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  1. Preface
  2. Unit 1. Mechanics
    1. 1 Units and Measurement
      1. Introduction
      2. 1.1 The Scope and Scale of Physics
      3. 1.2 Units and Standards
      4. 1.3 Unit Conversion
      5. 1.4 Dimensional Analysis
      6. 1.5 Estimates and Fermi Calculations
      7. 1.6 Significant Figures
      8. 1.7 Solving Problems in Physics
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Vectors
      1. Introduction
      2. 2.1 Scalars and Vectors
      3. 2.2 Coordinate Systems and Components of a Vector
      4. 2.3 Algebra of Vectors
      5. 2.4 Products of Vectors
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 Motion Along a Straight Line
      1. Introduction
      2. 3.1 Position, Displacement, and Average Velocity
      3. 3.2 Instantaneous Velocity and Speed
      4. 3.3 Average and Instantaneous Acceleration
      5. 3.4 Motion with Constant Acceleration
      6. 3.5 Free Fall
      7. 3.6 Finding Velocity and Displacement from Acceleration
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Motion in Two and Three Dimensions
      1. Introduction
      2. 4.1 Displacement and Velocity Vectors
      3. 4.2 Acceleration Vector
      4. 4.3 Projectile Motion
      5. 4.4 Uniform Circular Motion
      6. 4.5 Relative Motion in One and Two Dimensions
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 5 Newton's Laws of Motion
      1. Introduction
      2. 5.1 Forces
      3. 5.2 Newton's First Law
      4. 5.3 Newton's Second Law
      5. 5.4 Mass and Weight
      6. 5.5 Newton’s Third Law
      7. 5.6 Common Forces
      8. 5.7 Drawing Free-Body Diagrams
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 6 Applications of Newton's Laws
      1. Introduction
      2. 6.1 Solving Problems with Newton’s Laws
      3. 6.2 Friction
      4. 6.3 Centripetal Force
      5. 6.4 Drag Force and Terminal Speed
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 7 Work and Kinetic Energy
      1. Introduction
      2. 7.1 Work
      3. 7.2 Kinetic Energy
      4. 7.3 Work-Energy Theorem
      5. 7.4 Power
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 8 Potential Energy and Conservation of Energy
      1. Introduction
      2. 8.1 Potential Energy of a System
      3. 8.2 Conservative and Non-Conservative Forces
      4. 8.3 Conservation of Energy
      5. 8.4 Potential Energy Diagrams and Stability
      6. 8.5 Sources of Energy
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    9. 9 Linear Momentum and Collisions
      1. Introduction
      2. 9.1 Linear Momentum
      3. 9.2 Impulse and Collisions
      4. 9.3 Conservation of Linear Momentum
      5. 9.4 Types of Collisions
      6. 9.5 Collisions in Multiple Dimensions
      7. 9.6 Center of Mass
      8. 9.7 Rocket Propulsion
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 10 Fixed-Axis Rotation
      1. Introduction
      2. 10.1 Rotational Variables
      3. 10.2 Rotation with Constant Angular Acceleration
      4. 10.3 Relating Angular and Translational Quantities
      5. 10.4 Moment of Inertia and Rotational Kinetic Energy
      6. 10.5 Calculating Moments of Inertia
      7. 10.6 Torque
      8. 10.7 Newton’s Second Law for Rotation
      9. 10.8 Work and Power for Rotational Motion
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 11 Angular Momentum
      1. Introduction
      2. 11.1 Rolling Motion
      3. 11.2 Angular Momentum
      4. 11.3 Conservation of Angular Momentum
      5. 11.4 Precession of a Gyroscope
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 12 Static Equilibrium and Elasticity
      1. Introduction
      2. 12.1 Conditions for Static Equilibrium
      3. 12.2 Examples of Static Equilibrium
      4. 12.3 Stress, Strain, and Elastic Modulus
      5. 12.4 Elasticity and Plasticity
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    13. 13 Gravitation
      1. Introduction
      2. 13.1 Newton's Law of Universal Gravitation
      3. 13.2 Gravitation Near Earth's Surface
      4. 13.3 Gravitational Potential Energy and Total Energy
      5. 13.4 Satellite Orbits and Energy
      6. 13.5 Kepler's Laws of Planetary Motion
      7. 13.6 Tidal Forces
      8. 13.7 Einstein's Theory of Gravity
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    14. 14 Fluid Mechanics
      1. Introduction
      2. 14.1 Fluids, Density, and Pressure
      3. 14.2 Measuring Pressure
      4. 14.3 Pascal's Principle and Hydraulics
      5. 14.4 Archimedes’ Principle and Buoyancy
      6. 14.5 Fluid Dynamics
      7. 14.6 Bernoulli’s Equation
      8. 14.7 Viscosity and Turbulence
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Waves and Acoustics
    1. 15 Oscillations
      1. Introduction
      2. 15.1 Simple Harmonic Motion
      3. 15.2 Energy in Simple Harmonic Motion
      4. 15.3 Comparing Simple Harmonic Motion and Circular Motion
      5. 15.4 Pendulums
      6. 15.5 Damped Oscillations
      7. 15.6 Forced Oscillations
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 16 Waves
      1. Introduction
      2. 16.1 Traveling Waves
      3. 16.2 Mathematics of Waves
      4. 16.3 Wave Speed on a Stretched String
      5. 16.4 Energy and Power of a Wave
      6. 16.5 Interference of Waves
      7. 16.6 Standing Waves and Resonance
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 17 Sound
      1. Introduction
      2. 17.1 Sound Waves
      3. 17.2 Speed of Sound
      4. 17.3 Sound Intensity
      5. 17.4 Normal Modes of a Standing Sound Wave
      6. 17.5 Sources of Musical Sound
      7. 17.6 Beats
      8. 17.7 The Doppler Effect
      9. 17.8 Shock Waves
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
    17. Chapter 17
  12. Index

Check Your Understanding

16.1

The wavelength of the waves depends on the frequency and the velocity of the wave. The frequency of the sound wave is equal to the frequency of the wave on the string. The wavelengths of the sound waves and the waves on the string are equal only if the velocities of the waves are the same, which is not always the case. If the speed of the sound wave is different from the speed of the wave on the string, the wavelengths are different. This velocity of sound waves will be discussed in Sound.

16.2

In a transverse wave, the wave may move at a constant propagation velocity through the medium, but the medium oscillates perpendicular to the motion of the wave. If the wave moves in the positive x-direction, the medium oscillates up and down in the y-direction. The velocity of the medium is therefore not constant, but the medium’s velocity and acceleration are similar to that of the simple harmonic motion of a mass on a spring.

16.3

Yes, a cosine function is equal to a sine function with a phase shift, and either function can be used in a wave function. Which function is more convenient to use depends on the initial conditions. In Figure 16.11, the wave has an initial height of y(0.00,0.00)=0y(0.00,0.00)=0 and then the wave height increases to the maximum height at the crest. If the initial height at the initial time was equal to the amplitude of the wave y(0.00,0.00)=+A,y(0.00,0.00)=+A, then it might be more convenient to model the wave with a cosine function.

16.4

This wave, with amplitude A=0.5m,A=0.5m, wavelength λ=10.00m,λ=10.00m, period T=0.50s,T=0.50s, is a solution to the wave equation with a wave velocity v=20.00m/s.v=20.00m/s.

16.5

Since the speed of a wave on a taunt string is proportional to the square root of the tension divided by the linear density, the wave speed would increase by 2.2.

16.6

At first glance, the time-averaged power of a sinusoidal wave on a string may look proportional to the linear density of the string because P=12μA2ω2v;P=12μA2ω2v; however, the speed of the wave depends on the linear density. Replacing the wave speed with FTμFTμ shows that the power is proportional to the square root of tension and proportional to the square root of the linear mass density:
P=12μA2ω2v=12μA2ω2FTμ=12A2ω2μFT.P=12μA2ω2v=12μA2ω2FTμ=12A2ω2μFT.

16.7

Yes, the equations would work equally well for symmetric boundary conditions of a medium free to oscillate on each end where there was an antinode on each end. The normal modes of the first three modes are shown below. The dotted line shows the equilibrium position of the medium.

Three figures of a string of length L are shown. Each has two waves. The first one has 1 node. It is labeled lambda 1 = 2 by 1 times L, f1 = vw by lambda 1 = vw by 2L. The second figure has 2 nodes. It is labeled lambda 2 = 2 by 2 times L, f2 = vw by lambda 2 = vw by L. The third figure has three nodes. It is labeled lambda 3 = 2 by 3 times L, f3 = vw by lambda 3 equal to 3 times vw by 2L.

Note that the first mode is two quarters, or one half, of a wavelength. The second mode is one quarter of a wavelength, followed by one half of a wavelength, followed by one quarter of a wavelength, or one full wavelength. The third mode is one and a half wavelengths. These are the same result as the string with a node on each end. The equations for symmetrical boundary conditions work equally well for fixed boundary conditions and free boundary conditions. These results will be revisited in the next chapter when discussing sound wave in an open tube.

Conceptual Questions

1.

A wave on a guitar string is an example of a transverse wave. The disturbance of the string moves perpendicular to the propagation of the wave. The sound produced by the string is a longitudinal wave where the disturbance of the air moves parallel to the propagation of the wave.

3.

Propagation speed is the speed of the wave propagating through the medium. If the wave speed is constant, the speed can be found by v=λT=λf.v=λT=λf. The frequency is the number of wave that pass a point per unit time. The wavelength is directly proportional to the wave speed and inversely proportional to the frequency.

5.

No, the distance you move your hand up and down will determine the amplitude of the wave. The wavelength will depend on the frequency you move your hand up and down, and the speed of the wave through the spring.

7.

Light from the Sun and stars reach Earth through empty space where there is no medium present.

9.

The wavelength is equal to the velocity of the wave times the frequency and the wave number is equal to k=2πλ,k=2πλ, so yes, the wave number will depend on the frequency and also depend on the velocity of the wave propagating through the spring.

11.

The medium moves in simple harmonic motion as the wave propagates through the medium, continuously changing speed, therefore it accelerates. The acceleration of the medium is due to the restoring force of the medium, which acts in the opposite direction of the displacement.

13.

The wave speed is proportional to the square root of the tension, so the speed is doubled.

15.

Since the speed of a wave on a string is inversely proportional to the square root of the linear mass density, the speed would be higher in the low linear mass density of the string.

17.

The tension in the wire is due to the weight of the electrical power cable.

19.

The time averaged power is P=EλT=12μA2ω2λT=12μA2ω2v.P=EλT=12μA2ω2λT=12μA2ω2v. If the frequency or amplitude is halved, the power decreases by a factor of 4.

21.

As a portion on the string moves vertically, it exerts a force on the neighboring portion of the string, doing work on the portion and transferring the energy.

23.

The intensity of a spherical wave is I=P4πr2,I=P4πr2, if no energy is dissipated the intensity will decrease by a factor of nine at three meters.

25.

At the interface, the incident pulse produces a reflected pulse and a transmitted pulse. The reflected pulse would be out of phase with respect to the incident pulse, and would move at the same propagation speed as the incident pulse, but would move in the opposite direction. The transmitted pulse would travel in the same direction as the incident pulse, but at half the speed. The transmitted pulse would be in phase with the incident pulse. Both the reflected pulse and the transmitted pulse would have amplitudes less than the amplitude of the incident pulse.

27.
Figure shows the graph of a wave. Its y value is 0 at x=0. At x=3, the y value rises to 4 and stays constant till x=5. Here, it rises to 5 and stays constant till x=7. Here, it dips to 2 and stays constant till x=8. Here, it dips to -3 and stays constant till x=9. Here, it rises to 0 and stays constant.
29.

It may be as easy as changing the length and/or the density a small amount so that the parts do not resonate at the frequency of the motor.

31.

Energy is supplied to the glass by the work done by the force of your finger on the glass. When supplied at the right frequency, standing waves form. The glass resonates and the vibrations produce sound.

33.

For the equation y(x,t)=4.00cmsin(3m−1x)cos(4s−1t),y(x,t)=4.00cmsin(3m−1x)cos(4s−1t), there is a node because when x=0.00mx=0.00m, sin(3m−1(0.00m))=0.00,sin(3m−1(0.00m))=0.00, so y(0.00m,t)=0.00my(0.00m,t)=0.00m for all time. For the equation y(x,t)=4.00cmsin(3m−1x+π2)cos(4s−1t),y(x,t)=4.00cmsin(3m−1x+π2)cos(4s−1t), there is an antinode because when x=0.00mx=0.00m, sin(3m−1(0.00m)+π2)=+1.00sin(3m−1(0.00m)+π2)=+1.00, so y(0.00m,t)y(0.00m,t) oscillates between +A and −A as the cosine term oscillates between +1 and -1.

Problems

35.

2d=vtd=11.25m2d=vtd=11.25m

37.

v=fλ,so thatf=0.125Hz, so thatN=7.50timesv=fλ,so thatf=0.125Hz, so thatN=7.50times

39.

v=fλλ=0.400mv=fλλ=0.400m

41.

v=fλf=2.50×109Hzv=fλf=2.50×109Hz

43.

a. The P-waves outrun the S-waves by a speed of v=3.20km/s;v=3.20km/s; therefore, Δd=0.320km.Δd=0.320km. b. Since the uncertainty in the distance is less than a kilometer, our answer to part (a) does not seem to limit the detection of nuclear bomb detonations. However, if the velocities are uncertain, then the uncertainty in the distance would increase and could then make it difficult to identify the source of the seismic waves.

45.

v=1900m/sΔt=1.05μsv=1900m/sΔt=1.05μs

47.

y(x,t)=−0.037cmy(x,t)=−0.037cm

49.
Figure shows two pulse waves. Both have y values varying from 0 to 1. The first wave, shown as a dotted line is marked t=0 seconds. The crest of the wave is at x=0. The second wave, shown as a solid line is marked t= 3 seconds. The crest of the wave is at x=6.


The pulse will move Δx=6.00mΔx=6.00m.

51.

a. A=0.25m;A=0.25m; b. k=0.30m−1;k=0.30m−1; c. ω=0.90s−1;ω=0.90s−1; d. v=3.0m/s;v=3.0m/s; e. ϕ=π/3rad;ϕ=π/3rad; f. λ=20.93mλ=20.93m; g. T=6.98sT=6.98s

53.

A=0.30m,λ=4.50m,v=18.00m/s,f=4.00Hz,T=0.25sA=0.30m,λ=4.50m,v=18.00m/s,f=4.00Hz,T=0.25s

55.

y(x,t)=0.23msin(3.49m−1x0.63s−1t)y(x,t)=0.23msin(3.49m−1x0.63s−1t)

57.

They have the same angular frequency, frequency, and period. They are traveling in opposite directions and y2(x,t)y2(x,t) has twice the wavelength as y1(x,t)y1(x,t) and is moving at half the wave speed.

59.

Each particle of the medium moves a distance of 4A each period. The period can be found by dividing the velocity by the wavelength: t=10.42st=10.42s

61.

a. μ=0.040kg/m;μ=0.040kg/m; b. v=15.75m/sv=15.75m/s

63.

v=180m/sv=180m/s

65.

v=547.723m/s,Δt=5.48msv=547.723m/s,Δt=5.48ms

67.

0.707

69.

v1t+v2t=2.00m,t=1.69msv1t+v2t=2.00m,t=1.69ms

71.

v=288.68m/s,λ=0.73mv=288.68m/s,λ=0.73m

73.

a. A=0.0125cm;A=0.0125cm; b. FT=0.96NFT=0.96N

75.

v=74.54m/s,v=74.54m/s, Pλ=91.85WPλ=91.85W

77.

a. I=20.0W/m2;I=20.0W/m2; b. I=PA,A=10.0m2A=4πr2,r=0.892mI=PA,A=10.0m2A=4πr2,r=0.892m

79.

I=650W/m2I=650W/m2

81.

PEIX2P2P1=(X2X1)2P2=2.50kWPEIX2P2P1=(X2X1)2P2=2.50kW

83.

IX2I1I2=(X1X2)2I2=3.38×10−5W/m2IX2I1I2=(X1X2)2I2=3.38×10−5W/m2

85.

f=100.00Hz,A=1.10cmf=100.00Hz,A=1.10cm

87.

a. I2=0.063I1I2=0.063I1; b. I14πr12=I24πr22r2=3.16mI14πr12=I24πr22r2=3.16m

89.

2πr1A12=2πr2A22,A1=(r2r1)1/2A1=0.17m2πr1A12=2πr2A22,A1=(r2r1)1/2A1=0.17m

91.

y(x,t)=0.63my(x,t)=0.63m

93.

AR=2Acos(ϕ2),ϕ=1.17radAR=2Acos(ϕ2),ϕ=1.17rad

95.

yR=1.90cmyR=1.90cm

97.

ω=6.28s−1,k=3.00m−1,ϕ=π8rad,AR=2Acos(ϕ2),A=0.37mω=6.28s−1,k=3.00m−1,ϕ=π8rad,AR=2Acos(ϕ2),A=0.37m

99.

a.

Figure shows a blue wave labeled y1, a red wave labeled y2 and a black wave labeled yR on the same graph. The red and the blue waves have the same wavelength and amplitude, but are out of phase. The black wave has the same wavelength as the other two, but is greater in amplitude.

;
b. λ=2.0m,A=4mλ=2.0m,A=4m; c. λR=2.0m,AR=6.93mλR=2.0m,AR=6.93m

101.

yR(x,t)=2Acos(ϕ2)cos(kxωt+ϕ2);yR(x,t)=2Acos(ϕ2)cos(kxωt+ϕ2); The result is not surprising because cos(θ)=sin(θ+π2).cos(θ)=sin(θ+π2).

103.

λn=2.00nL,fn=vλnλ1=4.00m,f1=12.5Hzλ2=2.00m,f2=25.00Hzλ3=1.33m,f3=37.59Hzλn=2.00nL,fn=vλnλ1=4.00m,f1=12.5Hzλ2=2.00m,f2=25.00Hzλ3=1.33m,f3=37.59Hz

105.

v=158.11m/s,λ=4.44m,f=35.61Hzλs=9.63mv=158.11m/s,λ=4.44m,f=35.61Hzλs=9.63m

107.

y(x,t)=[0.60cmsin(3m−1x)]cos(4s−1t)y(x,t)=[0.60cmsin(3m−1x)]cos(4s−1t)

109.

λ100=0.06mv=56.8m/s,fn=nf1,n=1,2,3,4,5...f100=947Hzλ100=0.06mv=56.8m/s,fn=nf1,n=1,2,3,4,5...f100=947Hz

111.

T=2Δt,v=λT,λ=2.12mT=2Δt,v=λT,λ=2.12m

113.

λ1=6.00m,λ2=3.00m,λ3=2.00m,λ4=1.50mv=258.20m/s=λff1=43.03Hz,f2=86.07Hz,f3=129.10Hz,f4=172.13Hzλ1=6.00m,λ2=3.00m,λ3=2.00m,λ4=1.50mv=258.20m/s=λff1=43.03Hz,f2=86.07Hz,f3=129.10Hz,f4=172.13Hz

115.

v=134.16ms,λ=1.4m,f=95.83Hz,T=0.0104sv=134.16ms,λ=1.4m,f=95.83Hz,T=0.0104s

Additional Problems

117.

λ=0.10mλ=0.10m

119.

a. f=4.74×1014Hz;f=4.74×1014Hz; b. λ=422nmλ=422nm

121.

λ=16.00m,f=0.10Hz,T=10.00s,v=1.6m/sλ=16.00m,f=0.10Hz,T=10.00s,v=1.6m/s

123.

λ=(vb+v)tb,v=3.75m/s,λ=3.00mλ=(vb+v)tb,v=3.75m/s,λ=3.00m

125.

2(y1+y2)t2=Aω2sin(kxωt)Aω2sin(kxωt+ϕ)2(y1+y2)x2=Ak2sin(kxωt)Ak2sin(kxωt+ϕ)2y(x,t)x2=1v22y(x,t)t2Aω2sin(kxωt)Aω2sin(kxωt+ϕ)=(1v2)(Ak2sin(kxωt)Ak2sin(kxωt+ϕ))v=ωk2(y1+y2)t2=Aω2sin(kxωt)Aω2sin(kxωt+ϕ)2(y1+y2)x2=Ak2sin(kxωt)Ak2sin(kxωt+ϕ)2y(x,t)x2=1v22y(x,t)t2Aω2sin(kxωt)Aω2sin(kxωt+ϕ)=(1v2)(Ak2sin(kxωt)Ak2sin(kxωt+ϕ))v=ωk

127.

y(x,t)=0.40msin(0.015m−1x+1.5s−1t)y(x,t)=0.40msin(0.015m−1x+1.5s−1t)

129.

v=223.61m/s,k=1.57m−1,ω=142.43s−1v=223.61m/s,k=1.57m−1,ω=142.43s−1

131.

P=12A2(2πf)2μFTμ=2.00×10−4kg/mP=12A2(2πf)2μFTμ=2.00×10−4kg/m

133.

P=12μA2ω2λT,μ=0.0018kg/mP=12μA2ω2λT,μ=0.0018kg/m

135.

a. AR=2Acos(ϕ2),cos(ϕ2)=1,ϕ=0,2π,4π,...AR=2Acos(ϕ2),cos(ϕ2)=1,ϕ=0,2π,4π,...; b. AR=2Acos(ϕ2),cos(ϕ2)=0,ϕ=0,π,3π,5π...AR=2Acos(ϕ2),cos(ϕ2)=0,ϕ=0,π,3π,5π...

137.

yR(x,t)=0.6msin(4m−1x)cos(3s−1t)yR(x,t)=0.6msin(4m−1x)cos(3s−1t)

139.

a. (1)FT20.00kg(9.80m/s2)cos45°=0(2)m(9.80m/s2)FT=0m=14.14kg(1)FT20.00kg(9.80m/s2)cos45°=0(2)m(9.80m/s2)FT=0m=14.14kg; b. FT=138.57Nv=74.45m/sFT=138.57Nv=74.45m/s

141.

FT=12N,v=16.49m/sFT=12N,v=16.49m/s

143.

a. fn=nv2L,v=2Lfn+1n+1,n+1n=2Lfn+12Lfn,1+1n=1.2,n=5λn=2nL,λ5=1.6m,λ6=1.33mfn=nv2L,v=2Lfn+1n+1,n+1n=2Lfn+12Lfn,1+1n=1.2,n=5λn=2nL,λ5=1.6m,λ6=1.33m; b. FT=245.76NFT=245.76N

Challenge Problems

145.

a. Moves in the negative x direction at a propagation speed of v=2.00m/sv=2.00m/s. b. Δx=−6.00m;Δx=−6.00m; c.

Figure shows a graph labeled wave function versus time. Two identical pulse waves are shown on the graph. The red wave, labeled y parentheses x, 3, peaks at x = -6 m. The blue wave, labeled y parentheses x, 0, peaks at x = 0 m. The distance between the two peaks is labeled delta x = -6 m.
147.

sin(kxωt)=sin(kx+ϕ2)cos(ωt+ϕ2)cos(kx+ϕ2)sin(ωt+ϕ2)sin(kxωt+ϕ)=sin(kx+ϕ2)cos(ωt+ϕ2)+cos(kx+ϕ2)sin(ωt+ϕ2)sin(kxωt)+sin(kx+ωt+ϕ)=2sin(kx+ϕ2)cos(ωt+ϕ2)yR=2Asin(kx+ϕ2)cos(ωt+ϕ2)sin(kxωt)=sin(kx+ϕ2)cos(ωt+ϕ2)cos(kx+ϕ2)sin(ωt+ϕ2)sin(kxωt+ϕ)=sin(kx+ϕ2)cos(ωt+ϕ2)+cos(kx+ϕ2)sin(ωt+ϕ2)sin(kxωt)+sin(kx+ωt+ϕ)=2sin(kx+ϕ2)cos(ωt+ϕ2)yR=2Asin(kx+ϕ2)cos(ωt+ϕ2)

149.

sin(kx+ϕ2)=0,kx+ϕ2=0,π,2π,1.26m−1x+π20=π,2π,3πx=2.37m,4.86m,7.35msin(kx+ϕ2)=0,kx+ϕ2=0,π,2π,1.26m−1x+π20=π,2π,3πx=2.37m,4.86m,7.35m;

Figure shows a graph with wave y1 in blue, wave y2 in red and wave y1 plus y2 in black. All three have a wavelength of 5 m. Waves y1 and y2 have the same amplitude and are slightly out of phase with each other. The amplitude of the black wave is almost twice that of the other two.
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