University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 14

Periodic Motion - all with Video Answers

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Chapter Questions

Problem 1

BIO (a) Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern that has the same frequency as the note that is sung. If someone sings the note $\mathrm{B}$ flat, which has a frequency of $466 \mathrm{~Hz}$, how much time does it take the person's vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords? (b) Hearing. When sound waves strike the eardrum, this membrane vibrates with the same frequency as the sound. The highest pitch that young humans can hear has a period of $50.0 \mu \mathrm{s}$. What are the frequency and angular frequency of the vibrating eardrum for this sound? (c) Vision. When light having vibrations with angular frequency ranging from $2.7 \times 10^{15} \mathrm{rad} / \mathrm{s}$ to $4.7 \times 10^{15} \mathrm{rad} / \mathrm{s}$ strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light?
(d) Ultrasound. High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as x rays do. To detect small objects such as tumors, a frequency of around $5.0 \mathrm{MHz}$ is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

Donya Dobbin

Donya Dobbin

Numerade Educator

Problem 2

If an object on a horizontal, frictionless surface is attached to a spring, displaced, and then released, it will oscillate. If it is displaced $0.120 \mathrm{~m}$ from its equilibrium position and released with zero initial speed, then after $0.800 \mathrm{~s}$ its displacement is found to be $0.120 \mathrm{~m}$ on the opposite side, and it has passed the equilibrium position once during this interval. Find (a) the amplitude; (b) the period; (c) the frequency.

Jilin Wang

Boston University

Problem 3

The tip of a tuning fork goes through 440 complete vibrations in $0.500 \mathrm{~s}$. Find the angular frequency and the period of the motion.

Derek Walkama

Numerade Educator

Problem 4

The displacement of an oscillating object as a function of time is shown in Fig. $\mathbf{E} \mathbf{1 4 . 4}$. What are (a) the frequency; (b) the amplitude; (c) the period; (d) the angular frequency of this motion?

Donya Dobbin

Numerade Educator

Problem 5

A machine part is undergoing SHM with a frequency of $4.00 \mathrm{~Hz}$ and amplitude $1.80 \mathrm{~cm} .$ How long does it take the part to go from $x=0$ to $x=-1.80 \mathrm{~cm} ?$

Donya Dobbin

Numerade Educator

Problem 6

You are pushing your nephew on a playground swing. The swing seat is suspended from a horizontal bar by two light chains. Based on your experience with swings, estimate the length of each chain. Treat the motion of the child as that of a simple pendulum and assume that for safety the amplitude of the motion is kept small. You give your nephew a light push each time he reaches his closest distance from you. How much time elapses between your pushes?

Donya Dobbin

Numerade Educator

Problem 7

A $2.40 \mathrm{~kg}$ ball is attached to an unknown spring and allowed to oscillate. Figure $\mathbf{E} \mathbf{1 4 . 7}$ shows a graph of the ball's position $x$ as a function of time $t$. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

Donya Dobbin

Numerade Educator

Problem 8

In a physics lab, you attach a $0.200 \mathrm{~kg}$ air-track glider to the end of an ideal spring of negligible mass and start it oscillating. The elapsed time from when the glider first moves through the equilibrium point to the second time it moves through that point is $2.60 \mathrm{~s}$. Find the spring's force constant.

Jilin Wang

Boston University

Problem 9

When an object of unknown mass is attached to an ideal spring with force constant $120 \mathrm{~N} / \mathrm{m},$ it is found to vibrate with a frequency of $6.00 \mathrm{~Hz}$. Find (a) the period of the motion; (b) the angular frequency; (c) the mass of the object.

Donya Dobbin

Numerade Educator

Problem 10

When a $0.750 \mathrm{~kg}$ mass oscillates on an ideal spring, the frequency is $1.75 \mathrm{~Hz}$. What will the frequency be if $0.220 \mathrm{~kg}$ are (a) added to the original mass and (b) subtracted from the original mass? Try to solve this problem without finding the force constant of the spring.

Jilin Wang

Boston University

Problem 11

An object is undergoing SHM with period $0.900 \mathrm{~s}$ and amplitude $0.320 \mathrm{~m} .$ At $t=0$ the object is at $x=0.320 \mathrm{~m}$ and is instantaneously at rest. Calculate the time it takes the object to go (a) from $x=0.320 \mathrm{~m}$ to $x=0.160 \mathrm{~m}$ and (b) from $x=0.160 \mathrm{~m}$ to $x=0$

Derek Walkama

Numerade Educator

Problem 12

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the block is at $x=0.280 \mathrm{~m},$ the acceleration of the block is $-5.30 \mathrm{~m} / \mathrm{s}^{2} .$ What is the

Donya Dobbin

Numerade Educator

Problem 13

A $2.00 \mathrm{~kg},$ frictionless block is attached to an ideal spring with force constant $300 \mathrm{~N} / \mathrm{m}$. At $t=0$ the spring is neither stretched nor compressed and the block is moving in the negative direction at $12.0 \mathrm{~m} / \mathrm{s} .$ Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

Repeat Exercise $14.13,$ but assume that at $t=0$ the block has velocity $-4.00 \mathrm{~m} / \mathrm{s}$ and displacement $+0.200 \mathrm{~m}$ away from equilibrium.

Donya Dobbin

Numerade Educator

Problem 15

A block of mass $m$ is undergoing SHM on a horizontal, frictionless surface while attached to a light, horizontal spring. The spring has force constant $k$, and the amplitude of the $\mathrm{SHM}$ is $A$. The block has $v=0,$ and $x=+A$ at $t=0 .$ It first reaches $x=0$ when $t=T / 4$ where $T$ is the period of the motion. (a) In terms of $T,$ what is the time $t$ when the block first reaches $x=A / 2 ?$ (b) The block has its maximum speed when $t=T / 4$. What is the value of $t$ when the speed of the block first reaches the value $v_{\max } / 2 ?$ (c) Does $v=v_{\max } / 2$ when $x=A / 2 ?$

Donya Dobbin

Numerade Educator

Problem 16

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is $0.090 \mathrm{~m},$ it takes the block $2.70 \mathrm{~s}$ to travel from $x=0.090 \mathrm{~m}$ to $x=-0.090 \mathrm{~m} .$ If the amplitude is doubled, to $0.180 \mathrm{~m},$ how long does it take the block to travel (a) from $x=0.180 \mathrm{~m}$ to $x=-0.180 \mathrm{~m}$ and (b) from $x=0.090 \mathrm{~m}$ to $x=-0.090 \mathrm{~m} ?$

Jilin Wang

Boston University

Problem 17

BIO Weighing Astronauts. This procedure has been used to "weigh" astronauts in space: A $42.5 \mathrm{~kg}$ chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes $1.30 \mathrm{~s}$ to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes $2.54 \mathrm{~s}$ for one cycle. What is the mass of the astronaut?

Donya Dobbin

Numerade Educator

Problem 18

A $0.400 \mathrm{~kg}$ object undergoing $\mathrm{SHM}$ has $a_{x}=-1.80 \mathrm{~m} / \mathrm{s}^{2}$ when $x=0.300 \mathrm{~m} .$ What is the time for one oscillation?

Jilin Wang

Boston University

Problem 19

On a frictionless, horizontal air track, a glider oscillates at the end of an ideal spring of force constant $2.50 \mathrm{~N} / \mathrm{cm} .$ The graph in Fig. $\mathrm{E} 14.19$ shows the acceleration of the glider as a function of time. Find (a) the mass of the glider; (b) the maximum displacement of the glider from the equilibrium point; (c) the maximum force the spring exerts on the glider.

Derek Walkama

Numerade Educator

Problem 20

A $0.500 \mathrm{~kg}$ mass on a spring has velocity as a function of time given by $v_{x}(t)=-(3.60 \mathrm{~cm} / \mathrm{s}) \sin [(4.71 \mathrm{rad} / \mathrm{s}) t-(\pi / 2)] .$ What are (a) the period; (b) the amplitude; (c) the maximum acceleration of the mass; (d) the force constant of the spring?

Jilin Wang

Boston University

Problem 21

A $1.50 \mathrm{~kg}$ mass on a spring has displacement as a function of time given by $$ x(t)=(7.40 \mathrm{~cm}) \cos [(4.16 \mathrm{rad} / \mathrm{s}) t-2.42] $$
Find (a) the time for one complete vibration; (b) the force constant of the spring; (c) the maximum speed of the mass; (d) the maximum force on the mass; (e) the position, speed, and acceleration of the mass at $t=1.00 \mathrm{~s} ;$ (f) the force on the mass at that time.

Derek Walkama

Numerade Educator

Problem 22

BIO Weighing a Virus. In February 2004, scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just $30 \mathrm{nm}$ long) with a laser, first without the virus and then after the virus had attached itself to the silicon. The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached $\left(f_{\mathrm{S}+\mathrm{V}}\right)$ to the frequency without the virus $\left(f_{\mathrm{S}}\right)$ is given by $f_{\mathrm{S}+\mathrm{V}} / f_{\mathrm{S}}=1 / \sqrt{1+\left(m_{\mathrm{V}} / m_{\mathrm{S}}\right)},$ where $m_{\mathrm{V}}$ is the mass of the
virus and $m_{\mathrm{S}}$ is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of $2.10 \times 10^{-16} \mathrm{~g}$ and a frequency of $2.00 \times 10^{15} \mathrm{~Hz}$ without the virus and $2.87 \times 10^{14} \mathrm{~Hz}$ with the virus. What is the mass of the virus, in grams and in femtograms?

Brian Steward

Numerade Educator

Problem 23

The jerk is defined to be the time rate of change of the acceleration. (a) If the velocity of an object undergoing SHM is given by $v_{x}=-\omega A \sin (\omega t),$ what is the equation for the $x$ -component of the jerk as a function of time? (b) What is the value of $x$ for the object when the $x$ -component of the jerk has its largest positive value? (c) What is $x$ when the $x$ -component of the jerk is most negative? (d) When it is zero? (e) If $v_{x}$ equals $-0.040 \mathrm{~s}^{2}$ times the $x$ -component of the jerk for all $t,$ what is the period of the motion?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

For the oscillating object in Fig. $\mathrm{E} 14.4,$ what are (a) its maximum speed and (b) its maximum acceleration?

Donya Dobbin

Numerade Educator

Problem 25

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. The amplitude of the motion is $0.165 \mathrm{~m}$. The maximum speed of the block is $3.90 \mathrm{~m} / \mathrm{s}$. What is the maximum magnitude of the acceleration of the block?

Derek Walkama

Numerade Educator

Problem 26

A small block is attached to an ideal spring and is moving in SHM on a horizontal, friction less surface. The amplitude of the motion is $0.250 \mathrm{~m}$ and the period is $3.20 \mathrm{~s}$. What are the speed and acceleration of the block when $x=0.160 \mathrm{~m} ?$

Donya Dobbin

Numerade Educator

Problem 27

A $0.150 \mathrm{~kg}$ toy is undergoing $\mathrm{SHM}$ on the end of a horizontal spring with force constant $k=300 \mathrm{~N} / \mathrm{m}$. When the toy is $0.0120 \mathrm{~m}$ from its equilibrium position, it is observed to have a speed of $0.400 \mathrm{~m} / \mathrm{s} .$ What are the toy's (a) total energy at any point of its motion;
(b) amplitude of motion; (c) maximum speed during its motion?

Derek Walkama

Numerade Educator

Problem 28

A harmonic oscillator has angular frequency $\omega$ and amplitude $A$. (a) What are the magnitudes of the displacement and velocity when the elastic potential energy is equal to the kinetic energy? (Assume that $U=0$ at equilibrium.) (b) How often does this occur in each cycle? What is the time between occurrences? (c) At an instant when the displacement is equal to $A / 2,$ what fraction of the total energy of the system is kinetic and what fraction is potential?

Dading Chen

Numerade Educator

Problem 29

A $0.500 \mathrm{~kg}$ glider, attached to the end of an ideal spring with force constant $k=450 \mathrm{~N} / \mathrm{m},$ undergoes $\mathrm{SHM}$ with an amplitude of $0.040 \mathrm{~m}$. Compute (a) the maximum speed of the glider; (b) the speed of the glider when it is at $x=-0.015 \mathrm{~m} ;$ (c) the magnitude of the maximum acceleration of the glider; (d) the acceleration of the glider at $x=-0.015 \mathrm{~m} ;$ (e) the total mechanical energy of the glider at any point in its motion.

Jacob Shpiece

Numerade Educator

Problem 30

A block of mass $m$ is undergoing SHM on a horizontal, friction less surface while attached to a light, horizontal spring. The spring has force constant $k,$ and the amplitude of the motion of the block is $A$. (a) The average speed is the total distance traveled by the block divided by the time it takes it to travel this distance. Calculate the average speed for one cycle of the SHM. (b) How does the average speed for one cycle compare to the maximum speed $v_{\max } ?$ (c) Is the average speed more or less than half the maximum speed? Based on your answer, does the block spend more time while traveling at speeds greater than $v_{\max } / 2$ or less than $v_{\max } / 2 ?$

Donya Dobbin

Numerade Educator

Problem 31

A block of mass $m$ is undergoing SHM on a horizontal, friction less surface while it is attached to a light, horizontal spring that has force constant $k .$ The amplitude of the $\mathrm{SHM}$ of the block is $A .$ What is the distance $|x|$ of the block from its equilibrium position when its speed $v$ is half its maximum speed $v_{\max } ?$ Is this distance larger or smaller than $A / 2 ?$

Donya Dobbin

Numerade Educator

Problem 32

A block with mass $m=0.300 \mathrm{~kg}$ is attached to one end of an ideal spring and moves on a horizontal friction less surface. The other end of the spring is attached to a wall. When the block is at $x=+0.240 \mathrm{~m},$ its acceleration is $a_{x}=-12.0 \mathrm{~m} / \mathrm{s}^{2}$ and its velocity is $v_{x}=+4.00 \mathrm{~m} / \mathrm{s} .$ What are (a) the spring's force constant $k ;$ (b) the amplitude of the motion; (c) the maximum speed of the block during its motion; and (d) the maximum magnitude of the block's acceleration during its motion?

Donya Dobbin

Numerade Educator

Problem 33

You are watching an object that is moving in SHM. When the object is displaced $0.600 \mathrm{~m}$ to the right of its equilibrium position, it has a velocity of $2.20 \mathrm{~m} / \mathrm{s}$ to the right and an acceleration of $8.40 \mathrm{~m} / \mathrm{s}^{2}$ to the left. How much farther from this point will the object move before it stops momentarily and then starts to move back to the left?

Ryan Hood

Numerade Educator

Problem 34

$\mathrm{A}$ mass is oscillating with amplitude $A$ at the end of a spring. How far (in terms of $A$ ) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?

Jilin Wang

Boston University

Problem 35

A $2.00 \mathrm{~kg}$ frictionless block attached to an ideal spring with force constant $315 \mathrm{~N} / \mathrm{m}$ is undergoing simple harmonic motion. When the block has displacement $+0.200 \mathrm{~m},$ it is moving in the negative $x$ direction with a speed of $4.00 \mathrm{~m} / \mathrm{s}$. Find (a) the amplitude of the motion;
(b) the block's maximum acceleration; and (c) the maximum force the spring exerts on the block.

Derek Walkama

Numerade Educator

Problem 36

A proud deep-sea fisherman hangs a $65.0 \mathrm{~kg}$ fish from an ideal spring having negligible mass. The fish stretches the spring $0.180 \mathrm{~m}$. (a) Find the force constant of the spring. The fish is now pulled down $5.00 \mathrm{~cm}$ and released. (b) What is the period of oscillation of the fish? (c) What is the maximum speed it will reach?

Jilin Wang

Boston University

Problem 37

A $175 \mathrm{~g}$ glider on a horizontal, frictionless air track is attached to a fixed ideal spring with force constant $155 \mathrm{~N} / \mathrm{m}$. At the instant you make measurements on the glider, it is moving at $0.815 \mathrm{~m} / \mathrm{s}$ and is $3.00 \mathrm{~cm}$ from its equilibrium point. Use energy conservation to find (a) the amplitude of the motion and (b) the maximum speed of the glider. (c) What is the angular frequency of the oscillations?

Derek Walkama

Numerade Educator

Problem 38

A uniform, solid metal disk of mass $6.50 \mathrm{~kg}$ and diameter $24.0 \mathrm{~cm}$ hangs in a horizontal plane, supported at its center by a vertical metal wire. You find that it requires a horizontal force of $4.23 \mathrm{~N}$ tangent to the rim of the disk to turn it by $3.34^{\circ},$ thus twisting the wire. You now remove this force and release the disk from rest. (a) What is the torsion constant for the metal wire? (b) What are the frequency and period of the torsional oscillations of the disk? (c) Write the equation of motion for $\theta(t)$ for the disk.

Jilin Wang

Boston University

Problem 39

A thrill-seeking cat with mass $4.00 \mathrm{~kg}$ is attached by a harness to an ideal spring of negligible mass and oscillates vertically in SHM. The amplitude is $0.050 \mathrm{~m},$ and at the highest point of the motion the spring has its natural unstretched length. Calculate the elastic potential energy of the spring (take it to be zero for the unstretched spring), the kinetic energy of the cat, the gravitational potential energy of the system relative to the lowest point of the motion, and the sum of these three energies when the cat is (a) at its highest point; (b) at its lowest point; (c) at its equilibrium position.

Derek Walkama

Numerade Educator

Problem 40

A thin metal disk with mass $2.00 \times 10^{-3} \mathrm{~kg}$ and radius $2.20 \mathrm{~cm}$ is attached at its center to a long fiber (Fig. $\mathbf{E 1 4 . 4 0}$ ). The disk, when twisted and released, oscillates with a period of $1.00 \mathrm{~s}$. Find the torsion constant of the fiber.

Donya Dobbin

Numerade Educator

Problem 41

A certain alarm clock ticks four times each second, with each tick representing half a period. The balance wheel consists of a thin rim with radius $0.55 \mathrm{~cm}$, connected to the balance shaft by thin spokes of negligible mass. The total mass of the balance wheel is $0.90 \mathrm{~g}$. (a) What is the moment of inertia of the balance wheel about its shaft? (b) What is the torsion constant of the coil spring (Fig. 14.19)?

Derek Walkama

Numerade Educator

Problem 42

You want to find the moment of inertia of a complicated machine part about an axis through its center of mass. You suspend it from a wire along this axis. The wire has a torsion constant of $0.450 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}$. You twist the part a small amount about this axis and let it go, timing 165 oscillations in $265 \mathrm{~s}$. What is its moment of inertia?

Donya Dobbin

Numerade Educator

Problem 43

You pull a simple pendulum $0.240 \mathrm{~m}$ long to the side through an angle of $3.50^{\circ}$ and release it. (a) How much time does it take the pendulum bob to reach its highest speed? (b) How much time does it take if the pendulum is released at an angle of $1.75^{\circ}$ instead of $3.50^{\circ} ?$

Derek Walkama

Numerade Educator

Problem 44

Equation (14.35) shows that the equation $T=2 \pi \sqrt{L / g}$ for the period of a simple pendulum is an approximation that is accurate only when the angular displacement $\Theta$ of the pendulum is small. For what value of $\Theta$ is $T=2 \pi \sqrt{L / g}$ in error by $2.0 \%$ ? In your calculation consider only the first correction term in Eq. (14.35).

Donya Dobbin

Numerade Educator

Problem 45

A building in San Francisco has light fixtures consisting of small $2.35 \mathrm{~kg}$ bulbs with shades hanging from the ceiling at the end of light, thin cords $1.50 \mathrm{~m}$ long. If a minor earthquake occurs, how many swings per second will these fixtures make?

Derek Walkama

Numerade Educator

Problem 46

A Pendulum on Mars. A certain simple pendulum has a period on the earth of 1.60 s. What is its period on the surface of Mars, where $g=3.71 \mathrm{~m} / \mathrm{s}^{2} ?$

Bret Rosen

Numerade Educator

Problem 47

After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length $50.0 \mathrm{~cm} .$ She finds that the pendulum makes 100 complete swings in 136 s. What is the value of $g$ on this planet?

Derek Walkama

Numerade Educator

Problem 48

In the laboratory, a student studies a pendulum by graphing the angle $\theta$ that the string makes with the vertical as a function of time $t$, obtaining the graph shown in Fig. $\mathbf{E 1 4 . 4 8 .}$ (a) What are the period, frequency, angular frequency, and amplitude of the pendulum's motion? (b) How long is the pendulum? (c) Is it possible to determine the mass of the bob?

Donya Dobbin

Numerade Educator

Problem 49

A small sphere with mass $m$ is attached to a massless rod of length $L$ that is pivoted at the top, forming a simple pendulum. The pendulum is pulled to one side so that the rod is at an angle $\theta$ from the vertical, and released from rest. (a) In a diagram, show the pendulum just after it is released. Draw vectors representing the forces acting on the small sphere and the acceleration of the sphere. Accuracy counts! At this point, what is the linear acceleration of the sphere? (b) Repeat part (a) for the instant when the pendulum rod is at an angle $\theta / 2$ from the vertical. (c) Repeat part (a) for the instant when the pendulum rod is vertical. At this point, what is the linear speed of the sphere?

Jilin Wang

Boston University

Problem 50

We want to hang a thin hoop on a horizontal nail and have the hoop make one complete small-angle oscillation each $2.0 \mathrm{~s}$. What must the hoop's radius be?

Jilin Wang

Boston University

Problem 51

Two pendulums have the same dimensions (length $L$ ) and total mass $(m) .$ Pendulum $A$ is a very small ball swinging at the end of a uniform massless bar. In pendulum $B$, half the mass is in the ball and half is in the uniform bar. Find the period of each pendulum for small oscillations. Which one takes longer for a swing?

DW

Duncan Wood

Numerade Educator

Problem 52

A $1.80 \mathrm{~kg}$ monkey wrench is pivoted $0.250 \mathrm{~m}$ from its center of mass and allowed to swing as a physical pendulum. The period for small-angle oscillations is $0.940 \mathrm{~s}$. (a) What is the moment of inertia of the wrench about an axis through the pivot? (b) If the wrench is initially displaced 0.400 rad from its equilibrium position, what is the angular speed of the wrench as it passes through the equilibrium position?

Jilin Wang

Boston University

Problem 53

The two pendulums shown in Fig. $\mathbf{E} \mathbf{1 4 . 5 3}$ each consist of a uniform solid ball of mass $M$ supported by a rigid massless rod, but the ball for pendulum $A$ is very tiny while the ball for pendulum $B$ is much larger. Find the period of each pendulum for small displacements. Which ball takes longer to complete a swing?

Donya Dobbin

Numerade Educator

Problem 54

A holiday ornament in the shape of a hollow sphere with mass $M=0.015 \mathrm{~kg}$ and radius $R=0.050 \mathrm{~m}$ is hung from a tree limb by a small loop of wire attached to the surface of the sphere. If the ornament is displaced a small distance and released, it swings back and forth as a physical pendulum with negligible friction. Calculate its period. (Hint: Use the parallel-axis theorem to find the moment of inertia of the sphere about the pivot at the tree limb.)

Jilin Wang

Boston University

Problem 55

An object is moving in damped SHM, and the damping constant can be varied. If the angular frequency of the motion is $\omega$ when the damping constant is zero, what is the angular frequency, expressed in terms of $\omega$, when the damping constant is one-half the critical damping value?

Supratim Pal

Numerade Educator

Problem 56

$\mathrm{A} 50.0 \mathrm{~g}$ hard-boiled egg moves on the end of a spring with force constant $k=25.0 \mathrm{~N} / \mathrm{m} .$ Its initial displacement is $0.300 \mathrm{~m} . \mathrm{A}$ damping force $F_{x}=-b v_{x}$ acts on the egg, and the amplitude of the motion decreases to $0.100 \mathrm{~m}$ in $5.00 \mathrm{~s}$. Calculate the magnitude of the damping constant $b$.

Brian Steward

Numerade Educator

Problem 57

An unhappy $0.300 \mathrm{~kg}$ rodent, moving on the end of a spring with force constant $k=2.50 \mathrm{~N} / \mathrm{m},$ is acted on by a damping force $F_{x}=-b v_{x}$. (a) If the constant $b$ has the value $0.900 \mathrm{~kg} / \mathrm{s},$ what is the frequency of oscillation of the rodent? (b) For what value of the constant $b$ will the motion be critically damped?

Vishal Gupta

Numerade Educator

Problem 58

A mass is vibrating at the end of a spring of force constant $225 \mathrm{~N} / \mathrm{m}$. Figure $\mathbf{E} \mathbf{1 4 . 5 8}$ shows a graph of its position $x$ as a function of time $t$. (a) At what times is the mass not moving? (b) How much energy did this system originally contain? (c) How much energy did the system lose between $t=1.0 \mathrm{~s}$ and $t=4.0 \mathrm{~s} ?$ Where did this energy go?

Bradley Abell

Numerade Educator

Problem 59

A sinusoidally varying driving force is applied to a damped harmonic oscillator of force constant $k$ and mass $m .$ If the damping constant has a value $b_{1},$ the amplitude is $A_{1}$ when the driving angular frequency equals $\sqrt{k / m}$. In terms of $A_{1}$, what is the amplitude for the same driving frequency and the same driving force amplitude $F_{\max },$ if the damping constant is (a) $3 b_{1}$ and (b) $b_{1} / 2 ?$

Derek Walkama

Numerade Educator

Problem 60

Equation (14.46) and Fig. 14.28 describe a damped and driven oscillator. (a) For a damping constant $b=0.20 \sqrt{\mathrm{km}}$, confirm that the amplitude $A$ is $5 F_{\max } / k$ when $\omega_{\mathrm{d}}=\omega,$ where $\omega=\sqrt{k / m}$ is the natural angular frequency. (b) Repeat part (a) for $b=0.40 \sqrt{\mathrm{km}}$, and confirm that the amplitude $A$ is $2.5 F_{\max } / k$ when $\omega_{\mathrm{d}}=\omega .$ (c) As a measure of the width of the resonance peak, calculate $A$ when $\omega_{\mathrm{d}}=\omega / 2$ for $b=0.20 \sqrt{\mathrm{km}}$ and for $b=0.40 \sqrt{k m}$. In each case, what is the ratio of the amplitude for $\omega_{\mathrm{d}}=\omega$ to the amplitude for $\omega_{\mathrm{d}}=\omega / 2 ?$ For which value of the damping constant does the amplitude increase by the larger factor?

Donya Dobbin

Numerade Educator

Problem 61

Object $A$ has mass $m_{A}$ and is in $\mathrm{SHM}$ on the end of a spring with force constant $k_{A} .$ Object $B$ has mass $m_{B}$ and is in $\mathrm{SHM}$ on the end of a spring with force constant $k_{B}$. The amplitude $A_{A}$ for object $A$ is twice the amplitude $A_{B}$ for the motion of object $B$. Also, $m_{B}=4 m_{A}$ and $k_{A}=9 k_{B}$. (a) What is the ratio of the maximum speeds of the two objects, $v_{\max , A} / v_{\max , B} ?$ (b) What is the ratio of their maximum accelerations, $a_{\max , A} / a_{\max , B} ?$

Donya Dobbin

Numerade Educator

Problem 62

An object is undergoing SHM with period $0.300 \mathrm{~s}$ and amplitude $6.00 \mathrm{~cm} .$ At $t=0$ the object is instantaneously at rest at $x=6.00 \mathrm{~cm}$. Calculate the time it takes the object to go from $x=6.00 \mathrm{~cm}$ to $x=-1.50 \mathrm{~cm}$

Jilin Wang

Boston University

Problem 63

An object is undergoing SHM with period $1.200 \mathrm{~s}$ and amplitude $0.600 \mathrm{~m} .$ At $t=0$ the object is at $x=0$ and is moving in the negative $x$ -direction. How far is the object from the equilibrium position when $t=0.480 \mathrm{~s} ?$

Derek Walkama

Numerade Educator

Problem 64

Four passengers with combined mass $250 \mathrm{~kg}$ compress the springs of a car with worn-out shock absorbers by $4.00 \mathrm{~cm}$ when they get in. Model the car and passengers as a single object on a single ideal spring. If the loaded car has a period of vibration of $1.92 \mathrm{~s}$, what is the period of vibration of the empty car?

Donya Dobbin

Numerade Educator

Problem 65

An object with mass $m$ is moving in SHM. It has amplitude $A_{1}$ and total mechanical energy $E_{1}$ when the spring has force constant $k_{1}$ You want to quadruple the total mechanical energy, so $E_{2}=4 E_{1}$, and halve the amplitude, so $A_{2}=A_{1} / 2,$ by using a different spring, one with force constant $k_{2}$. (a) How is $k_{2}$ related to $k_{1}$ ? (b) What effect will the change in spring constant and amplitude have on the maximum speed of the moving object?

Donya Dobbin

Numerade Educator

Problem 66

A block with mass $M$ rests on a friction less surface and is connected to a horizontal spring of force constant $k .$ The other end of the spring is attached to a wall (Fig. $\mathbf{P 1 4 . 6 6 )} .$ A second block with mass $m$ rests on top of the first block. The coefficient of static friction between the blocks is $\mu_{\mathrm{s}}$. Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

Donya Dobbin

Numerade Educator

Problem 67

$\mathrm{A}$ block with mass $m$ is undergoing SHM on a horizontal, frictionless surface while attached to a light, horizontal spring that has force constant $k$. You use motion sensor equipment to measure the maximum speed of the block during its oscillations. You repeat the measurement for the same spring and blocks of different masses while keeping the amplitude $A$ at a constant value of $12.0 \mathrm{~cm}$. You plot your data as $v_{\max }^{2}$ versus $1 / m$ and find that the data lie close to a straight line that has slope $8.62 \mathrm{~N} \cdot \mathrm{m} .$ What is the force constant $k$ of the spring?

Narayan Hari

Numerade Educator

Problem 68

Consider the system of two blocks and a spring shown in Fig. $\mathrm{P} 14.66 .$ The horizontal surface is friction less, but there is static friction between the two blocks. The spring has force constant $k=150 \mathrm{~N} / \mathrm{m} .$ The masses of the two blocks are $m=0.500 \mathrm{~kg}$ and $M=4.00 \mathrm{~kg} .$ You set the blocks into motion by releasing block $M$ with the spring stretched a distance $d$ from equilibrium. You start with small values of $d,$ and then repeat with successively larger values. For small values of $d,$ the blocks move together in SHM. But for larger values of $d$ the top block slips relative to the bottom block when the bottom block is released. (a) What is the period of the motion of the two blocks when $d$ is small enough to have no slipping? (b) The largest value $d$ can have and there be no slipping is $d=8.8 \mathrm{~cm} .$ What is the coefficient of static friction $\mu_{\mathrm{s}}$ between the surfaces of the two blocks?

Donya Dobbin

Numerade Educator

Problem 69

A $1.50 \mathrm{~kg},$ horizontal, uniform tray is attached to a vertical ideal spring of force constant $185 \mathrm{~N} / \mathrm{m}$ and a $275 \mathrm{~g}$ metal ball is in the tray. The spring is below the tray, so it can oscillate up and down. The tray is then pushed down to point $A,$ which is $15.0 \mathrm{~cm}$ below the equilibrium point, and released from rest. (a) How high above point $A$ will the tray be when the metal ball leaves the tray? (Hint: This does not occur when the ball and tray reach their maximum speeds.) (b) How much time elapses between releasing the system at point $A$ and the ball leaving the tray? (c) How fast is the ball moving just as it leaves the tray?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

A $10.0 \mathrm{~kg}$ mass is traveling to the right with a speed of $2.00 \mathrm{~m} / \mathrm{s}$ on a smooth horizontal surface when it collides with and sticks to a second $10.0 \mathrm{~kg}$ mass that is initially at rest but is attached to one end of a light, horizontal spring with force constant $170.0 \mathrm{~N} / \mathrm{m}$. The other end of the spring is fixed to a wall to the right of the second mass. (a) Find the frequency, amplitude, and period of the subsequent oscillations. (b) How long does it take the system to return the first time to the position it had immediately after the collision?

Donya Dobbin

Numerade Educator

Problem 71

An apple weighs $1.00 \mathrm{~N}$. When you hang it from the end of a long spring of force constant $1.50 \mathrm{~N} / \mathrm{m}$ and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back-and-forth swings do not cause any appreciable change in the length of the spring.) What is the unstretched length of the spring (with the apple removed)?

Derek Walkama

Numerade Educator

Problem 72

SHM of a Floating Object. An object with height $h$mass $M$, and a uniform cross-sectional area $A$ floats upright in a liquid with density $\rho$. (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force with magnitude $F$ is applied to the top of the object. At the new equilibrium position, how much farther below the surface of the liquid is the bottom of the object than it was in part (a)? (Assume that some of the object remains above the surface of the liquid.) (c) Your result in part (b) shows that if the force is suddenly removed, the object will oscillate up and down in SHM. Calculate the period of this motion in terms of the density $\rho$ of the liquid, the mass $M,$ and the cross-sectional area $A$ of the object. You can ignore the damping due to fluid friction (see Section 14.7).

Donya Dobbin

Numerade Educator

Problem 73

A square object of mass $m$ is constructed of four identi- cal uniform thin sticks, each of length $L,$ attached together. This object is hung on a hook at its upper corner (Fig. $\mathbf{P 1 4 . 7 3}$ ). If it is rotated slightly to the left and then released, at what frequency will it swing back and forth?

Inder Jeet

Numerade Educator

Problem 74

An object with mass $0.200 \mathrm{~kg}$ is acted on by an elastic re- storing force with force constant $10.0 \mathrm{~N} / \mathrm{m}$. (a) Graph elastic potential energy $U$ as a function of displacement $x$ over a range of $x$ from $-0.300 \mathrm{~m}$ to $+0.300 \mathrm{~m}$. On your graph, let $1 \mathrm{~cm}=0.05 \mathrm{~J}$ vertically and $1 \mathrm{~cm}=0.05 \mathrm{~m}$ horizontally. The object is set into oscillation with an initial potential energy of $0.140 \mathrm{~J}$ and an initial kinetic energy of $0.060 \mathrm{~J}$. Answer the following questions by referring to the graph. (b) What is the amplitude of oscillation? (c) What is the potential energy when the displacement is one-half the amplitude? (d) At what displacement are the kinetic and potential energies equal? (e) What is the value of the phase angle $\phi$ if the initial velocity is positive and the initial displacement is negative?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

A $2.00 \mathrm{~kg}$ bucket containing $10.0 \mathrm{~kg}$ of water is hanging from a vertical ideal spring of force constant $450 \mathrm{~N} / \mathrm{m}$ and oscillating up and down with an amplitude of $3.00 \mathrm{~cm}$. Suddenly the bucket springs a leak in the bottom such that water drops out at a steady rate of $2.00 \mathrm{~g} / \mathrm{s}$. When the bucket is half full, find (a) the period of oscillation and (b) the rate at which the period is changing with respect to time. Is the period getting longer or shorter? (c) What is the shortest period this system can have?

Derek Walkama

Numerade Educator

Problem 76

Quantum mechanics is used to describe the vibrational motion of molecules, but analysis using classical physics gives some useful insight. In a classical model the vibrational motion can be treated as SHM of the atoms connected by a spring. The two atoms in a diatomic molecule vibrate about their center of mass, but in the molecule HI, where one atom is much more massive than the other, we can treat the hydrogen atom as oscillating in SHM while the iodine atom remains at rest. (a) A classical estimate of the vibrational frequency is $f=7 \times 10^{13} \mathrm{~Hz}$. The mass of a hydrogen atom differs little from the mass of a proton. If the HI molecule is modeled as two atoms connected by a spring, what is the force constant of the spring? (b) The vibrational energy of the molecule is measured to be about $5 \times 10^{-20} \mathrm{~J}$. In the classical model, what is the maximum speed of the H atom during its SHM? (c) What is the amplitude of the vibrational motion? How does your result compare to the equilibrium distance between the two atoms in the HI molecule, which is about $1.6 \times 10^{-10} \mathrm{~m} ?$

Donya Dobbin

Numerade Educator

Problem 77

A $5.00 \mathrm{~kg}$ partridge is suspended from a pear tree by an ideal spring of negligible mass. When the partridge is pulled down $0.100 \mathrm{~m}$ below its equilibrium position and released, it vibrates with a period of $4.20 \mathrm{~s}$. (a) What is its speed as it passes through the equilibrium position? (b) What is its acceleration when it is $0.050 \mathrm{~m}$ above the equilibrium position? (c) When it is moving upward, how much time is required for it to move from a point $0.050 \mathrm{~m}$ below its equilibrium position to a point $0.050 \mathrm{~m}$ above it? (d) The motion of the partridge is stopped, and then it is removed from the spring. How much does the spring shorten?

Derek Walkama

Numerade Educator

Problem 78

A slender rod of length $80.0 \mathrm{~cm}$ and mass $0.400 \mathrm{~kg}$ has its center of gravity at its geometrical center. But its density is not uniform; it increases by the same amount from the center of the rod out to either end. You want to determine the moment of inertia $I_{\mathrm{cm}}$ of the rod for an axis perpendicular to the rod at its center, but you don't know its density as a function of distance along the rod, so you can't use an integration method to calculate $I_{\mathrm{cm}}$. Therefore, you make the following measurements: You suspend the rod about an axis that is a distance $d$ (measured in meters) above the center of the rod and measure the period $T$ (measured in seconds) for small-amplitude oscillations about the axis. You repeat this for several values of $d$. When you plot your data as $T^{2}-4 \pi^{2} d / g$ versus $1 / d$, the data lie close to a straight line that has slope $0.320 \mathrm{~m} \cdot \mathrm{s}^{2} .$ What is the value of $I_{\mathrm{cm}}$ for the rod?

David González Cornejo

David González Cornejo

Numerade Educator

Problem 79

SHM of a Butcher's Scale. A spring of negligible mass and force constant $k=400 \mathrm{~N} / \mathrm{m}$ is hung vertically, and a $0.200 \mathrm{~kg}$ pan is suspended from its lower end. A butcher drops a $2.2 \mathrm{~kg}$ steak onto the pan from a height of $0.40 \mathrm{~m} .$ The steak makes a totally inelastic collision with the pan and sets the system into vertical SHM. What are (a) the speed of the pan and steak immediately after the collision; (b) the amplitude of the subsequent motion; (c) the period of that motion?

Donya Dobbin

Numerade Educator

Problem 80

A $40.0 \mathrm{~N}$ force stretches a vertical spring $0.250 \mathrm{~m}$. (a) What mass must be suspended from the spring so that the system will oscillate with a period of $1.00 \mathrm{~s} ?$ (b) If the amplitude of the motion is $0.050 \mathrm{~m}$ and the period is that specified in part (a), where is the object and in what direction is it moving $0.35 \mathrm{~s}$ after it has passed the equilibrium position, moving downward? (c) What force (magnitude and direction) does the spring exert on the object when it is $0.030 \mathrm{~m}$ below the equilibrium position, moving upward?

Jilin Wang

Boston University

Problem 81

Don't Miss the Boat. While on a visit to Minnesota ("Land of 10,000 Lakes"), you sign up to take an excursion around one of the larger lakes. When you go to the dock where the $1500 \mathrm{~kg}$ boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude $20 \mathrm{~cm}$ The boat takes $3.5 \mathrm{~s}$ to make one complete up-and-down cycle. When the boat is at its highest point, its deck is at the same height as the stationary dock. As you watch the boat bob up and down, you (mass $60 \mathrm{~kg}$ ) begin to feel a bit woozy, due in part to the previous night's dinner of lutefisk. As a result, you refuse to board the boat unless the level of the boat's deck is within $10 \mathrm{~cm}$ of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion?

Jincy M Saji

Numerade Educator

Problem 82

An interesting, though highly impractical example of oscillation is the motion of an object dropped down a hole that extends from one side of the earth, through its center, to the other side. With the assumption (not realistic) that the earth is a sphere of uniform density, prove that the motion is simple harmonic and find the period. [Note: The gravitational force on the object as a function of the object's distance $r$ from the center of the earth was derived in Example 13.10 (Section 13.6 ). The motion is simple harmonic if the acceleration $a_{x}$ and the displacement from equilibrium $x$ are related by Eq. (14.8), and the period is then $T=2 \pi / \omega .]$

Jilin Wang

Boston University

Problem 83

A rifle bullet with mass $8.00 \mathrm{~g}$ and initial horizontal velocity $280 \mathrm{~m} / \mathrm{s}$ strikes and embeds itself in a block with mass $0.992 \mathrm{~kg}$ that rests on a friction less surface and is attached to one end of an ideal spring. The other end of the spring is attached to the wall. The impact compresses the spring a maximum distance of $15.0 \mathrm{~cm} .$ After the impact, the block moves in SHM. Calculate the period of this motion.

Donya Dobbin

Numerade Educator

Problem 84

Two uniform solid spheres, each with mass $M=0.800 \mathrm{~kg}$ and radius $R=0.0800 \mathrm{~m},$ are connected by a short, light rod that is along a diameter of each sphere and are at rest on a horizontal tabletop. A spring with force constant $k=160 \mathrm{~N} / \mathrm{m}$ has one end attached to the wall and the other end attached to a friction less ring that passes over the rod at the center of mass of the spheres, which is midway between the centers of the two spheres. The spheres are each pulled the same distance from the wall, stretching the spring, and released. There is sufficient friction between the tabletop and the spheres for the spheres to roll without slipping as they move back and forth on the end of the spring. Show that the motion of the center of mass of the spheres is simple harmonic and calculate the period.

Linda Winkler

Numerade Educator

Problem 85

In Fig. $\mathbf{P 1 4 . 8 5}$ the upper ball is released from rest, collides with the stationary lower ball, and sticks to it. The strings are both $50.0 \mathrm{~cm}$ long. The upper ball has mass $2.00 \mathrm{~kg},$ and it is initially $10.0 \mathrm{~cm}$ higher than the lower ball, which has mass $3.00 \mathrm{~kg}$. Find the frequency and maximum angular displacement of the motion after the collision.

Donya Dobbin

Numerade Educator

Problem 86

The Silently Ringing Bell. A large, $34.0 \mathrm{~kg}$ bell is hung from a wooden beam so it can swing back and forth with negligible friction. The bell's center of mass is $0.60 \mathrm{~m}$ below the pivot. The bell's moment of inertia about an axis at the pivot is $18.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$. The clapper is a small, $1.8 \mathrm{~kg}$ mass attached to one end of a slender rod of length $L$ and negligible mass. The other end of the rod is attached to the inside of the bell; the rod can swing freely about the same axis as the bell. What should be the length $L$ of the clapper rod for the bell to ring silently - that is, for the period of oscillation for the bell to equal that of the clapper?

Donya Dobbin

Numerade Educator

Problem 87

A slender, uniform, metal rod with mass $M$ is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant $k$ is attached to the lower end of the rod, with the other end of the spring attached to a rigid support. If the rod is displaced by a small angle $\Theta$ from the vertical (Fig. $\mathbf{P 1 4 . 8 7}$ ) and released, show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle $\Theta$ is small enough for the ap-

Linda Winkler

Numerade Educator

Problem 88

Two identical thin rods, each with mass $m$ and length $L$, are joined at right angles to form an L-shaped object. This object is balanced on top of a sharp edge (Fig. $\mathbf{P 1 4 . 8 8}$ ). If the L-shaped object is deflected slightly, it oscillates. Find the frequency of oscillation.

Donya Dobbin

Numerade Educator

Problem 89

DATA A mass $m$ is attached to a spring of force constant $75 \mathrm{~N} / \mathrm{m}$ and allowed to oscillate. Figure $\mathrm{P} 14.89$ shows a graph of its velocity component $v_{x}$ as a function of time $t$. Find (a) the period, (b) the frequency, and (c) the angular frequency of this motion. (d) What is the amplitude (in $\mathrm{cm}$ ), and at what times does the mass reach this position? (e) Find the maximum acceleration magnitude of the mass and the times at which it occurs. (f) What is the value of $m ?$

Donya Dobbin

Numerade Educator

Problem 90

DATA You hang various masses $m$ from the end of a vertical, $0.250 \mathrm{~kg}$ spring that obeys Hooke's law and is tapered, which means the diameter changes along the length of the spring. since the mass of the spring is not negligible, you must replace $m$ in the equation $T=2 \pi \sqrt{m / k}$ with $m+m_{\text {eff }},$ where $m_{\text {eff }}$ is the effective mass of the oscillating spring. (See Challenge Problem 14.93.) You vary the mass $m$ and measure the time for 10 complete oscillations, obtaining these data:
$$
\begin{array}{l|lcccc}
\boldsymbol{m}(\mathbf{k g}) & 0.100 & 0.200 & 0.300 & 0.400 & 0.500 \\
\hline \text { Time (s) } & 8.7 & 10.5 & 12.2 & 13.9 & 15.1
\end{array}
$$
(a) Graph the square of the period $T$ versus the mass suspended from the spring, and find the straight line of best fit. (b) From the slope of that line, determine the force constant of the spring. (c) From the vertical intercept of the line, determine the spring's effective mass. (d) What fraction is $m_{\text {eff }}$ of the spring's mass? (e) If a $0.450 \mathrm{~kg}$ mass oscillates on the end of the spring, find its period, frequency, and angular frequency.

Donya Dobbin

Numerade Educator

Problem 91

DATA Experimenting with pendulums, you attach a light string to the ceiling and attach a small metal sphere to the lower end of the string. When you displace the sphere $2.00 \mathrm{~m}$ to the left, it nearly touches a vertical wall; with the string taut, you release the sphere from rest. The sphere swings back and forth as a simple pendulum, and you measure its period $T$. You repeat this act for strings of various lengths $L$, each time starting the motion with the sphere displaced $2.00 \mathrm{~m}$ to the left of the vertical position of the string. In each case the sphere's radius is very small compared with $L$. Your results are given in the table:
$$
\begin{array}{l|rrrrrrrr}
\boldsymbol{L}(\mathbf{m}) & 12.00 & 10.00 & 8.00 & 6.00 & 5.00 & 4.00 & 3.00 & 2.50 & 2.30 \\
\hline \boldsymbol{T}(\mathbf{s}) & 6.96 & 6.36 & 5.70 & 4.95 & 4.54 & 4.08 & 3.60 & 3.35 & 3.27
\end{array}
$$
(a) For the five largest values of $L,$ graph $T^{2}$ versus $L$. Explain why the data points fall close to a straight line. Does the slope of this line have the value you expected? (b) Add the remaining data to your graph. Explain why the data start to deviate from the straight-line fit as $L$ decreases. To see this effect more clearly, plot $T / T_{0}$ versus $L,$ where $T_{0}=2 \pi \sqrt{L / g}$ and $g=9.80 \mathrm{~m} / \mathrm{s}^{2} .$ (c) Use your graph of $T / T_{0}$ versus $L$ to estimate the angular amplitude of the pendulum (in degrees) for which the equation $T=2 \pi \sqrt{L / g}$ is in error by $5 \%$.

Linda Winkler

Numerade Educator

Problem 92

The Effective Force Constant of Two Springs. Two springs with the same unstretched length but different force constants $k_{1}$ and $k_{2}$ are attached to a block with mass $m$ on a level, frictionless surface. Calculate the effective force constant $k_{\text {eff }}$ in each of the three cases (a), (b), and (c) depicted in Fig. P14.92. (The effective force constant is defined by $\Sigma F_{x}=-k_{\text {eff }} x$.) (d) An object with mass $m$, suspended from a uniform spring with a force constant $k,$ vibrates with a frequency $f_{1}$. When the spring is cut in half and the same object is suspended from one of the halves, the frequency is $f_{2}$. What is the ratio $f_{1} / f_{2} ?$

Linda Winkler

Numerade Educator

Problem 93

A Spring with Mass. The preceding problems in this chapter have assumed that the springs had negligible mass. But of course no spring is completely massless. To find the effect of the spring's mass, consider a spring with mass $M,$ equilibrium length $L_{0},$ and spring constant $k$. When stretched or compressed to a length $L,$ the potential energy is $\frac{1}{2} k x^{2},$ where $x=L-L_{0}$. (a) Consider a spring, as described above, that has one end fixed and the other end moving with speed $v$. Assume that the speed of points along the length of the spring varies linearly with distance $l$ from the fixed end. Assume also that the mass $M$ of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of $M$ and $v .$ (Hint: Divide the spring into pieces of length $d l ;$ find the speed of each piece in terms of $l, v,$ and $L ;$ find the mass of each piece in terms of $d l, M,$ and $L ;$ and integrate from 0 to $L .$ The result is $n o t \frac{1}{2} M v^{2},$ since not all of the spring moves with the same speed.) (b) Take the time derivative of the conservation of energy equation, Eq. (14.21), for a mass $m$ moving on the end of a massless spring. By comparing your results to Eq. (14.8), which defines $\omega$, show that the angular frequency of oscillation is $\omega=\sqrt{k / m}$. (c) Apply the procedure of part (b) to obtain the angular frequency of oscillation $\omega$ of the spring considered in part (a). If the effective mass $M^{\prime}$ of the spring is defined by $\omega=\sqrt{k / M^{\prime}},$ what is $M^{\prime}$ in terms of $M ?$

Linda Winkler

Numerade Educator

Problem 94

BIO "Seeing" Surfaces at the Nanoscale. One technique for making images of surfaces at the nanometer scale, including membranes and biomolecules, is dynamic atomic force microscopy. In this technique, a small tip is attached to a cantilever, which is a flexible, rectangular slab supported at one end, like a diving board. The cantilever vibrates, so the tip moves up and down in simple harmonic motion. In one operating mode, the resonant frequency for a cantilever with force constant $k=1000 \mathrm{~N} / \mathrm{m}$ is $100 \mathrm{kHz}$. As the oscillating tip is brought within a few nanometers of the surface of a sample (as shown in the figure), it experiences an attractive force from the surface. For an oscillation with a small amplitude (typically, $0.050 \mathrm{nm}),$ the force $F$ that the sample surface exerts on the tip varies linearly with the displacement $x$ of the tip, $|F|=k_{\text {surf }} x,$ where $k_{\text {surf }}$ is the effective force constant for this force. The net force on the tip is therefore $\left(k+k_{\text {surf }}\right) x$, and the frequency of the oscillation changes slightly due to the interaction with the surface. Measurements of the frequency as the tip moves over different parts of the sample's surface can provide information about the sample.

Donya Dobbin

Numerade Educator

Problem 94

If we model the vibrating system as a mass on a spring, what is the mass necessary to achieve the desired resonant frequency when the tip is not interacting with the surface?
(a) $25 \mathrm{ng} ;$ (b) $100 \mathrm{ng} ;$ (c) $2.5 \mu \mathrm{g}$ (d) $100 \mu \mathrm{g}$.

Salamat Ali

Numerade Educator

Problem 95

In the model of Problem $14.94,$ what is the total mechanical energy of the vibration when the tip is not interacting with the surface? (a) $1.2 \times 10^{-18} \mathrm{~J} ;$ (b) $1.2 \times 10^{-16} \mathrm{~J} ;$ (c) $1.2 \times 10^{-9} \mathrm{~J}$ (d) $5.0 \times 10^{-8} \mathrm{~J}$.

Linda Winkler

Numerade Educator

Problem 96

By what percentage does the frequency of oscillation change if $k_{\text {surf }}=5 \mathrm{~N} / \mathrm{m} ?(\mathrm{a}) 0.1 \% ;(\mathrm{b}) 0.2 \%$
(c) $0.5 \%$ (d) $1.0 \%$

Linda Winkler

Numerade Educator