillation?
c. What is the book's maximum speed? At what position or positions does it have this speed?

Averell Hause

Carnegie Mellon University

Problem 17

A spring is hung from the ceiling. When a block is attached to its end, it stretches $2.0 \mathrm{cm}$ before reaching its new equilibrium length. The block is then pulled down slightly and released. What is the frequency of oscillation?

Averell Hause

Carnegie Mellon University

Problem 18

A spring with spring constant $15 \mathrm{N} / \mathrm{m}$ hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down $6.0 \mathrm{cm}$ and released. If the ball makes 30 oscillations in $20 \mathrm{s}$, what are its (a) mass and (b) maximum speed?

Averell Hause

Carnegie Mellon University

Problem 19

A mass on a string of unknown length oscillates as a pendulum with a period of 4.0 s. What is the period if
a. The mass is doubled?
b. The string length is doubled?
c. The string length is halved? Parts a to d are independent questions, each referring to the initial situation.

Averell Hause

Carnegie Mellon University

Problem 20

The angle of a pendulum is $\theta(t)=(0.10 \text { rad ) } \cos (5 t+\pi)$ where $t$ is in $\mathrm{s}$. Determine:
a. The amplitude.
b. The frequency.
c. The phase constant.
d. The length of the string.
e. The initial angle.
f. The angle at $t=2.0 \mathrm{s}$

Vishal Gupta

Numerade Educator

Problem 21

A 200 \mathrm{g} ball is tied to a string. It is pulled to an angle of $8.0^{\circ}$ and released to swing as a pendulum. A student with a stopwatch finds that 10 oscillations take 12 s. How long is the string?

Averell Hause

Carnegie Mellon University

Problem 22

What is the period of a 1.0 -m-long pendulum on (a) the earth and (b) Venus?

Averell Hause

Carnegie Mellon University

Problem 23

What is the length of a pendulum whose period on the moon matches the period of a 2.0 -m-long pendulum on the earth?

Averell Hause

Carnegie Mellon University

Problem 24

I Astronauts on the first trip to Mars take along a pendulum that has a period on earth of 1.50 s. The period on Mars turns out to be $2.45 \mathrm{s}$. What is the free-fall acceleration on Mars?

Averell Hause

Carnegie Mellon University

Problem 25

The 20 -cm-long wrench swings on its hook with a period of $0.90 \mathrm{s}$. When the wrench hangs from a spring of spring constant $360 \mathrm{N} / \mathrm{m},$ it stretches the spring $3.0 \mathrm{cm} .$ What is the wrench's moment of inertia about the hook?
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 26

A 2.0 \mathrm{g}$ spider is dangling at the end of a silk thread. You can make the spider bounce up and down on the thread by tapping lightly on his feet with a pencil. You soon discover that you can give the spider the largest amplitude on his little bungee cord if you tap exactly once every second. What is the spring constant of the silk thread?

Averell Hause

Carnegie Mellon University

Problem 27

The amplitude of an oscillator decreases to $36.8 \%$ of its initial value in $10.0 \mathrm{s}$. What is the value of the time constant?

Averell Hause

Carnegie Mellon University

Problem 28

Calculate and draw an accurate position graph from $t=0$ s to $t=10 \mathrm{s}$ of a damped oscillator having a frequency of $1.0 \mathrm{Hz}$ and a time constant of $4.0 \mathrm{s}$.

Averell Hause

Carnegie Mellon University

Problem 29

In a science museum, a $110 \mathrm{kg}$ brass pendulum bob swings at the end of a 15.0 -m-long wire. The pendulum is started at exactly 8: 00 A.M. every morning by pulling it 1.5 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulum's damping constant is only $0.010 \mathrm{kg} / \mathrm{s} .$ At exactly 12: 00 noon, how many oscillations will the pendulum have completed and what is its amplitude?

Averell Hause

Carnegie Mellon University

Problem 30

A spring with spring constant $15.0 \mathrm{N} / \mathrm{m}$ hangs from the ceiling. A $500 \mathrm{g}$ ball is attached to the spring and allowed to come to rest. It is then pulled down $6.0 \mathrm{cm}$ and released. What is the time constant if the ball's amplitude has decreased to $3.0 \mathrm{cm}$ after 30 oscillations?

Anand Jangid

Numerade Educator

Problem 31

Figure P14.31 is the position-versus-time graph of a particle in simple harmonic motion.
a. What is the phase constant?
b. What is the velocity at $t=0$ s?
c. What is $v_{\max } ?$
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 32

Figure P14.32 is the velocity-versus-time graph of a particle in simple harmonic motion.
a. What is the amplitude of the oscillation?
b. What is the phase constant?
c. What is the position at $t=0$ s?
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 33

The two graphs in Figure P14.33 are for two different vertical mass-spring systems.
a. What is the frequency of system A? What is the first time at which the mass has maximum speed while traveling in the upward direction?
b. What is the period of system B? What is the first time at which the energy is all potential?
c. If both systems have the same mass, what is the ratio $k_{\mathrm{A}} / k_{\mathrm{B}}$ of their spring constants?
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 34

An object in SHM oscillates with a period of 4.0 $s$ and an amplitude of $10 \mathrm{cm} .$ How long does the object take to move from $x=0.0 \mathrm{cm}$ to $x=6.0 \mathrm{cm} ?$

Averell Hause

Carnegie Mellon University

Problem 35

A $1.0 \mathrm{kg}$ block oscillates on a spring with spring constant 20 N/m. At $t=0$ s the block is $20 \mathrm{cm}$ to the right of the equilibrium position and moving to the left at a speed of $100 \mathrm{cm} / \mathrm{s}$ Determine the period of oscillation and draw a graph of position versus time.

Averell Hause

Carnegie Mellon University

Problem 36

Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The spring's length as a function of time is shown in Figure P14.36.
a. What is her mass if the spring constant is $240 \mathrm{N} / \mathrm{m} ?$
b. What is her speed when the spring's length is $1.2 \mathrm{m} ?$
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 37

The motion of a particle is given by $x(t)=(25 \mathrm{cm}) \cos (10 t)$ where $t$ is in $\mathrm{s}$. At what time is the kinetic energy twice the potential energy?

Averell Hause

Carnegie Mellon University

Problem 38

a. When the displacement of a mass on a spring is $\frac{1}{2} A,$ what fraction of the energy is kinetic energy and what fraction is potential energy?
b. At what displacement, as a fraction of $A$, is the energy half kinetic and half potential?

Averell Hause

Carnegie Mellon University

Problem 39

For a particle in simple harmonic motion, show that $v_{\max }=$ $(\pi / 2) v_{\mathrm{av}_{2}}$ where $v_{\mathrm{arg}}$ is the average speed during one cycle of the motion.

Averell Hause

Carnegie Mellon University

Problem 40

A $100 \mathrm{g}$ ball attached to a spring with spring constant $2.5 \mathrm{N} / \mathrm{m}$ oscillates horizontally on a frictionless table. Its velocity is
$20 \mathrm{cm} / \mathrm{s}$ when $x=-5.0 \mathrm{cm}$
a. What is the amplitude of oscillation?
b. What is the ball's maximum acceleration?
c. What is the ball's position when the acceleration is maximum?
d. What is the speed of the ball when $x=3.0 \mathrm{cm} ?$

Averell Hause

Carnegie Mellon University

Problem 41

A block on a spring is pulled to the right and released at $t=0$ s. It passes $x=3.00 \mathrm{cm}$ at $t=0.685 \mathrm{s},$ and it passes $x=-3.00 \mathrm{cm}$ at $t=0.886 \mathrm{s}$
a. What is the angular frequency?
b. What is the amplitude? Hint: $\cos (\pi-\theta)=-\cos \theta$
Hint: $\cos (\pi-\theta)=-\cos \theta$.

Supratim Pal

Numerade Educator

Problem 42

A 300 dollars g oscillator has a speed of $95.4 \mathrm{cm} / \mathrm{s}$ when its displacement is $3.0 \mathrm{cm}$ and $71.4 \mathrm{cm} / \mathrm{s}$ when its displacement is $6.0 \mathrm{cm} .$ What is the oscillator's maximum speed?

Averell Hause

Carnegie Mellon University

Problem 43

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk $(m=0.10 \mathrm{g})$ driven back and forth in SHM at $1.0 \mathrm{MHz}$ by an electromagnetic coil.
a. The maximum restoring force that can be applied to the disk without breaking it is $40,000 \mathrm{N}$. What is the maximum oscillation amplitude that won't rupture the disk?
b. What is the disk's maximum speed at this amplitude?

Averell Hause

Carnegie Mellon University

Problem 44

A $5.0 \mathrm{kg}$ block hangs from a spring with spring constant $2000 \mathrm{N} / \mathrm{m} .$ The block is pulled down $5.0 \mathrm{cm}$ from the equilibrium position and given an initial velocity of $1.0 \mathrm{m} / \mathrm{s}$ back toward equilibrium. What are the (a) frequency, (b) amplitude, and (c) total mechanical energy of the motion?

Averell Hause

Carnegie Mellon University

Problem 45

The prongs of a tuning fork each vibrate with an amplitude of
$-0.50 \mathrm{mm}$ at the tuning fork's frequency of $440 \mathrm{Hz}$
a. What is the maximum speed of the tip of one prong?
b. A $10 \mu \mathrm{g}$ flea was sitting on the tip of the prong when the tuning fork was sounded. Surface tension allows a flea's feet to hold onto a smooth surface with a force of up to $1.0 \mathrm{mN}$. Will the flea be able to hold onto the vibrating prong, or will it be thrown off?

Averell Hause

Carnegie Mellon University

Problem 46

A $200 \mathrm{g}$ block hangs from a spring with spring constant $10 \mathrm{N} / \mathrm{m}$ At $t=0$ s the block is $20 \mathrm{cm}$ below the equilibrium point and moving upward with a speed of $100 \mathrm{cm} / \mathrm{s}$. What are the block's
a. Oscillation frequency?
b. Distance from cquilibrium when the speed is $50 \mathrm{cm} / \mathrm{s} ?$
c. Position at $t=1.0 \mathrm{s} ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 47

A spring with spring constant $k$ is suspended vertically from a support and a mass $m$ is attached. The mass is held at the point where the spring is not stretched. Then the mass is released and begins to oscillate. The lowest point in the oscillation is $20 \mathrm{cm}$ below the point where the mass was released. What is the oscillation frequency?

Averell Hause

Carnegie Mellon University

Problem 48

While grocery shopping, you put several apples in the spring scale in the produce department. The scale reads $20 \mathrm{N},$ and you use your ruler (which you always carry with you) to discover that the pan goes down $9.0 \mathrm{cm}$ when the apples are added. If you tap the bottom of the apple-filled pan to make it bounce up and down a little, what is its oscillation frequency? Ignore the mass of the pan.

Averell Hause

Carnegie Mellon University

Problem 49

A compact car has a mass of 1200 kg. Assume that the car has one spring on each wheel, that the springs are identical, and that the mass is equally distributed over the four springs.
a. What is the spring constant of each spring if the empty car bounces up and down 2.0 times each second?
b. What will be the car's oscillation frequency while carrying four 70 kg passengers?

Averell Hause

Carnegie Mellon University

Problem 50

A 500 g block slides along a frictionless surface at a speed of $0.35 \mathrm{m} / \mathrm{s} .$ It runs into a horizontal massless spring with spring constant $50 \mathrm{N} / \mathrm{m}$ that extends outward from a wall. It compresses the spring, then is pushed back in the opposite direction by the spring, eventually losing contact with the spring.
a. How long does the block remain in contact with the spring?
b. How would your answer to part a change if the block's initial speed were doubled?

Averell Hause

Carnegie Mellon University

Problem 51

Figure $P14.51$ shows a $1.0 \mathrm{kg}$ mass riding on top of a $5.0 \mathrm{kg}$ mass as it oscillates on a frictionless surface. The spring constant is $50 \mathrm{N} / \mathrm{m}$ and the coefficient of static friction between the two blocks is $0.50 .$ What is the maximum oscillation amplitude for which the upper block does not slip?
(FIGURE CANT COPY)

Keshav Singh

Numerade Educator

Problem 52

The two blocks in Figure P14.51 oscillate on a frictionless surface with a period of $1.5 \mathrm{s}$. The upper block just begins to slip when the amplitude is increased to $40 \mathrm{cm} .$ What is the coefficient of static friction between the two blocks?

Averell Hause

Carnegie Mellon University

Problem 53

It has recently become possible to "weigh" DNA molecules by measuring the influence of their mass on a nano-oscillator. Shows a thin rectangular cantilever etched out of silicon (density $2300 \mathrm{kg} / \mathrm{m}^{3}$ ) with a small gold dot at the end. If pulled down and released, the end of the cantilever vibrates with simple harmonic motion, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot. The addition of their mass causes a very slight- -but measurable-decrease in the oscillation frequency.
(FIGURE CANT COPY).
A vibrating cantilever of mass $M$ can be modeled as a block of mass $\frac{1}{3} M$ attached to a spring. (The factor of $\frac{1}{3}$ arises from the moment of inertia of a bar pivoted at one end.) Neither the mass nor the spring constant can be determined very accurately-perhaps to only two significant figures- -but the oscillation frequency can be measured with very high precision simply by counting the oscillations. In one experiment, the cantilever was initially vibrating at exactly 12 MHz. Attachment of a DNA molecule caused the frequency to decrease by $50 \mathrm{Hz}$. What was the mass of the DNA?

Brandy Heflin

Numerade Educator

Problem 54

It is said that Galileo discovered a basic principle of the pendulum- that the period is independent of the amplitude- by using his pulse to time the period of swinging lamps in the cathedral as they swayed in the breeze. Suppose that one oscillation of a swinging lamp takes $5.5 \mathrm{s}$
a. How long is the lamp chain?
b. What maximum speed does the lamp have if its maximum angle from vertical is $3.0^{\circ} ?$

Averell Hause

Carnegie Mellon University

Problem 55

A $100 \mathrm{g}$ mass on a 1.0 -m-long string is pulled $8.0^{\circ}$ to one side and released. How long does it take for the pendulum to reach $4.0^{\circ}$ on the opposite side?

Averell Hause

Carnegie Mellon University

Problem 56

The earth's free-fall acceleration varies from $9.78 \mathrm{m} / \mathrm{s}^{2}$ at the $64 .$ equator to $9.83 \mathrm{m} / \mathrm{s}^{2}$ at the poles, both because the earth is rotating and it's not a perfect sphere. A pendulum whose length is precisely $1.000 \mathrm{m}$ can be used to measure $g .$ Such a device is called a gravimeter
a. How long do 100 oscillations take at the equator?
b. How long do 100 oscillations take at the north pole?
c. Is the difference between your answers to parts a and b measurable? What kind of instrument could you use to measure the difference?
d. Suppose you take your gravimeter to the top of a high mountain peak near the equator. There you find that 100 oscillations take $201.0 \mathrm{s}$. What is $g$ on the mountain top?

Averell Hause

Carnegie Mellon University

Problem 57

Show that Equation 14.52 for the angular frequency of a physical pendulum gives Equation 14.49 when applied to a simple pendulum of a mass on a string.

Averell Hause

Carnegie Mellon University

Problem 58

A $15-$ cm-long, $200 \mathrm{g}$ rod is pivoted at one end. A $20 \mathrm{g}$ ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?

Averell Hause

Carnegie Mellon University

Problem 59

A circular hoop of mass $M$ and radius $R$ is pivoted on an axle passing through one edge, as shown in figure P14.59. Find an expression for the frequency of small oscillations.

Averell Hause

Carnegie Mellon University

Problem 60

A $250 \mathrm{g}$ air-track glider is attached to a spring with spring constant $4.0 \mathrm{N} / \mathrm{m}$. The damping constant due to air resistance is $0.015 \mathrm{kg} / \mathrm{s} .$ The glider is pulled out $20 \mathrm{cm}$ from equilibrium and released. How many oscillations will it make during the time in which the amplitude decays to $e^{-1}$ of its initial value?

Averell Hause

Carnegie Mellon University

Problem 61

A $500 \mathrm{g}$ air-track glider attached to a spring with spring constant $10 \mathrm{N} / \mathrm{m}$ is sitting at rest on a frictionless air track. $\mathrm{A} 250 \mathrm{g}$ glider is pushed toward it from the far end of the track at a speed of $120 \mathrm{cm} / \mathrm{s}$. It collides with and sticks to the $500 \mathrm{g}$ glider. What are the amplitude and period of the subsequent oscillations?

Averell Hause

Carnegie Mellon University

Problem 62

A $200 \mathrm{g}$ block attached to a horizontal spring is oscillating with an amplitude of $2.0 \mathrm{cm}$ and a frequency of $2.0 \mathrm{Hz}$. Just as it passes through the equilibrium point, moving to the right, a sharp blow directed to the left exerts a $20 \mathrm{N}$ force for $1.0 \mathrm{ms}$. What are the new (a) frequency and (b) amplitude?

Averell Hause

Carnegie Mellon University

Problem 63

A pendulum consists of a massless, rigid rod with a mass at one end. The other end is pivoted on a frictionless pivot so that it can turn through a complete circle. The pendulum is inverted, so the mass is directly above the pivot point, then released. The speed of the mass as it passes through the lowest point is $5.0 \mathrm{m} / \mathrm{s} .$ If the pendulum undergoes small-amplitude oscillations at the bottom of the arc, what will the frequency be?

Averell Hause

Carnegie Mellon University

Problem 64

Figure P14.64 is a top view of an object of mass $m$ connected between two stretched rubber bands of length $L$. The object rests on a frictionless surface. At equilibrium, the tension in each rubber band is $T .$ Find an expression for the frequency of oscillations perpendicular to the rubber bands. Assume the amplitude is sufficiently small that the magnitude of the tension in the rubber bands is essentially unchanged as the mass oscillates.
(FIGURE CANT COPY)

Samuel Smith

Numerade Educator

Problem 65

A molecular bond can be modeled as a spring between two atoms that vibrate with simple harmonic motion. Shows an SHM approximation for the potential energy of an HCl molecule. For $E<4 \times 10^{-19} \mathrm{J}$ it is a good approximation to the more accurate HCl potential-energy curve that was shown in Figure $10.37 .$ Because the chlorine atom is so much more massive than the hydrogen atom, it is reasonable to assume that the hydrogen atom $\left(m=1.67 \times 10^{-27} \mathrm{kg}\right)$ vibrates back and forth while the chlorine atom remains at rest. Use the graph to estimate the vibrational frequency of the HC1 molecule.
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 66

An ice cube can slide around the inside of a vertical circular hoop of radius $R$. It undergoes small-amplitude oscillations if displaced slightly from the equilibrium position at the lowest point. Find an expression for the period of these small-amplitude oscillations.

Averell Hause

Carnegie Mellon University

Problem 67

A penny rides on top of a piston as it undergoes vertical simple harmonic motion with an amplitude of $4.0 \mathrm{cm} .$ If the frequency is low, the penny rides up and down without difficulty. If the frequency is steadily increased, there comes a point at which the penny leaves the surface.
a. At what point in the cycle does the penny first lose contact with the piston?
b. What is the maximum frequency for which the penny just barely remains in place for the full cycle?

Averell Hause

Carnegie Mellon University

Problem 68

On your first trip to Planet X you happen to take along a $200 \mathrm{g}$ mass, a 40 -cm-long spring, a meter stick, and a stopwatch. You're curious about the free-fall acceleration on Planet X, where ordinary tasks seem easier than on earth, but you can't find this information in your Visitor's Guide. One night you suspend the spring from the ceiling in your room and hang the mass from it. You find that the mass stretches the spring by $31.2 \mathrm{cm} .$ You then pull the mass down $10.0 \mathrm{cm}$ and release it. With the stopwatch you find that 10 oscillations take 14.5 s. Can you now satisfy your curiosity?

Averell Hause

Carnegie Mellon University

Problem 69

The 15 g head of a bobble-head doll oscillates in $\mathrm{SHM}$ at a frequency of $4.0 \mathrm{Hz}$
a. What is the spring constant of the spring on which the head is mounted?
b. Suppose the head is pushed $2.0 \mathrm{cm}$ against the spring, then released. What is the head's maximum speed as it oscillates?
c. The amplitude of the head's oscillations decreases to $0.5 \mathrm{cm}$ in $4.0 \mathrm{s}$. What is the head's damping constant?

Averell Hause

Carnegie Mellon University

Problem 70

An oscillator with a mass of $500 \mathrm{g}$ and a period of 0.50 s has an amplitude that decreases by $2.0 \%$ during each complete oscillation.
a. If the initial amplitude is $10 \mathrm{cm},$ what will be the amplitude after 25 oscillations?
b. At what time will energy be reduced to $60 \%$ of its initial value?

Averell Hause

Carnegie Mellon University

Problem 71

A 200 g oscillator in a vacuum chamber has a frequency of 2.0 Hz. When air is admitted, the oscillation decreases to $60 \%$ of its initial amplitude in 50 s. How many oscillations will have been completed when the amplitude is $30 \%$ of its initial value?

Averell Hause

Carnegie Mellon University

Problem 72

Prove that the expression for $x(t)$ in Equation 14.56 is a solution to the equation of motion for a damped oscillator, Equation $14.55,$ if and only if the angular frequency $\omega$ is given by the expression in equation 14.57.

Averell Hause

Carnegie Mellon University

Problem 73

A block on a frictionless table is connected as shown in Figure P14.73 to two springs having spring constants $k_{1}$ and $k_{2}$ Show that the block's oscillation frequency is given by $$f=\sqrt{f_{1}^{2}+f_{2}^{2}}$$,
where $f_{1}$ and $f_{2}$ are the frequencies at which it would oscillate if attached to spring 1 or spring 2 alone.
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 74

A block on a frictionless table is connected as shown in Figure P14.74 to two springs having spring constants $k_{1}$ and $k_{2} .$ Find an expression for the block's oscillation frequency $f$ in terms of the frequencies $f_{1}$ and $f_{2}$ at which it would oscillate if attached to spring 1 or spring 2 alone.
(FIGURE CANT COPY)

Averell Hause

Carnegie Mellon University

Problem 75

A block hangs in equilibrium from a vertical spring. When a second identical block is added, the original block sags by $5.0 \mathrm{cm} .$ What is the oscillation frequency of the two-block system?

Averell Hause

Carnegie Mellon University

Problem 76

A $1.00 \mathrm{kg}$ block is attached to a horizontal spring with spring constant $2500 \mathrm{N} / \mathrm{m}$. The block is at rest on a friction less surface.
A 10 g bullet is fired into the block, in the face opposite the spring, and sticks.
a. What was the bullet's speed if the subsequent oscillations have an amplitude of $10.0 \mathrm{cm} ?$
b. Could you determine the bullet's speed by measuring the oscillation frequency? If so, how? If not, why not?

Averell Hause

Carnegie Mellon University

Problem 77

A spring is standing upright on a table with its bottom end fastened to the table. A block is dropped from a height $3.0 \mathrm{cm}$ above the top of the spring. The block sticks to the top end of the spring and then oscillates with an amplitude of $10 \mathrm{cm} .$ What is the oscillation frequency?

Averell Hause

Carnegie Mellon University

Problem 78

Jose, whose mass is $75 \mathrm{kg}$, has just completed his first bungee jump and is now bouncing up and down at the end of the cord. His oscillations have an initial amplitude of $11.0 \mathrm{m}$ and a period of $4.0 \mathrm{s}$
a. What is the spring constant of the bungee cord?
b. What is Jose's maximum speed while oscillating?
c. From what height above the lowest point did Jose jump?
d. If the damping constant due to air resistance is $6.0 \mathrm{kg} / \mathrm{s}$, how many oscillations will Jose make before his amplitude has decreased to $2.0 \mathrm{m} ?$ Hint: Although not entirely realistic, treat the bungee cord as an ideal spring that can be compressed to a shorter length as well as stretched to a longer length.

Averell Hause

Carnegie Mellon University

Problem 79

A 1000 kg car carrying two 100 kg football players travels over a bumpy "washboard" road with the bumps spaced 3.0 m apart. The driver finds that the car bounces up and down with maximum amplitude when he drives at a speed of $5.0 \mathrm{m} / \mathrm{s}$ $(\approx 11 \mathrm{mph}) .$ The car then stops and picks up three more $100 \mathrm{kg}$ passengers. By how much does the car body sag on its suspension when these three additional passengers get in?

Vishal Gupta

Numerade Educator

Problem 80

Figure CPI 4.80 shows a $200 \mathrm{g}$ uniform rod pivoted at one end. The other end is attached to a horizontal spring. The spring is neither stretched nor compressed when the rod hangs straight down. What is the rod's oscillation period? You can assume that the rod's angle from vertical is always small.

Averell Hause

Carnegie Mellon University