Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 14

Oscillators - all with Video Answers

Educators

+ 5 more educators

Chapter Questions

Problem 1

(I) If a particle undergoes SHM with amplitude $0.18 \mathrm{m},$ what is the total distance it travels in one period?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 2

(I) An elastic cord is 65 $\mathrm{cm}$ long when a weight of 75 $\mathrm{N}$ hangs from it but is 85 $\mathrm{cm}$ long when a weight of 180 $\mathrm{N}$ hangs from it. What is the "spring" constant $k$ of this elastic cord?

Averell Hause

Carnegie Mellon University

Problem 3

(1) The springs of a 1500 -kg car compress 5.0 $\mathrm{mm}$ when its 68 $\mathrm{kg}$ driver gets into the driver's seat. If the car goes over a bump, what will be the frequency of oscillations? Ignore damping.

Surendra Kumar

Numerade Educator

Problem 4

(1) $(a)$ What is the equation describing the motion of a mass on the end of a spring which is stretched 8.8 $\mathrm{cm}$ from equilibrium and then released from rest, and whose period is 0.66 $\mathrm{s} ?$ (b) What will be its displacement after 1.8 $\mathrm{s} ?$

Averell Hause

Carnegie Mellon University

Problem 5

(II) Estimate the stiffness of the spring in a child's pogo stick if the child has a mass of 35 $\mathrm{kg}$ and bounces once every 2.0 seconds.

Keshav Singh

Numerade Educator

Problem 6

(II) A fisherman's scale stretches 3.6 $\mathrm{cm}$ when a 2.4 -kg fish hangs from it. (a) What is the spring stiffness constant and (b) what will be the amplitude and frequency of oscillation if the fish is pulled down 2.5 $\mathrm{cm}$ more and released so that it oscillates up and down?

Averell Hause

Carnegie Mellon University

Problem 7

(II) Tall buildings are designed to sway in the wind. In a 100 -km/h wind, for example, the top of the 110 -story Sears Tower oscillates horizontally with an amplitude of 15 $\mathrm{cm}$ . The building oscillates at its natural frequency, which has a period of 7.0 s. Assuming SHM, find the maximum horizontal velocity and acceleration experienced by a Sears employee as she sits working at her desk located on the top floor. Compare the maximum acceleration (as a percentage) with the acceleration due to gravity.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 8

(II) Construct a Table indicating the position $x$ of the mass in Fig. 2 at times $t=0,\left\{T, \frac{1}{2} T, \frac{3}{4} T, T,$ and $\frac{5}{4} T,$ where $T$ is \right. the period of oscillation. On a graph of $x$ vs. $t,$ plot these six points. Now connect these points with a smooth curve. Based on these simple considerations, does your curve resemble that of a cosine or sine wave?

Averell Hause

Carnegie Mellon University

Problem 9

(II) A small fly of mass 0.25 $\mathrm{g}$ is caught in a spider's web. The web oscillates predominately with a frequency of 4.0 $\mathrm{Hz}$ . (a) What is the value of the effective spring stiffness constant $k$ for the web? (b) At what frequency would you expect the web to oscillate if an insect of mass 0.50 $\mathrm{g}$ were trapped?

Keshav Singh

Numerade Educator

Problem 10

(II) A mass $m$ at the end of a spring oscillates with a frequency of 0.83 Hz. When an additional $680-g$ mass is added to $m,$ the frequency is 0.60 $\mathrm{Hz}$ . What is the value of $m ?$

Averell Hause

Carnegie Mellon University

Problem 11

(II) A uniform meter stick of mass $M$ is pivoted on a hinge at one end and held horizontal by a spring with spring constant $k$ attached at the other end (Fig. 28$)$ . If the stick oscillates up, and down slightly, what is its frequency? [Hint. Write a torque equation about the hinge.

Surendra Kumar

Numerade Educator

Problem 12

(II) A balsa wood block of mass 55 $\mathrm{g}$ floats on a lake, bobbing up and down at a frequency of 3.0 $\mathrm{Hz}$ (a) What is the value of the effective spring constant of the water? (b) A partially filled water bottle of mass 0.25 $\mathrm{kg}$ and almost the same size and shape of the balsa block is tossed into the water. At what frequency would you expect the bottle to bob up and down? Assume SHM.

Averell Hause

Carnegie Mellon University

Problem 13

(II) Figure 29 shows two examples of SHM, labeled $A$ and B. For each, what is $(a)$ the amplitude, $(b)$ the frequency, and $(c)$ the period? $(d)$ Write the equations for both $A$ and $B$ in the form of a sine or cosine.

Keshav Singh

Numerade Educator

Problem 14

(II) Determine the phase constant $\phi$ in Eq. 4 if, at $t=0$ , the oscillating mass is at $(a) \quad x=-A, \quad(b) \quad x=0$ , $(c) x=A,(d) x=\frac{1}{2} A,(e) x=-\frac{1}{2} A,(f) x=A / \sqrt{2}$
$$x=A \cos (\omega t+\phi)$$

Averell Hause

Carnegie Mellon University

Problem 15

(II) A vertical spring with spring stiffness constant 305 $\mathrm{N} / \mathrm{m}$ oscillates with an amplitude of 28.0 $\mathrm{cm}$ when 0.260 $\mathrm{kg}$ hangs from it. The mass passes through the equilibrium point $(y=0)$ with positive velocity at $t=0 .$ (a) What equation describes this motion as a function of time? (b) At what times will the spring be longest and shortest?

Keshav Singh

Numerade Educator

Problem 16

(II) The graph of displacement vs. time for a small mass $m$ at the end of a spring is shown in Fig. $30 .$ At $t=0, x=0.43 \mathrm{cm} .$ (a) If $m=9.5 \mathrm{g}$ , find the spring constant, $k .$ (b) Write the equation for displacement $x$ as a function of time.

Averell Hause

Carnegie Mellon University

Problem 17

(II) The position of a SHO as a function of time is given by $x=3.8 \cos (5 \pi t / 4+\pi / 6)$ where $t$ is in seconds and $x$ in meters. Find (a) the period and frequency, (b) the position and velocity at $t=0,$ and $(c)$ the velocity and acceleration at $t=2.0 \mathrm{s}$

Surendra Kumar

Numerade Educator

Problem 18

(II) A tuning fork oscillates at a frequency of 441 $\mathrm{Hz}$ and the tip of each prong moves 1.5 $\mathrm{mm}$ to either side of center. Calculate $(a)$ the maximum speed and $(b)$ the maximum acceleration of the tip of a prong.

Averell Hause

Carnegie Mellon University

Problem 19

(II) An object of unknown mass $m$ is hung from a vertical spring of unknown spring constant $k,$ and the object is observed to be at rest when the spring has extended by 14 $\mathrm{cm} .$ The object is then given a slight push and executes SHM. Determine the period $T$ of this oscillation.

Keshav Singh

Numerade Educator

Problem 20

(II) A 1.25 -kg mass stretches a vertical spring 0.215 $\mathrm{m}$ . If the spring is stretched an additional 0.130 $\mathrm{m}$ and released, how long does it take to reach the (new) equilibrium position again?

Averell Hause

Carnegie Mellon University

Problem 21

(1I) Consider two objects, A and B, both undergoing SHM, but with different frequencies, as described by the equations $x_{\mathrm{A}}=(2.0 \mathrm{m}) \sin (2.0 t)$ and $x_{\mathrm{B}}=(5.0 \mathrm{m}) \sin (3.0 t),$ where $t$ is in seconds. After $t=0,$ find the next three times $t$ at which both objects simultaneously pass through the origin.

Keshav Singh

Numerade Educator

Problem 22

(II) A 1.60 -kg object oscillates from a vertically hanging light spring once every 0.55 s. (a) Write down the equation giving its position $y(+$ upward) as a function of time $t,$ assuming it started by being compressed 16 $\mathrm{cm}$ from the equilibrium position (where $y=0 )$ , and released. $(b)$ How
long will it take to get to the equilibrium position for the first time? (c) What will be its maximum speed? (d) What will be its maximum acceleration, and where will it first be attained?

Averell Hause

Carnegie Mellon University

Problem 23

(II) A bungee jumper with mass 65.0 $\mathrm{kg}$ jumps from a high bridge. After reaching his lowest point, he oscillates up and down, hitting a low point eight more times in 43.0 $\mathrm{s}$ . He finally comes to rest 25.0 $\mathrm{m}$ below the level of the bridge. Estimate the spring stiffness constant and the unstretched length of the bungee cord assuming SHM.

Keshav Singh

Numerade Educator

Problem 24

(II) A block of mass $m$ is supported by two identical parallel vertical springs, each with spring stiffness constant $k($ Fig. 31$)$ . What will be the frequency of vertical oscillation?

Averell Hause

Carnegie Mellon University

Problem 25

(III) A mass $m$ is connected to two springs, with spring constants $k_{1}$ and $k_{2},$ in two different ways as shown in Fig. 32 a and b. Show that the period for the configuration shown in part (a) is given by
$$T=2 \pi \sqrt{m\left(\frac{1}{k_{1}}+\frac{1}{k_{2}}\right)}$$
and for that in part (b) is given by
$T=2 \pi \sqrt{\frac{m}{k_{1}+k_{2}}}$
Ignore friction.

Keshav Singh

Numerade Educator

Problem 26

(III) A mass $m$ is at rest on the end of a spring of spring constant $k .$ At $t=0$ it is given an impulse $J$ by a hammer. Write the formula for the subsequent motion in terms of $m, k, J,$ and $t$

Averell Hause

Carnegie Mellon University

Problem 27

(I) A 1.15 -kg mass oscillates according to the equation $x=0.650 \cos 7.40 t$ where $x$ is in meters and $t$ in seconds. Determine $(a)$ the amplitude, $(b)$ the frequency, $(c)$ the total energy, and $(d)$ the kinetic energy and potential energy when $x=0.260 \mathrm{m} .$

Keshav Singh

Numerade Educator

Problem 28

(I) $(a)$ At what displacement of a $S H O$ is the energy half kinetic and half potential? (b) What fraction of the total energy of a SHO is kinetic and what fraction potential when the displacement is one third the amplitude?

Averell Hause

Carnegie Mellon University

Problem 29

(II) Draw a graph like Fig. 11 for a horizontal spring whose spring constant is 95 $\mathrm{N} / \mathrm{m}$ and which has a mass of 55 $\mathrm{g}$ on the end of it. Assume the spring was started with an initial amplitude of 2.0 $\mathrm{cm} .$ Neglect the mass of the spring and any friction with the horizontal surface. Use your graph to estimate $(a)$ the potential energy, $(b)$ the kinetic energy, and
$(c)$ the speed of the mass, for $x=1.5 \mathrm{cm} .$

Keshav Singh

Numerade Educator

Problem 30

(II) $\mathrm{A} .3 .35$ -kg mass at the end of a spring oscillates 2.5 times per second with an amplitude of 0.15 $\mathrm{m} .$ Determine $(a)$ the velocity when it passes the equilibrium point, (b) the velocity when it is 0.10 $\mathrm{m}$ from equilibrium, (c) the total energy of the system, and $(d)$ the equation describing the motion of the mass, assuming that at $t=0, x$ was a maximum.

Averell Hause

Carnegie Mellon University

Problem 31

(II) It takes a force of 95.0 $\mathrm{N}$ to compress the spring of a toy popgun 0.175 $\mathrm{m}$ to "load" a $0.160-\mathrm{kg}$ ball. With what speed will the ball leave the gun if fired horizontally?

Keshav Singh

Numerade Educator

Problem 32

(II) A 0.0125 -kg bullet strikes a 0.240 -kg block attached to a fixed horizontal spring whose spring constant is $2.25 \times 10^{3} \mathrm{N} / \mathrm{m}$ and sets it into oscillation with an amplitude of 12.4 $\mathrm{cm} .$ What was the initial speed of the bullet if the two objects move together after impact?

Averell Hause

Carnegie Mellon University

Problem 33

(II) If one oscillation has 5.0 times the energy of a second one of equal frequency and mass, what is the ratio of their amplitudes?

Keshav Singh

Numerade Educator

Problem 34

(II) A mass of 240 $\mathrm{g}$ oscillates on a horizontal frictionless surface at a frequency of 3.0 $\mathrm{Hz}$ and with amplitude of 4.5 $\mathrm{cm} .(a)$ What is the effective spring constant for this motion? (b) How much energy is involved in this motion?

Averell Hause

Carnegie Mellon University

Problem 35

(II) A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. It takes 3.6 $\mathrm{J}$ of work to compress the spring by 0.13 $\mathrm{m}$ . If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 15 $\mathrm{m} / \mathrm{s}^{2} .$ Find the value of $(a)$ the spring constant and $(b)$ the mass.

Keshav Singh

Numerade Educator

Problem 36

(II) An object with mass 2.7 $\mathrm{kg}$ is executing simple harmonic motion, attached to a spring with spring constant $k=280 \mathrm{N} / \mathrm{m}$ . When the object is 0.020 $\mathrm{m}$ from its equilibrium position, it is moving with a speed of 0.55 $\mathrm{m} / \mathrm{s}$ . (a) Calculate the amplitude of the motion. (b) Calculate the maximum speed attained by the object.

Averell Hause

Carnegie Mellon University

Problem 37

(II) Agent Arlene devised the following method of measuring the muzzle velocity of a rifle (Fig. $33 ) .$ She fires a bullet into a 4.648 -kg wooden block resting on a smooth surface, and attached to a spring of spring constant $k=142.7 \mathrm{N} / \mathrm{m} .$ The bullet, whose mass is $7.870 \mathrm{g},$ remains embedded in the wooden block. She measures the maximum distance that the block compresses the spring to be 9.460 $\mathrm{cm} .$ What is the speed $v$ of the bullet?

Keshav Singh

Numerade Educator

Problem 38

(II) Obtain the displacement $x$ as a function of time for the simple harmonic oscillator using the conservation of energy, Eqs. $10 .[$Hint. Integrate Eq. 11 a with $v=d x / d t .]$
$\begin{array}{rlr}{E} & {=\frac{1}{2} m(0)^{2}+\frac{1}{2} k A^{2}=\frac{1}{2} k A^{2}} \\ {E} & {=\frac{1}{2} m v^{2}+\frac{1}{2} k(0)^{2}=\frac{1}{2} m v_{\max }^{2}} \\ {E} & {=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2}}\end{array}$

Averell Hause

Carnegie Mellon University

Problem 39

(II) At $t=0,$ a $785-\mathrm{g}$ mass at rest on the end of a horizontal spring $(k=184 \mathrm{N} / \mathrm{m})$ is struck by a hammer which gives it an initial speed of 2.26 $\mathrm{m} / \mathrm{s} .$ Determine $(a)$ the period and frequency of the motion, $(b)$ the amplitude, (c) the maximum acceleration, (d) the position as a function of time, $(e)$ the total energy, and $(f)$ the kinetic energy when $x=0.40 A$ where $A$ is the amplitude.

Keshav Singh

Numerade Educator

Problem 40

(II) A pinball machine uses a spring launcher that is compressed 6.0 $\mathrm{cm}$ to launch a ball up a $15^{\circ}$ ramp. Assume that the pinball is a solid uniform sphere of radius $r=1.0 \mathrm{cm}$ and mass $m=25 \mathrm{g}$ . If it is rolling without slipping at a speed of 3.0 $\mathrm{m} / \mathrm{s}$ when it leaves the launcher, what is the spring constant of the spring launcher?

Averell Hause

Carnegie Mellon University

Problem 41

(1) A pendulum has a period of 1.35 s on Earth. What is its period on Mars, where the acceleration of gravity is about 0.37 that on Earth?

Keshav Singh

Numerade Educator

Problem 42

(I) A pendulum makes 32 oscillations in exactly 50 s. What is its $(a)$ period and $(b)$ frequency?

Averell Hause

Carnegie Mellon University

Problem 43

(II) A simple pendulum is 0.30 $\mathrm{m}$ long. At $t=0$ it is released from rest starting at an angle of $13^{\circ} .$ Ignoring friction, what will be angular position of the pendulum at (a) $t=0.35 \mathrm{s},$ (b) $t=3.45 \mathrm{s},$ and $(c) t=6.00 \mathrm{s?}$

Vipender Yadav

Numerade Educator

Problem 44

(II) What is the period of a simple pendulum 53 $\mathrm{cm}$ long (a) on the Earth, and $(b)$ when it is in a freely falling elevator?

Averell Hause

Carnegie Mellon University

Problem 45

(II) A simple pendulum oscillates with an amplitude of $10.0^{\circ} .$ What fraction of the time does it spend between $+5.0^{\circ}$ and $-5.0^{\circ} ?$ Assume SHM.

Keshav Singh

Numerade Educator

Problem 46

(II) Your grandfather clock's pendulum has a length of 0.9930 $\mathrm{m} .$ If the clock loses 26 $\mathrm{s}$ per day, how should you adjust the length of the pendulum?

Averell Hause

Carnegie Mellon University

Problem 47

(II) Derive a formula for the maximum speed $v_{\max }$ of a simple pendulum bob in terms of $g$ , the length $\ell$ , and the maximum angle of swing $\theta_{\max }$ .

Keshav Singh

Numerade Educator

Problem 48

(II) A pendulum consists of a tiny bob of mass $M$ and a uniform cord of mass $m$ and length $\ell$ (a) Determine a formula for the period using the small angle approximation.
(b) What would be the fractional error if you use the formula for a simple pendulum, Eq. 12 $\mathrm{c} ?$
$$T=\frac{1}{f}=2 \pi \sqrt{\frac{\ell}{g}} \quad[\theta$ small $] \quad(12 \mathrm{c})$$

Averell Hause

Carnegie Mellon University

Problem 49

(II) The balance wheel of a watch is a thin ring of radius 0.95 $\mathrm{cm}$ and oscillates with a frequency of 3.10 $\mathrm{Hz}$ . If a torque of $1.1 \times 10^{-5} \mathrm{m}$ . N causes the wheel to rotate $45^{\circ}$ calculate the mass of the balance wheel.

Keshav Singh

Numerade Educator

Problem 50

(II) The human leg can be compared to a physical pendulum, with a "natural" swinging period at which
walking is easiest. Consider the leg as two rods joined rigidly together at the knee; the axis for the leg is the hip joint. The length of each rod is about the same, 55 $\mathrm{cm} .$ The upper rod has a mass of 7.0 kg and the lower rod has a mass of $4.0 \mathrm{kg},(a)$ Calculate the natural swinging period of the system. $(b)$ Check your answer by standing on a chair and measuring the time for one or more complete back-and- forth swings. The effect of a shorter leg is a shorter swinging period, enabling a faster "natural" stride.

Averell Hause

Carnegie Mellon University

Problem 51

(II) $(a)$ Determine the equation of motion for $\theta$ as a a function of time) for a torsion pendulum, Fig. 18 , and show that the motion is simple harmonic. $(b)$ Show that the period $T$ is $T=2 \pi \sqrt{I / K}$ . [The balance wheel of a mechanical watch is an example of a torsion pendulum in which the restoring torque is applied by a coil spring.

Keshav Singh

Numerade Educator

Problem 52

(II) A student wants to use a meter stick as a pendulum. She plans to drill a small hole through the meter stick and suspend it from a smooth pin attached to the wall (Fig. 34) Where in the meter stick should she drill the hole to obtain the short an possible period? How she obtain with a meter stick in this way?

Averell Hause

Carnegie Mellon University

Problem 53

(II) A meter stick is hung at its center from a thin wire (Fig. 35 a). It is twisted and oscillates with a period of 5.0 $\mathrm{s}$ . The meter stick is sawed off to a length of 70.0 $\mathrm{cm} .$ This piece is again balanced at its center and set in oscillation (Fig, 35b). With what period does it oscillate?

Keshav Singh

Numerade Educator

Problem 54

(II) An aluminum disk, 12.5 $\mathrm{cm}$ in diameter and 375 $\mathrm{g}$ in mass, is mounted on a vertical shaft with very low friction (Fig, $36 ) .$ One end of a flat coil spring is attached to the disk, the other end to the base of the apparatus. The disk is set into rotational oscillation and the frequency is 0.331 $\mathrm{Hz}$ . What is the torsional spring constant $K(\tau=-K \theta) ?$

Averell Hause

Carnegie Mellon University

Problem 55

(II) A plywood disk of radius 20.0 $\mathrm{cm}$ and mass 2.20 $\mathrm{kg}$ has a small hole drilled through it, 2.00 $\mathrm{cm}$ from its edge (Fig. 37$)$ . The disk is hung from the wall by means of a metal pin through the hole, and is used as a pendulum. What is the period of this pendulum for small oscillations?

Keshav Singh

Numerade Educator

Problem 56

(II) A 0.835 -kg block oscillates on the end of a spring whose spring constant is $k=41.0 \mathrm{N} / \mathrm{m}$ . The mass moves in a fluid which offers a resistive force $F=-b v,$ where $b=0.662 \mathrm{N} \cdot \mathrm{s} / \mathrm{m} .$ (a) What is the period of the motion? (b) What is the fractional decrease in amplitude per cycle? (c) Write the displacement as a function of time if at $t=0, x=0,$ and at $t=1.00 \mathrm{s}, x=0.120 \mathrm{m}$

Averell Hause

Carnegie Mellon University

Problem 57

(II) Estimate how the damping constant changes when a car's shock absorbers get old and the car bounces three times after going over a speed bump.

MA

Manne Andergronde

Numerade Educator

Problem 58

(II) A physical pendulum consists of an 85 -cm-long, 240 -g-mass, uniform wooden rod hung from a nail near one end (Fig, 38 ). The motion is damped because of friction in the pivot; the damping force is approximately proportional to $d \theta / d t .$ The rod is set in oscillation by displacing it $15^{\circ}$ from its equilibrium position and releasing it. After 8.0 $\mathrm{s}$ the amplitude of the oscillation has been reduced to $5.5^{\circ} .$ If the angular displacement can be written as $\theta=A e^{-\gamma t} \cos \omega^{\prime} t,$ find $(a) \gamma,(b)$ the approximate period of the motion, and $(c)$ how long it takes for the amplitude to be reduced to $\frac{1}{2}$ of its original value.

Averell Hause

Carnegie Mellon University

Problem 59

(II) A damped harmonic oscillator loses 6.0$\%$ of its mechanical energy per cycle. $(a)$ By what percentage does its frequency differ from the natural frequency $f_{0}=(1 / 2 \pi) \sqrt{k / m} ?$ (b) After how many periods will the amplitude have decreased to 1$/ e$ of its original value?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 60

(II) A vertical spring of spring constant 115 $\mathrm{N} / \mathrm{m}$ supports a mass of 75 $\mathrm{g}$ . The mass oscillates in a tube of liquid. If the mass is initially given an amplitude of 5.0 $\mathrm{cm}$ , the mass is observed to have an amplitude of 2.0 $\mathrm{cm}$ after 3.5 $\mathrm{s}$ . Estimate the damping constant b. Neglect buoyant forces.

Suzanne W.

Numerade Educator

Problem 61

(III) $\quad(a)$ Show that the total mechanical energy, $E=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2},$ as a function of time for a lightly damped harmonic oscillator is
$E=\frac{1}{2} k A^{2} e^{-(b / m) t}=E_{0} e^{-(b / m) t}$
where $E_{0}$ is the total mechanical energy at $t=0$ . (Assume $\omega^{\prime}>b / 2 m . )(b)$ Show that the fractional energy lost per period is
$\frac{\Delta E}{E}=\frac{2 \pi b}{m \omega_{0}}=\frac{2 \pi}{Q}$
where $\omega_{0}=\sqrt{k} / m$ and $Q=m \omega_{0} / b$ is called the quality factor or $Q$ value of the system. A larger $Q$ value means the system can undergo oscillations for a longer time.

Keshav Singh

Numerade Educator

Problem 62

(III) A glider on an air track is connected by springs to either end of the track (Fig. $39 ) .$ Both springs have the same spring constant, $k,$ and the glider has mass $M .$ (a) Determine the frequency of the oscillation, assuming no damping, if $k=125 \mathrm{N} / \mathrm{m}$ and $M=215 \mathrm{g} .$ (b) It is observed that after 55 oscillations, the amplitude of the oscillation has dropped to one-half of its initial value. Estimate the value of $\gamma,$ using Eq. $16 .(c)$ How long does it take the amplitude to decrease to one-quarter of its initial value?
$x=A e^{-\gamma t} \cos \omega^{\prime} t$

Averell Hause

Carnegie Mellon University

Problem 63

(II) For a forced oscillation at resonance $\left(\omega=\omega_{0}\right),$ what is the value of the phase angle $\phi_{0}$ in Eq. 22$?$ (b) What, then, is the displacement at a time when the driving force $F_{\text { ext }}$ is a maximum, and at a time when $F_{\text { ext }}=0 ?$ (c) What is the phase difference (in degrees) between the driving force and the displacement in this case?
$x=A_{0} \sin \left(\omega t+\phi_{0}\right)$

Keshav Singh

Numerade Educator

Problem 64

(II) Differentiate Eq. 23 to show that the resonant amplitude peaks at
$\begin{aligned} \omega &=\sqrt{\omega_{0}^{2}-\frac{b^{2}}{2 m^{2}}} \\ A_{0} &=\frac{F_{0}}{m \sqrt{\left(\omega^{2}-\omega_{0}^{2}\right)^{2}+b^{2} \omega^{2} / m^{2}}} \end{aligned}$

Averell Hause

Carnegie Mellon University

Problem 65

(II) An 1150 kg automobile has springs with $k=16,000 \mathrm{N} / \mathrm{m}$ . One of the tires is not properly balanced; it has a little extra mass on one side compared to the other, causing the car to shake at certain speeds. If the tire radius is $42 \mathrm{cm},$ at what speed will the wheel shake most?

Keshav Singh

Numerade Educator

Problem 66

(II) Construct an accurate resonance curve, from $\omega=0$ to $\omega=2 \omega_{0},$ for $Q=6.0 .$

Averell Hause

Carnegie Mellon University

Problem 67

(1I) The amplitude of a driven harmonic oscillator reaches a value of 23.7$F_{0} / k$ at a resonant frequency of 382 Hz. What is the $Q$ value of this system?

Keshav Singh

Numerade Educator

Problem 68

(III) By direct substitution, show that Eq. $22,$ with Eqs. 23 and $24,$ is a solution of the equation of motion $(\mathrm{Eq}$ . 21) for the forced oscillator. [Hint: To find sin $\phi_{0}$ and cos $\phi_{0}$
from tan $\phi_{0},$ draw a right triangle.]
$m \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x=F_{0} \cos \omega t$
$\phi_{0}=\tan ^{-1} \frac{\omega_{0}^{2}-\omega^{2}}{\omega(b / m)}$

Averell Hause

Carnegie Mellon University

Problem 69

(III) Consider a simple pendulum (point mass bob) 0.50 m long with a $Q$ of 350 (a) How long does it take for the amplitude (assumed small) to decrease by two-thirds? (b) If the amplitude is 2.0 $\mathrm{cm}$ and the bob has mass 0.27 $\mathrm{kg}$ , what is the initial energy loss rate of the pendulum in watts? (c) If we are to stimulate resonance with a sinusoidal driving force, how close must the driving frequency be to the natural frequency of the pendulum (give $\Delta f=f-f_{0} ) ?$

Keshav Singh

Numerade Educator

Problem 70

A 62 -kg person jumps from a window to a fire net 20.0 $\mathrm{m}$ below, which stretches the net 1.1 $\mathrm{m}$ . Assume that the net behaves like a simple spring. (a) Calculate how much it would stretch if the same person were lying in it. (b) How much would it stretch if the person jumped from 38 $\mathrm{m}$ ?

Averell Hause

Carnegie Mellon University

Problem 71

An energy-absorbing car bumper has a spring constant of 430 $\mathrm{kN} / \mathrm{m}$ . Find the maximum compression of the bumper if the car, with mass 1300 $\mathrm{kg}$ , collides with a wall at a speed of 2.0 $\mathrm{m} / \mathrm{s}$ (approximately 5 $\mathrm{mi} / \mathrm{h} )$ .

Keshav Singh

Numerade Educator

Problem 72

The length of a simple pendulum is $0.63 \mathrm{m},$ the pendulum bob has a mass of 295 $\mathrm{g}$ , and it is released at an angle of $15^{\circ}$ to the vertical. (a) With what frequency does it oscillate? (b) What is the pendulum bob's speed when it passes through the lowest point of the swing? Assume SHM. (c) What is the total energy stored in this oscillation assuming no losses?

Averell Hause

Carnegie Mellon University

Problem 73

A simple pendulum oscillates with frequency $f .$ What is its frequency if the entire pendulum accelerates at 0.50$g$ (a) upward, and $(b)$ downward?

Keshav Singh

Numerade Educator

Problem 74

A $0.650-\mathrm{kg}$ mass oscillates according to the equation $x=0.25 \sin (5.50 t)$ where $x$ is in meters and $t$ is in seconds. Determine (a) the amplitude, (b) the frequency, (c) the period, (d) the total energy, and ( $e$ ) the kinetic energy and potential energy when $x$ is 15 $\mathrm{cm} .$

Averell Hause

Carnegie Mellon University

Problem 75

(a) A crane has hoisted a $1350-\mathrm{kg}$ car at the junkyard. The crane's steel cable is 20.0 $\mathrm{m}$ long and has a diameter of 6.4 $\mathrm{mm}$ . If the car starts bouncing at the end of the cable, what is the period of the bouncing? [Hint: Refer to Table 1.1 (b) What amplitude of bouncing will likely cause the cable to snap? (See Table $2,$ and assume Hooke's law holds all the way up to the breaking point.)

Keshav Singh

Numerade Educator

Problem 76

An oxygen atom at a particular site within a DNA molecule can be made to execute simple harmonic motion when illuminated by infrared light. The oxygen atom is bound with a spring-like chemical bond to a phosphorus atom, which is rigidly attached to the DNA backbone. The oscillation of the oxygen atom occurs with frequency $f=3.7 \times 10^{13}$ . If the oxygen atom at this site is chemically replaced
with a sulfur atom, the spring constant of the bond is unchanged (sulfur is just below oxygen in the Periodic Table). Predict the frequency for a DNA molecule after the sulfur substitution.

Averell Hause

Carnegie Mellon University

Problem 77

A "seconds" pendulum has a period of exactly 2.000 s That is, each one-way swing takes 1.000 s. What is the length of a seconds pendulum in Austin, Texas, where $g=9.793 \mathrm{m} / \mathrm{s}^{2}$ ? If the pendulum is moved to Paris, where $g=9.809 \mathrm{m} / \mathrm{s}^{2}$ , by how many millimeters must we lengthen the pendulum? What is the length of a seconds pendulum on the Moon, where $g=1.62 \mathrm{m} / \mathrm{s}^{2} ?$

Keshav Singh

Numerade Educator

Problem 78

A 320 -kg wooden raft floats on a lake. When a 75 -kg man stands on the raft, it sinks 3.5 $\mathrm{cm}$ deeper into the water. When he steps off, the raft oscillates for a while. (a) What is the frequency of oscillation? (b) What is the total energy of oscillation (ignoring damping)?

Averell Hause

Carnegie Mellon University

Problem 79

At what displacement from equilibrium is the speed of a SHO half the maximum value?

Keshav Singh

Numerade Educator

Problem 80

A diving board oscillates with simple harmonic motion of frequency 2.5 cycles per second. What is the maximum amplitude with which the end of the board can oscillate in order that a pebble placed there (Fig. 40$)$ does not lose contact with the board during the oscillation?

Averell Hause

Carnegie Mellon University

Problem 81

A rectangular block of wood floats in a calm lake. Show that, if friction is ignored, when the block is pushed gently down into the water and then released, it will then oscillate with SHM. Also, determine an equation for the force constant.

Shital Rijal

Numerade Educator

Problem 82

A 950 -kg car strikes a huge spring at a speed of 25 $\mathrm{m} / \mathrm{s}$ (Fig. 41$)$ , compressing the spring 5.0 $\mathrm{m}$ . (a) What is the spring stiffness constant of the spring? (b) How long is the car in contact with the spring before it bounces off in the opposite direction?

Averell Hause

Carnegie Mellon University

Problem 83

A $1.60-\mathrm{kg}$ table is supported on four springs. A $0.80-\mathrm{kg}$ chunk of modeling clay is held above the table and dropped so that it hits the table with a speed of 1.65 $\mathrm{m} / \mathrm{s}$ (Fig. 42). The clay makes an inelastic collision with the table, and the table and clay oscillate up and down. After a long time the table comes to rest 6.0 $\mathrm{cm}$ below its original position. $(a)$ What is the effective spring constant of all four springs taken together? (b) With what maximum amplitude does the plat- form oscillate?

Linda Winkler

Numerade Educator

Problem 84

In some diatomic molecules, the force each atom exerts on the other can be approximated by $F=-C / r^{2}+D / r^{3}$ , where $r$ is the atomic separation and $C$ and $D$ are positive constants. $(a)$ Graph $F$ vs. $r$ from $r=0.8 D / C$ to $r=4 D / C$ . (b) Show that equilibrium occurs at $r=r_{0}=D / C$ . (c) Let $\Delta r=r-r_{0}$ be a small displacement from equilibrium, where $\Delta r \ll r_{0} .$ Show that for such small displacements, the motion is approximately simple harmonic, and (d) deter- mine the force constant. (e) What is the period of such motion? [Hint: Assume one atom is kept at rest.]

Averell Hause

Carnegie Mellon University

Problem 85

A mass attached to the end of a spring is stretched a distance $x_{0}$ from equilibrium and released. At what distance from equilibrium will it have $(a)$ velocity equal to half its maximum velocity, and $(b)$ acceleration equal to half its maximum acceleration?

Shital Rijal

Numerade Educator

Problem 86

Carbon dioxide is a linear molecule. The carbon-oxygen bonds in this molecule act very much like springs. Figure 43 shows one possible way the oxygen atoms in this molecule can oscillate: the oxygen atoms oscillate symmetrically in and out, while the central carbon atom remains at rest. Hence each oxygen atom acts like a simple harmonic oscillator with a mass equal to the mass of an oxygen atom. It is observed that this oscillation occurs with a frequency of $f=2.83 \times 10^{13}$ Hz. What is the spring constant of the $\mathrm{C}-\mathrm{O}$ bond?

Averell Hause

Carnegie Mellon University

Problem 87

Imagine that a 10 -cm-diameter circular hole was drilled all the way through the center of the Earth (Fig, $44 ) .$ At one end of the hole, you drop an apple into the hole. Show that, if you assume that the Earth has a constant density, the subsquent motion of the apple is simple harmonic. How long will the apple take to return? Assume that we can ignore all frictional effects.

Keshav Singh

Numerade Educator

Problem 88

A thin, straight, uniform rod of length $\ell=1.00 \mathrm{m}$ and mass $m=215 \mathrm{g}$ hangs from a pivot at one end. $(a)$ What is its period for small-amplitude oscillations? (b) What is the length of a simple pendulum that will have the same period?

Averell Hause

Carnegie Mellon University

Problem 89

A mass $m$ is gently placed on the end of a freely hanging spring. The mass then falls 32.0 $\mathrm{cm}$ before it stops and begins to rise. What is the frequency of the oscillation?

Keshav Singh

Numerade Educator

Problem 90

A child of mass $m$ sits on top of a rectangular slab of mass $M=35 \mathrm{kg},$ which in turn rests on the frictionless horizontal floor at a pizza shop. The slab is attached to a horizontal spring with spring constant $k=430 \mathrm{N} / \mathrm{m}$ (the other end is attached to an immovable wall, Fig. 45. The coefficient of static friction between the child and the top of the slab is $\mu=0.40 .$ The shop owner's intention is that, when displaced from the equilibrium position and released, the slab and child (with no slippage between the two) execute SHM with amplitude $A=0.50 \mathrm{m} .$ Should there be a weight restriction for this ride? If so, what is it?

Averell Hause

Carnegie Mellon University

Problem 91

Estimate the effective spring constant of a trampoline.

Keshav Singh

Numerade Educator

Problem 92

In Section 5 of "oscillations," the oscillation of a simple pendulum (Fig. 46$)$ is viewed as linear motion along the arc length $x$ and analyzed via $F=m a$ . Alternatively, the pendulum's movement can be regarded as rotational motion about its point of support and analyzed using $\tau=I \alpha$ . Carry out this alternative analysis and show that
$\theta(t)=\theta_{\max } \cos \left(\sqrt{\frac{g}{\ell}} t+\phi\right)$
where $\quad \theta(t)$ is the angular displacement of the pendulum from the vertical at time $t,$ as long as its maximum value is less than about $15^{\circ}$ .

Averell Hause

Carnegie Mellon University

Problem 93

(II) A mass $m$ on a frictionless surface is attached to a spring with spring constant $k$ as shown in Fig. $47 .$ This mass-spring system is then observed to execute simple harmonic motion with a period $T .$ The mass $m$ is changed several times and the associated period $T$ is measured in each case, generating the following data Table:
(a) Starting with $\mathrm{Eq} .7 \mathrm{b}$ , show why a graph of $T^{2}$ vs. $m$ is expected to yield a straight line. How can $k$ be determined from the straight line's slope? What is the line's $y$ -intercept expected to be? (b) Using the data in the Table, plot $T^{2}$ vs. $T^{2}$ vs. $T^{2}$ vs. and show that this graph yields a straight line. Determine the slope and (nonzero) y-intercept. (c) Show that a nonzero $y$ -intercept can be expected in our plot theoretically if, rather than simply using $m$ for the mass in Eq. $7 b,$ we use $m+m_{0},$ where $m_{0}$ is a constant. That is, repeat part $(a)$ using $m+m_{0}$ for the mass in Eq. 7 $\mathrm{b}$ . Then use the result of this analysis to determine $k$ and $m_{0}$ from your graph's slope and $y$ -intercept. (d) Offer a physical interpretation for $m_{0},$ a mass that appears to be oscillating in addition to the attached mass $m .$
$T=2 \pi \sqrt{\frac{m}{k}}$

Shital Rijal

Numerade Educator

Problem 94

(III) Damping proportional to $v^{2}$ . Suppose the oscillator of Example 5 of "oscillations" is damped by a force proportional to the square of the velocity, $F_{\text { damping }}=-c v^{2},$ where $c=0.275 \mathrm{kg} / \mathrm{m}$ is a constant. Numerically integrate the differential equation from $t=0$ to $t=2.00 \mathrm{s}$ to an accuracy of $2 \%,$ and plot your results.

Averell Hause

Carnegie Mellon University