Principles And Practice Of Physics 1st Edition Eric Mazur - Solutions

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You have two string segments of equal length, segment 1 of linear mass density \(\mu_{1}\) and segment 2 of linear mass density \(\mu_{2}>\mu_{1}\). You splice the two together and then tie the combination tautly between two posts. You splice in such a way that, once attached to the posts, the
A guitar string has a length of \(0.650 \mathrm{~m}\) and a mass of \(4.00 \times 10^{-3} \mathrm{~kg}\). (a) If it is kept under a tension of \(126 \mathrm{~N}\), what is the fundamental (smallest harmonic) frequency at which it vibrates? (b) At what other frequencies (higher harmonics) could it
The equations for two waves traveling along the same string are\[f_{1}(x, t)=a \sin (b x-q t)\]and\[f_{2}(x, t)=a \sin \left(b x+q t+\frac{1}{3} \pi\right),\]with \(a=3.00 \times 10^{-2} \mathrm{~m}, b=4 \pi \mathrm{m}^{-1}\), and \(q=500 \mathrm{~s}^{-1}\). (a) Calculate the amplitude and
A string vibrates in a standing wave according to the function\[f(x, t)=a \sin (b x) \cos (q t),\]with \(a=6.00 \times 10^{-2} \mathrm{~m}, b=\frac{1}{3} \pi \mathrm{cm}^{-1}\), and \(q=40 \pi \mathrm{s}^{-1}\). What are the amplitude and wave speed of each of the component waves that form this
A traveling wave on a long string is described by the time-dependent wave function\[f(x, t)=a \sin (b x-q t),\]with \(a=6.00 \times 10^{-2} \mathrm{~m}, b=5 \pi \mathrm{m}^{-1}\), and \(q=314 \mathrm{~s}^{-1}\). You want a traveling wave of this frequency and wavelength but with amplitude \(0.0400
You have a trough \(3.60 \mathrm{~m}\) long. It is filled with water, and you want to create a standing wave such that a node occurs every \(0.30 \mathrm{~m}\). (a) What is the wavelength of this wave? (b) A leaf floating on the water \(0.60 \mathrm{~m}\) from one end of the trough is displaced a
Two sinusoidal waves travel along the same string. Their time-dependent wave functions are\[\begin{aligned}& f_{1}(x, t)=a \sin \left(b x-q t-\frac{1}{4} \pi\right) \\& f_{2}(x, t)=a \sin \left(b x-q t-\frac{1}{3} \pi\right)\end{aligned}\]with \(a=5.00 \times 10^{-2} \mathrm{~m}, \quad b=0.120
Two sinusoidal waves travel in opposite directions along the same string. The wavelength and frequency are the same in both waves, and each has amplitude \(0.0289 \mathrm{~m}\). If there is a phase difference \(\Delta \phi\) between the two waves, what is the maximum displacement of the wave
A piano wire has a linear mass density of \(5.00 \times 10^{-3} \mathrm{~kg} / \mathrm{m}\) and is under \(1.35 \mathrm{kN}\) of tension. At what speed does a wave travel along this wire?
Your professor uses a plastic chain \(10 \mathrm{~m}\) long to demonstrate transverse waves in class. If you measure the speed of a traveling wave to be \(15 \mathrm{~m} / \mathrm{s}\), and the professor measures the tension in the chain to be \(100 \mathrm{~N}\), what is the chain's mass?
A large crane is being used to lower a \(10,000-\mathrm{kg}\) storage unit into place on a barge. If the cable that supports the unit from the end of the crane is \(12 \mathrm{~m}\) long, and the linear mass density of the cable is \(1.78 \mathrm{~kg} / \mathrm{m}\), how long does it take a wave to
A violinist is bowing one string to produce a certain note. List three ways in which she can produce a note of higher frequency, either while playing or by preparing the violin differently before she begins to play.
How long after the propellers at the back of an oil tanker start turning does the front of the ship start to move?
A wave traveling along a string has a speed of \(24 \mathrm{~m} / \mathrm{s}\) when the tension in the string is \(120 \mathrm{~N}\). What is the speed of the same wave if the tension is reduced to \(100 \mathrm{~N}\) ?
Two wires that have different linear mass densities, \(0.45 \mathrm{~kg} / \mathrm{m}\) and \(0.29 \mathrm{~kg} / \mathrm{m}\), are spliced together. They are then used as a guy line to secure a telephone pole. If the tension is \(300 \mathrm{~N}\), what is the difference in the speed of a wave
A string of linear mass density \(\mu_{1}\) fixed at both ends is replaced by a string of linear mass density \(16 \mu_{1}\). By what factor does the frequency of the fundamental standing wave change if the tension and wavelength remain what they were in the original string?
A \(25.0-\mathrm{m}\) steel wire and a \(50.0-\mathrm{m}\) copper wire are attached end to end and stretched to a tension of \(145 \mathrm{~N}\). Both wires have a radius of \(0.450 \mathrm{~mm}\), and the densities are \(7.86 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) for the steel and \(8.92
A \(100-\mathrm{m}\) steel cable that helps support the Golden Gate Bridge is \(72.0 \mathrm{~mm}\) in diameter and composed of 100 steel wires twisted together. Approximate this as a single uniform steel cable that has a mass density of \(7.86 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). The
One end of the wire in Figure P16.56 is anchored to the side of a building, and the other end is attached at an angle of \(27^{\circ}\) to a light horizontal bar from which a \(75-\mathrm{kg}\) sign hangs. The linear mass density of the wire is \(0.067 \mathrm{~kg} / \mathrm{m}\). If the wind
A taut rope has a mass of \(0.128 \mathrm{~kg}\) and a length of \(3.60 \mathrm{~m}\). What average power must be supplied to the rope to generate sinusoidal waves that have amplitude \(0.200 \mathrm{~m}\) and wavelength \(0.600 \mathrm{~m}\) if the waves are to travel at \(25.0 \mathrm{~m} /
Consider two waves \(X\) and \(Y\) traveling in the same medium. The two carry the same amount of energy per unit time, but \(\mathrm{X}\) has half the amplitude of \(\mathrm{Y}\). What is the ratio of their wavelengths? Which wave has the longer wavelength?
You hold one end of a string that is attached to a wall by its other end. The string has a linear mass density of \(0.067 \mathrm{~kg} / \mathrm{m}\). You raise your end briskly at \(12 \mathrm{~m} / \mathrm{s}\) for \(0.016 \mathrm{~s}\), creating a transverse wave that moves at \(31 \mathrm{~m} /
One end of a horizontal string that has a linear mass density of \(3.5 \mathrm{~kg} / \mathrm{m}\) is displaced vertically at a speed of \(45 \mathrm{~m} / \mathrm{s}\) for \(6.7 \mathrm{~ms}\). The pulse created by this motion travels down the string at \(78 \mathrm{~m} / \mathrm{s}\). How much
A wave traveling along a string is described by the timedependent wave function \(f(x, t)=a \sin (b x+q t)\), with \(a=0.0268 \mathrm{~m}, b=5.85 \mathrm{~m}^{-1}\), and \(q=76.3 \mathrm{~s}^{-1}\). The linear mass density of the string is \(0.0456 \mathrm{~kg} / \mathrm{m}\). Calculate (a) the
An oscillator attached to a string transmits \(350 \mathrm{~mW}\) of power down the string. The wave created by this motion has amplitude \(0.0165 \mathrm{~m}\) and wavelength \(1.80 \mathrm{~m}\). If the string is under \(14.8 \mathrm{~N}\) of tension, what is the wave speed?
A cord of linear mass density \(0.360 \mathrm{~kg} / \mathrm{m}\) is attached to a harmonic oscillator that has a maximum output of \(200 \mathrm{~W}\).(a) With the oscillator operating at this maximum power and the tension in the cord set at \(30.0 \mathrm{~N}\), a wave of amplitude \(8.00 \times
One end of a 9. 00-m wire vibrates sinusoidally, creating a wave that travels in the positive \(x\) direction. The wave has a frequency of \(60.0 \mathrm{~Hz}\), a wavelength of \(3.00 \mathrm{~m}\), and an amplitude of \(0.0725 \mathrm{~m}\). (a) Write the time-dependent wave function that
Show that \(f(x, t)=e^{b(x-v t)}\), where \(b\) and \(v\) are constants, is a solution to Eq. 16. 51, the wave equation.Data from Eq. 16. 51 af dx ax 2 1 a Zat
What keeps a surfer riding the crest of a wave from dropping down to the wave's base?
A string initially hanging vertically is displaced to the right with a speed of \(25 \mathrm{~m} / \mathrm{s}\) for \(0.040 \mathrm{~s}\), then returned to its original position in \(0.010 \mathrm{~s}\). This motion is repeated, with no delays, numerous times. The wave created by this motion
Show that if \(f_{1}(x, t)=A_{1} \sin \left(k_{1} x-\omega_{1} t\right)\) and \(f_{2}(x, t)=\) \(A_{2} \sin \left(k_{2} x-\omega_{2} t\right)\) are both solutions to the wave equation (Eq. 16. 51), then \(f(x, t)=f_{1}(x, t)+f_{2}(x, t)\) is also a solution.Data from Eq. 16. 51 af dx ax 2 1 a Zat
(a) Show that the equations (i) \(A \cos (k x+\omega t)\), (ii) \(e^{-b|x-a t|^{2}}\), and \((i i i)-\left(b^{2} t-x\right)^{2}\) all satisfy the wave equation (Eq. 16. 51) provided that \(A, k, \omega, b\), and \(q\) are constants. (b) Calculate the speed of each wave of part \(a\).Data from Eq.
Two children are holding a rope taut between them when one suddenly yanks the rope. (a) Does the child at the other end instantly get pulled? (b) Answer the same question for two children holding a rigid steel bar between them.
A heavy rope is hanging by one end from the ceiling with a block hanging from the other end. Is the speed of a pulse in the rope greater at the top or the bottom of the rope?
First you stretch a spring to twice its unstretched length and send pulse A from one end to the other. Then you stretch the spring to three times its unstretched length and send pulse B from one end to the other. How does the time interval needed for \(A\) to travel through the spring compare with
You pluck a guitar string so as to produce a standing wave for which \(\lambda=1.3 \mathrm{~m}\). What is the maximum value of the string's displacement at a position \(0.500 \mathrm{~m}\) from the lower end of the string if the maximum resultant displacement of the wave is \(0.75 \mathrm{~mm}\) ?
Tossing rocks into the center of a pond, you notice that the ripples are evenly spaced \(170 \mathrm{~mm}\) apart. If it takes \(5.0 \mathrm{~s}\) for 15 ripples to pass a log sticking up out of the water, (a) what is the wave speed? (b) How long does it take a ripple to reach shore if the pond is
When a person sings in the shower, certain notes seem louder than others. What's going on?
A standing wave is created on a string that is \(1.75 \mathrm{~m}\) long. The wave has eight nodes counting the nodes at the two ends. (a) What is the wavelength? (b) What is the frequency if the wave speed is \(130 \mathrm{~m} / \mathrm{s}\) ? (c) If the resultant displacement is \(20.0
Would water waves traveling along a trough on the surface of Mars travel faster than, slower than, or at the same speed as water waves traveling along a similar trough on the surface of Earth?
You hold one end of a horizontal string and raise your hand during a time interval of \(0.18 \mathrm{~s}\), doing \(1.2 \mathrm{~J}\) of work on the string in the process. You then allow your hand to return to its original position. If the wave pulse you create travels along the string at \(67
Your friend wants to test your understanding of waves. She weaves together one end of two different ropes (one with greater mass density than the other, but each with the same radius and length) to make a single rope. Then she attaches one end of the combination to a pole. You are blindfolded and
Your job as assistant coach is to attach one end of several heavy, \(10-\mathrm{m}\) ropes to the trusses that support the gym ceiling, leaving the other end of each hanging free. The team is going to spend the afternoon climbing these ropes, but you begin to wonder about wave pulses traveling on
A circular wave travels such that a crest occurs at radius \(r_{1}\) at instant \(t_{1}\). The same crest has a radius \(r_{2}=2 r_{1}\) at \(t_{2}=2 t_{1}\). (a) What is the wave speed? (b) Where is the crest at \(t_{3}=3 t_{1}\) ? (c) What is the ratio (energy per unit length at \(t_{3}\)
A spherical wave travels such that a crest occurs at radius \(r_{1}\) at instant \(t_{1}\). The same crest has a radius \(r_{2}=2 r_{1}\) at \(t_{2}=2 t_{1}\).(a) What is the wave speed? (b) Where is the crest at \(t_{3}=3 t_{1}\) ?(c) What is the ratio (energy per unit area at
Two circular wavefronts interfere as shown in Figure 17. 24. (a) Draw the antinodal lines. (b) Is the medium displacement along these lines constant?Data from Figure 17. 24
Two coherent sources are placed two wavelengths apart. Point \(\mathrm{P}\) lies on a nodal line in the interference pattern of the two sources, and point Q lies on an antinodal line. (a) If one source is turned off, does the energy arriving at P increase, decrease, or remain the same? (b) If one
Because sound waves diffract around an open doorway, you can hear sounds coming from outside the doorway. You cannot, however, see objects outside the doorway unless you are directly in line with them. What does this observation imply about the wavelength of light?
The day after the incident described in Problem 44, the instructor finds herself in the same situation. This time, she tries a harder physics exercise. She keeps running at a constant \(6.0 \mathrm{~m} / \mathrm{s}\) after drawing even with the bus door and pulls ahead for a while, but the
A classmate leaves a message on your voice mail betting that you cannot throw a stone high enough so it lands on the roof of a 20 -m-high building. As you stare out of your window pondering whether to accept the challenge, the well in the courtyard suddenly gives you an idea. You drop a stone into
Indicate at least two possible choices of system in each of the following two situations. For each choice, make a sketch showing the system boundary and state which objects are inside the system and which are outside. (a) After you throw a ball upward, it accelerates downward toward Earth. (b) A
The \(x\) component of the velocity of a car changes from \(-10 \mathrm{~m} / \mathrm{s}\) to \(-2.0 \mathrm{~m} / \mathrm{s}\) in \(10 \mathrm{~s}\). (a) Is the car traveling in the positive or negative \(x\) direction? (b) Does \(\Delta \vec{v}\) point in the positive or negative \(x\) direction?
(a) In Figure 3 . 3, are the \(x\) components of the velocity represented by the \(x(t)\) curves positive or negative? (b) Are the speeds increasing or decreasing? (c) Are the \(x\) components of \(\Delta \vec{v}\) positive or negative? (d) Are the \(x\) components of the acceleration positive or
(a) A car is speeding up in the negative \(x\) direction. In what direction do \(\vec{a}\) and \(\vec{v}\) point? (b) To which of the four graphs in Figures 3 . 2 and 3 . 3 does the situation represented in Figure 3 . 1c correspond?
Hold a book and a sheet of paper, cut to the same size as the book, side by side \(1 \mathrm{~m}\) above the floor. Hold the paper parallel to the floor and the book with its covers parallel to the floor, and release them at the same instant. Which hits the floor first? Now put the paper on top of
What does Figure \(3 . 6 b\) tell you about the velocity of the ball from one exposure to the next?
(a) Does the speed of a falling object increase or decrease? ( \(b\) ) If the positive \(x\) axis points up, does \(v_{x}\) increase or decrease as the object falls? (c) Is the \(x\) component of the acceleration positive or negative?
Imagine throwing a ball downward so that it has an initial speed of \(10 \mathrm{~m} / \mathrm{s}\). What is its speed \(1 \mathrm{~s}\) after you release it? \(2 \mathrm{~s}\) after?
Make a motion diagram for the following situation: A seaside cliff rises \(30 \mathrm{~m}\) above the ocean surface, and a person standing at the edge of the cliff launches a rock vertically upward at a speed of \(15 \mathrm{~m} / \mathrm{s}\). After reaching the top of its trajectory, the rock
Determine the velocity of the stone dropped from the top of the Empire State Building in Example 3 . 2 just before the stone hits the ground.
Repeat Example 3.5 for an \(x\) axis that points downward (leaving the origin at the ball's initial position).
In Example 3 . 6, what is the \(x\) component of the velocity of the stone just before it hits the ground?
As the angle \(\theta\) of the incline used to collect the data of Figure 3 . 22 is increased beyond \(90^{\circ}\), what happens to the acceleration? Does this result make sense (provided you always put the cart on the top side of the track)?
What is the difference between velocity and acceleration?
Does nonzero acceleration mean the same thing as speeding up?
Does the acceleration vector always point in the direction in which an object is moving? If so, explain why. If not, describe a situation in which the direction of the acceleration is not the same as the direction of motion.
How is the curvature of an \(x(t)\) curve related to the sign of the \(x\) component of acceleration?
A cantaloupe and a plum fall from kitchen-counter height at the same instant. Which hits the floor first?
Is it correct to say that a stone dropped from a bridge into the water speeds up as it falls because the acceleration due to gravity increases as the stone gets closer to Earth?
You toss a rock straight up. Compare the acceleration of the rock at the instant just after it leaves your hand with its acceleration at the instant just before it lands back in your hand, which has remained at the point of release.
You throw a ball straight up. What is the ball's acceleration at the top of its trajectory?
List the information that should be included in a motion diagram.
What is the purpose of a motion diagram?
What can you say about a train's acceleration if its \(v(t)\) curve is (a) a straight line that is not parallel to the \(t\) axis and \((b)\) a horizontal line that is parallel to the \(t\) axis?
For an object experiencing constant acceleration, the expression for position as a function of time is \(x(t)=x_{\mathrm{i}}+v_{x, \mathrm{i}} t+\frac{1}{2} a_{x} t^{2}\). Explain, in terms of the area under the \(v(t)\) curve for the object, why the acceleration term includes the factor
For constant acceleration, describe the relationship among displacement, initial and final velocities, and acceleration when the time variable is algebraically eliminated.
You throw a ball straight up and then hold your hand at the release position. Compare the time interval between the release of the ball and its arrival at its highest position to the time interval between leaving its highest position and returning to your hand.
(a) For an object released from rest, describe how the distance the object falls varies with time. (b) Describe how the object's speed varies with time.
How is distance traveled related to the amount of time needed to travel that distance for a ball rolling down an inclined plane?
In what way does the motion of an object rolling down an inclined plane resemble that of an object in free fall?
On which of the following, if any, does the magnitude of the acceleration of a ball rolling down an inclined plane depend: angle of incline, speed of ball, direction of motion?
For what type of motion is it important to distinguish between instantaneous and average acceleration?
What does each of the following represent: (a) slope of an \(x(t)\) curve at a given point on the curve, \((b)\) curvature of an \(x(t)\) curve at a given point on the curve, \((c)\) slope of a \(v(t)\) curve at a given point on the curve, \((d)\) area under an \(a(t)\) curve, \((e)\) area under a
In the foreground of a side-view picture of a table, the legs touch the floor at a point that is \(12 \mathrm{~mm}\) from the bottom of the picture, and the tabletop is \(65 \mathrm{~mm}\) from the bottom of the picture. One end of the tabletop is \(14 \mathrm{~mm}\) from the left edge of the
(a) Which of the numbered lines 1-8 in the graphs in Figure 2.14 represents the greatest average speed? (b) For which of the numbered lines is the \(x\) component of the average velocity negative?
A friend constructs a graph of position as a function of frame number for a film clip of a moving object. She then challenges you to construct a graph identical to hers. She tells you that she measured distances in millimeters, but what two additional pieces of information must she give you?
How do you determine whether an object is moving or at rest?
Suppose you have a film clip showing the motion of several objects. If you want to calibrate distances for a graph, what information do you need to know about at least one of the objects?
What is the purpose of using the phrase \(x\) component of when describing some physical quantities?
How do you represent displacement in a drawing or diagram?
What does interpolation mean with regard to plotting data points?
What are the defining characteristics of a scalar? Of a vector?
Is displacement a scalar or a vector? Is distance a scalar or a vector?
Can distance be negative? Can distance traveled be negative?
How is distance traveled calculated when motion occurs in three segments: first in one direction along the \(x\) axis, then in the opposite direction, and finally in the initial direction?
Describe the graphical procedure for adding two vectors and for subtracting one vector from another.
If you multiply a vector \(\vec{A}\) by a scalar \(c\), is the result a scalar or a vector? If the result is a scalar, what is its magnitude? If the result is a vector, what are its magnitude and direction? What if the scalar \(c=0\) ?
Is average speed a scalar or a vector? Is average velocity a scalar or a vector?
What properties does velocity have that make it a vector?
How is an object's average velocity related to its displacement during a given time interval?

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Principles And Practice Of Physics 1st Edition Eric Mazur - Solutions

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