Physics 3 (SCI 32201 - SSRU

Demonstration School of

Suan Sunandha Rajabhat University

Physics 3 (SCI 32201)

Mathayom 5/1

1st semester of academic year 2022

Ariyaphol Jiwalak

Preface

The physics book that you are reading now is my attempt to refine the content from the standardized textbooks to suit Thai students. However, there are still some mistakes. And I welcome suggestions from students to improve the books for students in the next generation.

I would like students to realize that physics is a subject that provides a basic understanding of various fields of science. Physics can explain the nature and teaches us to think systematically and logically. Understanding physics is very useful for understanding things that occurs around us. I wrote this book, not for anyone to become a physicist. But I wrote for everyone who studies this course to have some basic physics knowledge and have good experience and good memories in physics.

Ariyaphol Jiwalak 18 February 2022

Content

Page

Chapter 1: Simple harmonic motion 1 1. An object attached to a spring 2 2. Simple pendulum 13 Assignment 17

Chapter 2: Wave 51 1. Types of waves 52 2. Description of waves 53 3. Speed of waves 55 4. Huygens’ principle and Principle of superposition 60 5. Reflection, Refraction, Interference, and Diffraction of waves 62 Assignment 72

Chapter 3: Optics 95 1. Double-slit interference 96 2. Single-slit diffraction 100 3. Diffraction gratings 102 4. Plane mirrors 106 5. Concave mirrors, Convex mirrors, Converging lens, and Diverging lens 108 6. Defects of vision 126 7. Refraction of light 127 8. Light color and Pigment color 139 Assignment 144

Chapter 4: Sound 177 1. Speed of sound waves 179 2. Reflection, Refraction, Interference, Diffraction of sound waves 180 3. Power of a sound, Sound intensity, and Sound level 182 4. Beats 189 5. Resonance 190 6. Shock waves 198 7. Doppler effect 201 Assignment 208

Credit 1.0 Total class hours 40 hours (20 weeks) Classroom Mathayom 5/1 Teacher Ariyaphol Jiwalak Contact Tel: 089-7525223 E-mail: [email protected] Website: http://elsd.ssru.ac.th/ariyaphol_ji Teaching Schedule

Week Topics Elective Physics

Secondary 5 (page number)

Time (hours)

CHAPTER 1: Simple Harmonic Motion - 6 1-2 Course Introduction, An object attached to a spring

3 Simple pendulum CHAPTER 2: Wave

72-84 8 4 Types of waves, Description of waves, Speed of waves 5 Huygens’ principle and Principle of superposition 6 Reflection and Refraction of waves 7 Interference and Diffraction of waves CHAPTER 3: Optics

- 4 8 Double-slit interference 9 Single-slit diffraction, Diffraction gratings 10 Midterm Exam 2

11-12 Mirrors and Lens, Defect of vision - 8 13 Refraction of light

14 Light color and Pigment color

CHAPTER 4: Sound

86-108 10

15 Speed of sound waves

16 Reflection, Refraction, Interference and Diffraction of sound waves 17 Power of a sound, Sound intensity, and Sound level 18 Beats, Resonance 19 Shock waves, Doppler effect

20 Final Exam 2 Total 40

Course Outline for Physics 3 (SCI 32201) 1st Semester, Academic Year 2022

For Grade 11, Demonstration School of Suan Sunandha Rajabhat University

Evaluation criteria Attendance 10% Assignment 40% Quiz 10% Midterm exam 20%

Final exam 20%

Physics 3 (SCI 32201) Chapter 1: Simple Harmonic Motion ___________________________________________________________________________________________

Ariyaphol Jiwalak | Demonstration School of Suan Sunandha Rajabhat University © 2022 1

7สียงแล

1. An object attached to a spring

- Horizontal SHM - Vertical SHM

2. Simple pendulum

Chapter 1: Simple Harmonic Motion (SHM)



1. An object attached to a spring .

Horizontal SHM.

x⃗

v⃗

a

F⃗

Ek

Ep

k

Equilibrium position

natural length

F⃗ ext

m

x⃗

https://www.google.co.th/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwj50vTcmprXAhVIgbwKHWHICwEQjRwIBw&url=https://www.sciencebuddies.org/science-fair-projects/references/linear-nonlinear-springs-tutorial&psig=AOvVaw1mWdhGC1UUXz0M0cKsJjes&ust=1509516824530921








x (m)

t (s)

v (m/s)

t (s)

a (m/s2)

t (s)

• Displacement, Velocity, and Acceleration in SHM

Displacement x (t) = A sin (ωt + ∅)

where A = amplitude (maximum displacement) ω = angular frequency (rad/s)

ω =√k

m

∅ = initial phase

Velocity v (t) = ωA cos (ωt + ∅)

v (x) = ± ω√A2 – x2

Acceleration a (t) = – ω2A sin (ωt + ∅)

a (x) = – ω2x

The acceleration a is directly proportional to the displacement x and always has the opposite sign.

• Period and Frequency Period T = 2π√m

k

Frequency f = 1

2π√

k

m

xmax

vmax

amax

phase



Derivation Angular frequency ω =√k

m

Derivation Period T = 2π√m

k

Derivation Velocity as a function of Displacement v (x) = ± ω√A2 – x2



Example 1.1: A 200-g block connected to a light spring for which the force constant is 5.00 N/m is free to oscillate on a frictionless, horizontal surface. The block is displaced 5.00 cm from equilibrium and released from rest as in figure below.

a) Find the period of its motion.

b) Determine the maximum speed of the block.

c) What is the maximum acceleration of the block? d) Express the position, velocity, and acceleration as functions of time.

Example 1.2: Consider again the block-spring system in Example 1.1.

a) Find the time at which the block is located at x = + 1

2 A.

b) Suppose we measure the speed of the block at x = + 1

2 A. Will the speed of the block

be half the maximum speed?

u⃗



Example 1.3: A spring is mounted horizontally, with its left end fixed. A spring balance attached to the free end and pulled toward the right indicates that the stretching force is proportional to the displacement, and a force of 6.0 N causes a displacement of 0.030 m. We replace the spring balance with a 0.50-kg glider, pull it 0.020 m to the right along a frictionless air track, and release it from rest. Find

a) the force constant k of the spring.

b) the angular frequency ω, frequency f, and period T of the resulting oscillation.

c) the maximum velocity.

d) the maximum acceleration.

e) the velocity and acceleration when the glider is halfway from its initial position to the

equilibrium position.

f) the total energy, potential energy, and kinetic energy when the glider is halfway from its initial position to the equilibrium position.



Example 1.4: A 0.500-kg cart connected to a light spring for which the force constant is 20.0 N/m oscillates on a frictionless, horizontal air track. a) Calculate the maximum speed of the cart if the amplitude of the motion is 3.00 cm. b) What is the velocity of the cart when the position is 2.00 cm? c) Compute the kinetic and potential energies of the system when the position of the

cart is 2.00 cm.

___________________________________________________________________________________________ Additional Problem 1.4: A block of mass m = 0.2 kg oscillates on a spring whose force constant k is 500 N/m. The amplitude of the oscillations is 4.0 cm. Calculate the maximum speed of the block.



Example 1.5: A 2.00-kg block is attached to a massless spring that has a force constant of k = 25.0 N/m. The spring is stretched 0.400 m from its equilibrium position and released from rest. Find a) the total energy of the system. b) the frequency of oscillation.



Example 1.6: A block oscillating on the end of a spring moves from its position of maximum spring stretch to maximum spring compression in 0.25 s. Determine the period and frequency of this motion.

Example 1.7: A student observing an oscillating block counts 45 cycles of oscillation in one minute.

Determine its frequency (in hertz) and period (in seconds). Example 1.8: A block is attached to a spring and set into oscillatory motion, and its frequency is

measured. If this block were removed and replaced by a second block with 1/4 the mass of the first block, how would the frequency of the oscillations compare to that of the first block?

Example 1.9: A block of mass m = 2.0 kg is attached to an ideal spring of force constant k = 200 N/m.

The block is at rest at its equilibrium position. An impulsive force acts on the block, giving it an initial speed of 2.0 m/s. Find the amplitude of the resulting oscillations.



equilibrium position

natu

ral l

ength

k

natu

ral l

ength

Vertical SHM.

m

x1

F⃗ ext

F⃗ ext

x1

k

x2

x2

m

m m

equilibrium position









Example 1.10: A block of mass m = 1.5 kg is attached to the end of a vertical spring of force constant k = 300 N/m. After the block comes to rest, it is pulled down a distance of 2.0 cm and released. a) How far does the weight of the block cause the spring to stretch initially?

b) What are the minimum and maximum amounts of stretch of the spring during the oscillations of the block?

c) At what point(s) will the speed of the block be zero?

d) At what point(s) will the acceleration of the block be zero?



Example 1.11: A 4.0-kilogram mass is suspended by a spring. In its equilibrium position, the mass has extended the spring 0.10 meters beyond its unstretched length. What is the spring constant of this spring?

Example 1.12: A 4.0-kilogram mass is suspended from an unstretched spring. When released from rest,

the mass moves a maximum distance of 0.20 meters before reversing direction. What are the period and frequency of this spring-mass oscillator?

Example 1.13: A plucked guitar string vibrates at a frequency of 100 Hertz. What is the period of vibration of the string?



2. Simple pendulum .

• Angular displacement θ (t) = θmax sin (ωt + ∅) where ω =√g

l

• Period and Frequency Period T = 2π√l

g

Frequency f = 1

2π√

g

l

m

θ l



Derivation Angular frequency ω =√g

l

Derivation Period T = 2π√l

g

m

θ l

s



Example 2.1: Find the period and frequency of a simple pendulum 1.000 m long at a location where g = 9.800 m/s2.

Example 2.2: A pendulum swings back and forth between points A and C as shown below. B is the

midpoint between points A and C.

Which of the following statements are true? _____ 1. The potential energy at A is equal to the kinetic energy at C. _____ 2. The kinetic energy at B is equal to the total energy of the pendulum. _____ 3. The pendulum has both kinetic energy and potential energy at point A. _____ 4. The potential energy is maximum at points A and C. _____ 5. The kinetic energy is maximum at point B.



Summary .

1. An object attached to a spring

Displacement x (t) = A sin (ωt + ∅)

Velocity v (t) = ωA cos (ωt + ∅)

v (x) = ± ω√A2 – x2

Acceleration a (t) = – ω2A sin (ωt + ∅)

a (x) = – ω2x

Period T = 2π√m

k

2. Simple pendulum

Angular displacement θ (t) = θmax sin (ωt + ∅)

Period T = 2π√l

g



Assignment .

1. An object attached to a spring .

Horizontal SHM.

Problem 1.1: The position of a particle of mass m = 2.0 kg is given by the expression

x = 0.1 sin (πt)

where x is in meters and t is in seconds. Determine a) the amplitude of the motion. b) the angular frequency. c) the frequency and period of the motion. d) the position of the particle at t = 1.0 s. e) the time at which the particle is located at x = 0.05 m.



f) the velocity as functions of time. g) the velocity of the particle at t = 1.0 s. h) the acceleration as functions of time. i) the acceleration of the particle at t = 1.0 s. j) the maximum velocity. k) the maximum acceleration.



l) the velocity of the particle at x = 0.05 m. m) the acceleration of the particle at x = 0.05 m. n) the total energy at the equilibrium position (x = 0 m).

Additional Problem 1.1:

The position of a particle is given by the expression

x = 4.00 cos (3.00πt+π)

where x is in meters and t is in seconds. Determine a) the frequency of the motion. b) the period of the motion. c) the amplitude of the motion. d) the position of the particle at t = 0.250 s.



Problem 1.2: The displacement of an oscillating object as a function of time is shown in Fig. below.

What are a) the frequency of this motion?

b) the amplitude of this motion?

c) the period of this motion?

d) the angular frequency of this motion?

e) the maximum speed?

f) the maximum acceleration?



Problem 1.3: A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2.00 cm, and the frequency is 1.50 Hz. a) Find an expression for the position of the particle as a function of time.

b) Determine the maximum speed of the particle the earliest time (t > 0) at which the particle has this speed.

c) Find the maximum positive acceleration of the particle and the earliest time (t > 0) at which the particle has this acceleration.

d) Find the total distance traveled by the particle between t = 0 and t = 1.00 s.



Problem 1.4: A 0.0200-kg bolt moves with SHM that has an amplitude of 0.245 m and a period of 1.485 s. The displacement of the bolt is +0.245 m when t = 0. Compute a) the displacement of the bolt when t = 0.510 s.

b) the magnitude and direction of the force acting on the bolt when t = 0.510 s.

c) the minimum time required for the bolt to move from its initial position to the point where x = -0.180 m.

d) the speed of the bolt when x = -0.180 m. Additional Problem 1.4: An object is undergoing SHM with period 1.230 s and amplitude 0.590 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 = 0.475 s?



Problem 1.5: An object is undergoing SHM with period 0.820 s and amplitude 0.320 m. At t = 0 the object is at x = 0.320 m and is instantaneously at rest. Calculate the time it takes the object to go a) from x = 0.320 m to x = 0.160 m.

b) from x = 0.160 m to x = 0 m.

Additional Problem 1.5: An object is undergoing SHM with period 0.340 s and amplitude 6.40 cm. At t = 0 the object is instantaneously at rest x = 6.40 cm. Calculate the time it takes the object to go from x = 6.40 cm to x = -1.50 cm ___________________________________________________________________________________________ Problem 1.6: A machine part is undergoing SHM with a frequency of 5.10 Hz and amplitude 1.80 cm.

How long does it take the part to go from x = 0 to x = -1.80 cm?



Problem 1.7: A particle executes simple harmonic motion with an amplitude of 3.00 cm. At what position does its speed equal half of its maximum speed?

Problem 1.8: A 0.450-kg object undergoing SHM has ax = -2.50 m/s2 when x = 0.400 m. What is the time

for one oscillation? Problem 1.9: You are watching an object that is moving in SHM. When the object is displaced 0.600 m

to the right of its equilibrium position, it has a velocity of 2.20 m/s to the right and an acceleration of 8.40 m/s2 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?



Problem 1.10: The amplitude of a system moving in simple harmonic motion is doubled. Determine the change in a) the total energy.

b) the maximum speed.

c) the maximum acceleration.

d) the period.

Problem 1.11: A simple harmonic oscillator of amplitude A has a total energy E. Determine a) the kinetic energy and the potential energy when the position is one-third the

amplitude.

b) For what values of the position does the kinetic energy equal one-half the potential energy?

c) Are there any values of the position where the kinetic energy is greater than the maximum potential energy? Explain.



Problem 1.12: A harmonic oscillator has angular frequency ω and amplitude A. a) What are the magnitudes of the displacement and velocity when the elastic potential

energy is equal to the kinetic energy?

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?



Problem 1.13: A 0.60-kg block attached to a spring with force constant 130 N/m is free to move on a frictionless, horizontal surface as in Figure below.

The block is released from rest when the spring is stretched 0.13 m. At the instant the block is released, find a) the force on the block.

b) its acceleration.



Problem 1.14: A 2.20-kg mass on a spring has displacement as a function of time given by

x (t) = (7.40 cm) cos [(4.16 rad/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 s.

f) the force on the mass at t = 1.00 s.



Problem 1.15: A 0.500-kg mass on a spring has velocity as a function of time given by

v (t) = - (3.60 cm/s) sin [(4.71 rad/s)t - (π/2)] What are a) the period?

b) the amplitude?

c) the maximum acceleration of the mass?

d) the force constant of the spring?



Problem 1.16: A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12 m/s. Find a) the amplitude.

b) the phase angle.

c) Write an equation for the position as a function of time.

Problem 1.17: A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the block has velocity -4.00 m/s and displacement +0.200 m. Find a) the amplitude.

b) the phase angle.

c) Write an equation for the position as a function of time.



Problem 1.18: An object attached to a spring vibrates with simple harmonic motion as described by Figure below.

For this motion, find a) the amplitude.

b) the period.

c) the angular frequency.

d) the maximum speed.

e) the maximum acceleration.

f) an equation for its position x as a function of time.



Additional Problem 1.18: A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure below 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?

d) amplitude?

e) What is the force constant of the spring?



Problem 1.19: A mass m is attached to a spring of force constant 75 N/m and allowed to oscillate. Figure below shows a graph of its velocity component vx as a function of time t.

a) Find the period.

b) Find the frequency.

c) Find the angular frequency of this motion.

d) What is the amplitude (in 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?



Problem 1.20: On a frictionless, horizontal air track, a glider oscillates at the end of an ideal spring of force constant 2.30 N/cm. The graph in Fig. below 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.



Problem 1.21: 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 m. The maximum speed of the block is 3.90 m/s. What is the maximum magnitude of the acceleration of the block?

Problem 1.22: 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 0.200 m and the period is 3.45 s. What are the speed and acceleration of the block when x = 0.160 m?



Problem 1.23: 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 = 0.310 m, the acceleration of the block is -5.96 m/s2. What is the frequency of the motion?

Problem 1.24: When a body of unknown mass is attached to an ideal spring with force constant 128 N/m,

it is found to vibrate with a frequency of 6.20 Hz. Find a) the period of the motion.

b) the angular frequency.

c) the mass of the body.



Problem 1.25: A 2.00-kg frictionless block is attached to an ideal spring with force constant 315 N/m undergoing simple harmonic motion. When the block has displacement +0.200 m, it is moving in the negative x-direction with a speed of 4.00 m/s. Find a) the amplitude of the motion.

b) the block’s maximum acceleration.

c) the maximum force the spring exerts on the block. Additional Problem 1.25: A 180-g glider on a horizontal, frictionless air track is attached to a fixed ideal spring with force constant 180 N/m. At the instant you make measurements on the glider, it is moving at 0.765 m/s and is 2.50 cm from its equilibrium point. Find a) the amplitude of the motion. b) the maximum speed of the glider. c) What is the angular frequency of the oscillations?



Problem 1.26: A 2.00-kg frictionless block is attached to an ideal spring with force constant 315 N/m. Initially the spring is neither stretched nor compressed, but the block is moving in the negative direction at 12.0 m/s. Find a) the amplitude of the motion.

b) the block’s maximum acceleration.

c) the maximum force the spring exerts on the block.



Problem 1.27: A block with mass m = 0.300 kg is attached to one end of an ideal spring and moves on a horizontal frictionless surface. The other end of the spring is attached to a wall. When the block is at x = +0.240 m, its acceleration is ax = -12.0 m/s2 and its velocity is vx = +4.00 m/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?

d) the maximum magnitude of the block’s acceleration during its motion?



Problem 1.28: If an object on a horizontal, frictionless surface is attached to a spring, displaced, and then released, it will oscillate. If it is displaced 0.115 m from its equilibrium position and released with zero initial speed, then after 0.815 s its displacement is found to be 0.115 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. Problem 1.29: 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 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel from x = 0.180 m to x = -0.180 m.



Problem 1.30: In a physics lab, you attach a 0.200-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 s. Find the spring’s force constant.

Problem 1.31: When a 0.800-kg mass oscillates on an ideal spring, the frequency is 1.41 Hz. What will the

frequency be if 0.260 kg are a) added to the original mass?

b) subtracted from the original mass?



Problem 1.32: A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose

force constant is 83.8 N/m as in Figure below. A horizontal force F causes the spring to stretch a distance of 5.46 cm from its equilibrium position.

a) Find the magnitude of F .

b) What is the total energy stored in the system when the spring is stretched?

c) Find the magnitude of the acceleration of the block just after the applied force is removed.

d) Find the speed of the block when it first reaches the equilibrium position.

e) If the surface is not frictionless but the block still reaches the equilibrium position, would your answer to part (d) be larger or smaller?

f) What other information would you need to know to find the actual answer to part (e) in this case?

g) What is the largest value of the coefficient friction that would allow the block to reach the equilibrium position?



Problem 1.33: A 0.500-kg glider, attached to the end of an ideal spring with force constant k = 450 N/m, undergoes SHM with an amplitude of 0.040 m. Compute a) the maximum speed of the glider.

b) the speed of the glider when it is at x = -0.015 m.

c) the magnitude of the maximum acceleration of the glider.

d) the acceleration of the glider at x = -0.015 m.

e) the total mechanical energy of the glider at any point in its motion.



Problem 1.34: A 0.155-kg toy is undergoing SHM on the end of a horizontal spring with force constant k

= 305 N/m. When the toy is 1.25 × 10-2 m from its equilibrium position, it is observed to have a speed of 0.295 m/s. What are the toy’s a) total energy at any point of its motion?

b) amplitude of the motion?

c) maximum speed during its motion?

Problem 1.35: 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?



Vertical SHM.

Problem 1.36: A 7.00-kg object is hung from the bottom end of a vertical spring fastened to an overhead beam. The object is set into vertical oscillations having a period of 2.60 s. Find the force constant of the spring.

Problem 1.37: A 40.0-N force stretches a vertical spring 0.250 m. What mass must be suspended from the

spring so that the system will oscillate with a period of 1.00 s?



Problem 1.38: A proud deep-sea fisherman hangs a 65.0-kg fish from an ideal spring having negligible mass. The fish stretches the spring 0.180 m. a) Find the force constant of the spring.

The fish is now pulled down 5.00 cm and released. b) What is the period of oscillation of the fish?

c) What is the maximum speed it will reach?



2. Simple pendulum .

Problem 2.1: A simple pendulum 2.00 m long swings through a maximum angle of 30.0o with the vertical. Calculate its period assuming a small amplitude.

Problem 2.2: In the laboratory, a student studies a pendulum by graphing the angle θ that the string

makes with the vertical as a function of time t, obtaining the graph shown in Figure below.

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?



Problem 2.3: A simple pendulum makes 120 complete oscillations in 3.00 min at a location where g = 9.80 m/s2. Find a) the period of the pendulum.

b) its length.

Problem 2.4: You pull a simple pendulum 0.255 m long to the side through an angle of 3.50o 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 1.75o instead of 3.50o?



Problem 2.5: After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 55.0 cm. She finds that the pendulum makes 105 complete swings in 135 s. What is the value of g on this planet?

Problem 2.6: A certain simple pendulum has a period on the earth of 1.40 s. What is its period on the

surface of Mars, where g = 3.71 m/s2?



Problem 2.7: A small rock with mass 0.12 kg is fastened to a massless string with length 0.80 m to form a pendulum. The pendulum is swinging so as to make a maximum angle of 45o with the vertical. Air resistance is negligible. a) What is the speed of the rock when the string passes through the vertical position?

b) What is the tension in the string when it makes an angle of 45o with the vertical?

c) What is the tension in the string as it passes through the vertical?

Problem 2.8: A simple pendulum has a mass of 0.250 kg and a length of 1.00 m. It is displaced through an angle of 15.0o and then released. What are a) the maximum speed of the bob?

b) the maximum angular acceleration of the bob?

c) the maximum restoring force on the bob?

Physics 3 (SCI 32201) Chapter 2: Wave ___________________________________________________________________________________________


7สียงแล

1. Types of waves - Mechanical waves and Electromagnetic waves - Longitudinal waves and Transverse waves - Pulse wave and Continuous wave

2. Description of waves - Displacement-Position graph - Displacement-Time graph - Period and Frequency

3. Speed of waves - Speed, Frequency, and Wavelength - Speed of waves in a medium

Speed of waves on strings Speed of waves on the surface of the water Speed of sound waves in air

4. Huygens’ principle and Principle of superposition - Huygens’ principle - Principle of superposition

Constructive interference Destructive interference

5. Reflection, Refraction, Interference, and Diffraction of waves - Reflection

Fixed end and Free end Law of reflection

- Refraction Law of refraction

- Interference Coherent sources Standing waves

- Diffraction Barrier Broad slit and Narrow slit Double slit

Chapter 2: Wave



Wave motion represents phenomena in which a disturbance propagates through a medium. The disturbance carries energy from one point to another. But there is no matter that moves over that distance.

1. Types of waves .

Mechanical waves and Electromagnetic waves.

• Mechanical waves - Mechanical wave is a disturbance that travels through the medium. - For example, waves on a string, waves on the surface of the water, sound waves, and waves

on a spring.

• Electromagnetic waves - Electromagnetic (EM) waves do not require a medium. - The electric field and magnetic fields can sustain each other, forming an EM wave that

propagates through empty space. - For example, radio waves, microwaves, infrared, visible light, ultraviolet, x-rays, and gamma

rays. Longitudinal waves and Transverse waves.

• Longitudinal waves - The motions of the particles of the medium are the same direction that the wave travels. - For example, sound waves and waves on a spring.

• Transverse waves - The motions of the particles of the medium are perpendicular to the direction of travel of the

wave. - For example, waves on a string, and waves on a spring, and electromagnetic waves.

Pulse wave and Continuous wave.

• Pulse wave

• Continuous wave



2. Description of waves .

Displacement-Position graph.

Displacement-Time graph.

Period and Frequency.

• Period (T) Period is the number of seconds between the arrivals of two adjacent crests at a given location

in space or the time interval required for an element of the medium to undergo a complete cycle and return to the same position. The SI unit of period is the seconds (s).

• Frequency (f) Frequency is the number of crests (or troughs, or any other point on the wave) that pass a

given point in a unit time interval. The SI unit of frequency is the number of waves per second (s-1) or Hertz (Hz).

Relationship between period and frequency

f = 1

T

180o

360o

90o (crest) A

270o (trough)

λ

Position 0o

Displacement

λ

T 2T Time

Displacement



Example 2.1: Consider the following diagram of a wave

a) What is the wavelength of the wave?

b) What is the amplitude of the wave?

Example 2.2: A sinusoidal wave traveling in the positive x direction has a frequency of 8.00 Hz. Find the

period T.



3. Speed of waves .

Speed, Frequency, and Wavelength.

The wave pattern travels with constant speed v and advances a distance of one wavelength λ in a time interval of one period T.

v = s

t

So the wave speed is

v = λ

T

because f = 1

T ,

v = fλ The speed of propagation equals the product of wavelength and frequency.

Example 3.1: A sinusoidal wave traveling in the positive x direction has a wavelength of 40.0 cm, and a frequency of 8.00 Hz. Find the speed v of the wave.



Example 3.2: Consider the following diagram of a wave. If the speed of the wave is 12 m/s,

a) What is the frequency of the wave?

b) What is the period of the wave? Example 3.3: The period of a traveling wave is 0.5 s and its wavelength is 0.4 m. What are its frequency

and wave speed? Example 3.4: A traveling wave on a rope has a frequency of 2.5 Hz. If the speed of the wave is 1.5 m/s,

what are its period and wavelength? ___________________________________________________________________________________________ Additional Problem 3.4: Cousin Throckmorton holds one end of the clothesline taut and wiggles it up and down sinusoidally with frequency 2.00 Hz. The wave speed on the clothesline is v = 12.0 m/s. Find a) the period T.

b) the wavelength λ.



Speed of wave in a medium.

• Speed of waves on strings

vstring =√T

μ

where T = tension in string (N)

μ = mass per unit length (kg/m)

• Speed of waves on the surface of the water

vwater ∝ √gd

where d = depth of water (m)

• Speed of sound waves in air

vsound = 331 + 0.6T where T = air temperature (oC) For example, sound waves travel through room-temperature air with a speed of about

vsound = 331 + 0.6 (25)

vsound ≈ 350 m/s



Example 3.5: A horizontal rope with linear mass density µ = 0.5 kg/m has a tension of 50 N. The non-attached end is oscillated vertically with a frequency of 2 Hz. a) What are the speed and wavelength of the resulting wave?

b) How would you answer these questions if f were increased to 5 Hz? Example 3.6: A uniform string has a mass of 0.300 kg and a length of 6.00 m. The string passes over a

pulley and supports a 2.00-kg object (figure below). Find the speed of a pulse traveling along this string.



Example 3.7: One end of a 2.00-kg rope is tied to a support at the top of a mine shaft 80.0 m deep. The rope is stretched taut by a 20.0-kg box of rocks attached at the bottom.

a) The geologist at the bottom of the shaft signals to a colleague at the top by jerking the rope sideways. What is the speed of a transverse wave on the rope?

b) If a point on the rope is in transverse SHM with f = 2.00 Hz, how many cycles of the

wave are there in the rope’s length?



4. Huygen’s principle and Principle of superposition .

Huygens’ principle.

“Every point of a wave front may be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the wave.”



Principle of superposition.

When two waves overlap, “the actual displacement of any point on the string at any time is obtained by adding the displacement the point would have if only the first wave were present and the displacement it would have if only the second wave were present”.

• Constructive interference: The displacements of the two waves are identical, giving a large resultant displacement

• Destructive interference: The displacements of the two waves are equal and opposite and cancel

each other out.

Note that when the pulses no longer overlap, they have not been permanently affected by the interference.

Example 4.1: Two waves, one with an amplitude of 8 cm and the other with an amplitude of 3 cm,

travel in the same direction on a single string and overlap. What are the maximum and minimum amplitudes of the string while these waves overlap?



5. Reflection, Refraction, Interference, and Diffraction of waves .

Reflection.

Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. The reflected waves have the same speed, frequency, and wavelength.

• Reflection of a wave pulse at a fixed end and free end of a string

Fixed end Free end The string is rigidly attached to a The string might be tied to a light ring that slides support at one end. on a frictionless rod perpendicular to the string

The reflected pulse is inverted. The reflected pulse is NOT inverted.

• Law of reflection 1. The incident and reflected rays and the normal to the surface all lie in the same plane. 2. The angle of reflection is equal to the angle of incidence for all wavelengths and for any pair

of materials (θ1 = θ2).

v1 = v2 f1 = f2 λ1 = λ2

(The normal is a line drawn perpendicular to the surface at the point where the incident ray strikes the surface.)

When a wave strikes a smooth interface separating two transparent materials (such as air and glass

or water and glass), the wave is in general partly reflected and partly refracted (transmitted) into the second material.



Refraction.

Refraction is the change in direction of a wave passing from one medium to another. The refracted waves is changed in velocity (and wavelength) but remain the same frequency.

• Refraction of a wave pulse at a fixed end and free end of a string

Two ropes of unequal linear densities are connected, and a wave is created in the rope on the left, which propagates to the right,

- toward the interface with the lighter rope.

- toward the interface with the heavier rope.

When a wave strikes the boundary to a new medium (in this case, the heavier rope), some of the wave’s energy is reflected and some is transmitted. How do the speed and wavelength of the incident wave compare with the speed and wavelength of the transmitted wave?



• Law of refraction “The ratio of the sines of the angle of incident and the angle of refraction is equal to the ratio

of the two velocities of waves.”

sin θ1

sin θ2 =

v1

v2 =

λ1

λ2

where θ = the angle between the rays and the normal or the angle between the wave front and the boundary between two media.

v1 > v2 v1 < v2

sin θ1 > sin θ2 sin θ1 < sin θ2 f1 = f2 f1 = f2 λ1 > λ2 λ1 < λ2

Conclusions: v1 ≠ v2

f1 = f2 λ1 ≠ λ2

Example 5.1: A wave travels through a medium with velocity v, frequency f, and wavelength λ. If the

wave then enters another medium that increases the velocity to 2v, what will be the corresponding frequency and wavelength?

medium 1

medium 2

medium 1

medium 2



Interference.

• Coherent sources Two sources are said to be coherent if they have a constant phase difference and the same

frequency (and wavelength), and the same amplitude.

Coherent sources are in phase.

The key concept of double-slit interference is the ‘path difference’. The condition for interference at any point:

|S

1P - S2P| = nλ

- Constructive interference: n = 0, 1, 2, 3, … → Antinode (A) - Destructive interference: n = 0.5, 1.5, 2.5, … → Node (N)

S1 S2



S1 S2

1λ

S1 S2

2λ

S1 S2

2.5λ

S1 S2

1.5λ

S1 S2

3λ

S1 S2

3.5λ



• Standing waves A standing wave is an oscillation pattern with a stationary outline that results from the

interference of two identical waves (with the same frequence, wavelength and amplitude) traveling in opposite directions.

Nodes: Points at which the string never moves. Antinodes: Points at which the amplitude of the string motion is greatest.

The principle of superposition explains how the two waves combine to form a standing wave.

The red curves show a wave traveling to the right. The blue curves show a wave traveling to the left. The black solid curves show the combination of two waves according to the superposition principle. Because at each position of the combined wave will only oscillate up and down, and the wave pattern doesn’t appear to be moving in either direction along the string, it is called a standing wave.

t0

Displacement

Position

t1

t2

t3

t4



Coherent sources are in phase Two identical waves traveling in opposite directions forms a standing wave. Midway between

the sources is the antinode.

Single source

The superposition of the incident and reflected waves forms a standing wave - Fixed end The fixed end is a node.

For example, the string is rigidly attached to a support at one end, sound waves.

- Free end The free end is an antinode. For example, waves on the surface of the water.

- 2 loop = ___ λ

- 1 loop = ___ λ

- 0.5 loop = ___ λ

- Distance between adjacent antinodes = ___ λ - Distance between adjacent nodes = ___ λ - Distance between a node and an adjacent antinode = ___ λ



Example 5.1: A guitar string lies along the x-axis when in equilibrium. The end of the string at x = 0 is fixed. A sinusoidal wave with frequency f = 440 Hz travels along the string in the -x direction at 143 m/s. It is reflected from the fixed end, and the superposition of the incident and reflected waves forms a standing wave. Locate the nodes.



Diffraction.

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of shadow of the obstacle/aperture. The wave passes through the barrier is reduced in amplitude but remain the same wavelength and frequency.

• Barrier

• Broad slit d >> λ Narrow slit d << λ

• Double slit d << λ

Note: Waves are different from particles in which waves can interfere and diffract.

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Summary .

2. Description of waves - Displacement-Position graph - Displacement-Time graph

- Period and Frequency: f = 1

T

3. Speed of waves - Speed, Frequency, and Wavelength

v = fλ - Speed of waves in a medium

Strings: vstring =√T

μ

Surface of the water: vwater = √gd

Sound waves in air: vsound = 331 + 0.6T 4. Huygens’ principle and Principle of superposition

- Huygens’ principle “Every point of a wave front may be considered the source of secondary wavelets

that spread out in all directions with a speed equal to the speed of propagation of the wave.” - Principle of superposition

When two waves overlap, “the actual displacement of any point on the string at any time is obtained by adding the displacement the point would have if only the first wave were present and the displacement it would have if only the second wave were present”.

5. Reflection, Refraction, Interference, and Diffraction of waves - Reflection

Fixed end The reflected pulse is inverted. Free end The reflected pulse is NOT inverted. Law of reflection: The angle of reflection is equal to the angle of incidence (θ1 = θ2).

- Refraction

Law of refraction: sin θ1

sin θ2 =

v1

v2 =

λ1

λ2

- Interference |S

1P – S2P| = nλ

- Diffraction d << λ



Assignment .

3. Speed of waves .

Problem 3.1 refer to the figure below, which represents a wave propagating along a string with a speed of 320 cm/s.

a) What is the wavelength of the wave? b) What is the amplitude of the wave? c) What is the frequency of the wave? d) What is the period of the wave?



Problem 3.2: A sinusoidal wave has a wavelength of 35.0 cm, and a frequency of 12.0 Hz. Find a) the period T from the frequency.

b) the wave speed v. Additional Problem 3.2: When a particular wire is vibrating with a frequency of 4.00 Hz, a transverse wave of wavelength 60.0 cm is produced. Determine the speed of waves along the wire. ___________________________________________________________________________________________ Problem 3.3: Transverse waves on a string have wave speed 8.00 m/s and wavelength 0.320 m. Find the

frequency and period of these waves. Additional Problem 3.3: A transverse sine wave with a wavelength of 1.80 m travels along a long, horizontal, stretched string with a speed of 36 m/s. What are the frequency of the wave?



Problem 3.4: A continuous succession of sinusoidal wave pulses are produced at one end of a very long string and travel along the length of the string. The wave has frequency 61.0 Hz and wavelength 0.585 m. How long does it take the wave to travel a distance of 8.50 m along the length of the string?

Problem 3.5: On December 26, 2004, a great earthquake occurred off the coast of Sumatra and triggered

immense waves (tsunami) that killed some 200,000 people. Satellites observing these waves from space measured 800 km from one wave crest to the next and a period between waves of 1.0 hour. What was the speed of these waves in m/s and in km/h.



Problem 3.6: A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 3.0 s for the boat to travel from its highest point to its lowest, a total distance of 0.69 m. The fisherman sees that the wave crests are spaced 8.0 m apart. a) How fast are the waves traveling?

b) What is the amplitude of each wave?

c) If the total vertical distance traveled by the boat were 0.35 m but the other data remained the same, how would the answers to parts (a) and (b) be affected?



Problem 3.7: A simple harmonic oscillator generates a wave on a rope. The oscillator operates at a frequency of 40.0 Hz and with an amplitude of 3.00 cm. The rope has a linear mass density of 50 g/m and is stretched with a tension of 5.00 N. a) Determine the speed of the wave.

b) Find the wavelength. Problem 3.8: With what tension must a rope with length 2.70 m and mass 0.145 kg be stretched for

transverse waves of frequency 45.0 Hz to have a wavelength of 0.770 m? Problem 3.9: Transverse waves travel with a speed of 20.0 m/s on a string under a tension of 6.00 N.

What tension is required for a wave speed of 30.0 m/s on the same string?



Problem 3.10: One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at 122 Hz. The other end passes over a pulley and

supports a 1.50-kg mass. The linear mass density of the rope is 5.7 × 10-2 kg/m. a) What is the speed of a transverse wave on the rope?

b) What is the wavelength?

c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 kg?

Problem 3.11: Tension is maintained in a string as in Figure below. The observed wave speed is v =

24.0 m/s when the suspended mass is m = 3.00 kg.

a) What is the mass per unit length of the string?

b) What is the wave speed when the suspended mass is m = 2.00 kg?



Problem 3.12: One end of a long taut string is tied to a distant pole while the other end of the string is held by a girl (see figure below). This girl quickly flicks her hand up and down to create a pulse moving towards the pole.

She now wants to produce a pulse that takes a shorter time to reach the pole. How can she do this?

a) Flick the string harder to push more force into the pulse. b) Flick the string faster to create a pulse with higher frequency. c) Flick the string further up and down to create a pulse with larger amplitude. d) Flick the string a shorter distant up and down to create a pulse with smaller

amplitude. e) Wait until the first pulse is reflected back then flick again to add the pulses

together. f) None of the above would produce a pulse that takes a shorter time to reach

the pole.

She still wants the pulse to reach the pole in a shorter time by changing the properties of the string. How can she do this?

a) She should use a lighter string, under the same tension, because the velocity increases as the density decreases.

b) She should use a heavier string, under the same tension, because the velocity increases as the density increases.

c) She should decrease the tension in the string because the velocity increases as the tension decreases.

d) None of the above would produce a pulse that takes a shorter time because the speed is determined by frequency and wavelength according to v f = .



4. Huygen’s principle and Principle of superposition .

Principle of superposition.

Problem 4.1: Two pulses are moving towards each other. Each pulse has a speed of 1 cm/s. The figure below shows the pulses at time t = 0 s. Each square width corresponds to 1 cm x 1 cm.

Draw the shape of the resultant pulse after 2 s.




Problem 4.2: Two pulses are moving in opposite directions at 1.0 cm/s on a taut string, as shown in Fig. below. Each square is 1.0 cm.

Sketch the shape of the string at the end of a) 6.0 s.

b) 7.0 s.

c) 8.0 s.

Additional Problem 4.2: Two triangular wave pulses are traveling toward each other on a stretched string as shown in Fig. below. Each pulse is identical to the other and travels at 2.00 cm/s. The leading edges of the pulses are 1.00 cm apart at t = 0.

Sketch the shape of the string at t = 0.250 s, t = 0.500 s, t = 0.750 s, t = 1.000 s, and t = 1.250 s.



Problem 4.3: Two pulses are moving towards each other. Each pulse has a speed of 1 cm/s. The figure below shows the pulses at time t = 0 s. Each square width corresponds to 1 cm x 1 cm.





Problem 4.4: Two wave pulses A and B are moving in opposite directions, each with a speed v = 2.00 cm/s. The amplitude of A is twice the amplitude of B. The pulses are shown in Figure below at t = 0. Sketch the resultant wave at t = 1.00 s, 1.50 s, 2.00 s, 2.50 s, and 3.00 s.

t = 1.00 s

t = 1.50 s t = 2.00 s

t = 2.50 s

t = 3.00 s



Problem 4.5: Suppose that the left-traveling pulse in Problem 4 is below the level of the unstretched string instead of above it. Make the same sketches that you did in that exercise.

t = 1.00 s

t = 1.50 s t = 2.00 s

t = 2.50 s

t = 3.00 s



Problem 4.6: Two pulses of different amplitudes approach each other, each having a speed of v = 1.00 m/s. Figure below shows the positions of the pulses at time t = 0.

a) Sketch the resultant wave at t = 2.00 s, 4.00 s, 5.00 s, and 6.00 s.

t = 2.00 s

t = 4.00 s

t = 5.00 s

t = 6.00 s



b) If the pulse on the right is inverted so that it is upright, how would your sketches of the resultant wave change?

t = 2.00 s

t = 4.00 s

t = 5.00 s

t = 6.00 s



Additional Problem 4.6: Figure below shows two rectangular wave pulses on a stretched string traveling toward each other. Each pulse is traveling with a speed of 1.00 mm/s and has the height and width shown in the figure. If the leading edges of the pulses are 8.00 mm apart at t = 0, sketch the shape of the string at and t = 4.00 s, t = 6.00 s, and t = 10.0 s.



5. Reflection, Refraction, Interference, and Diffraction of waves .

Reflection.

Problem 5.1: A girl is demonstrating wave motion on a string attached to a pole. The string can be either firmly attached so that the end cannot move or tied to a ring that can move loosely up and down on the pole. The girl flicks the string creating an asymmetric pulse moving towards the pole. The pulse has a speed of 1 cm/s. Each square in the figure corresponds to 1 cm x 1 cm. The figure below shows the pulse at t = 0 s.

a) Draw the shape of the resultant pulse after 4 s, assuming the string is firmly attached to the pole.

b) Draw the shape of the resultant pulse after 4 s, assuming the string is tied to a ring that can move loosely up and down on the pole.



Problem 5.2: A girl is demonstrating wave motion on a string attached to a pole. The string can be either firmly attached so that the end cannot move or tied to a ring that can move loosely up and down on the pole. The girl flicks the string creating a symmetric pulse moving towards the pole. The pulse has a speed of 1 cm/s. Each square in the figure corresponds to 1 cm x 1 cm. The figure on the right shows the pulse at t = 0 s.

a) Draw the shape of the resultant pulse after 2 s, assuming the string is firmly attached to the pole.

b) Draw the shape of the resultant pulse after 2 s, assuming the string is tied to a ring that can move loosely up and down on the pole.



Additional Problem 5.2.1: A wave pulse on a string has the dimensions shown in Fig. below at t = 0. The wave speed is 40 cm/s.

a) If point O is a fixed end, draw the total wave on the string at t = 15 ms, 20 ms, 25 ms, 30 ms, 35 ms, 40 ms, and 45 ms.

b) Repeat part (a) for the case in which point O is a free end. Additional Problem 5.2.2: A wave pulse on a string has the dimensions shown in Fig. below at t = 0. The wave speed is 5.0 m/s.

a) If point O is a fixed end, draw the total wave on the string at t = 1.0 ms, 2.0 ms, 3.0 ms, 4.0 ms, 5.0 ms, 6.0 ms, and 7.0 ms.

b) Repeat part (a) for the case in which point O is a free end.



Interference.

Problem 5.3: A 1.54-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 47.3 m/s. What are the wavelength and frequency of a) the fundamental?

b) the second overtone?

c) the fourth harmonic?

Additional Problem 5.3: A taut string has a length of 2.60 m and is fixed at both ends. a) Find the wavelength of the fundamental mode of vibration of the string. b) Can you find the frequency of this mode? Explain why or why not. ___________________________________________________________________________________________ Problem 5.4: A standing wave is established in a 1 2 0 - cm-long string fixed at both ends. The string

vibrates in four segments when driven at 120 Hz. a) Determine the wavelength.

b) What is the fundamental frequency of the string?



Problem 5.5: A string that is 30.0 cm long and has a mass per unit length of 9.00 × 10-3 kg/m is stretched to a tension of 20.0 N. Find a) the fundamental frequency

b) the next three frequencies of possible standing-wave patterns on the string. Problem 5.6: A 2.00-m-long wire having a mass of 0.100 kg is fixed at both ends. The tension in the wire

is maintained at 20.0 N. a) What are the frequencies of the first three allowed modes of vibration?

b) If a node is observed at a point 0.400 m from one end, in what mode and with what frequency is it vibrating?

Problem 5.7: A wire with mass 45.0 g is stretched so that its ends are tied down at points 78.0 cm apart.

The wire vibrates in its fundamental mode with frequency 56.0 Hz and with an amplitude at the antinodes of 0.300 cm. a) What is the speed of propagation of transverse waves in the wire?

b) Compute the tension in the wire.



Problem 5.8: A string with both ends held fixed is vibrating in its third harmonic. The waves have a speed of 192 m/s and a frequency of 240 Hz. The amplitude of the standing wave at an antinode is 0.400 cm. a) Calculate the amplitude at points on the string a distance of 20.0 cm from the left

end of the string.

b) At this point in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement?

Problem 5.9: Adjacent antinodes of a standing wave on a string are 15 . 0 cm apart. A particle at an

antinode oscillates in simple harmonic motion with amplitude 0.850 cm and period 0.0750 s. The string lies along the +x-axis and is fixed at x = 0. a) How far apart are the adjacent nodes?

b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern?

c) What is the shortest distance along the string between a node and an antinode?



Problem 5.10: In the arrangement shown in Figure below, an object of mass m = 5.00 kg hangs from a cord around a light pulley. The length of the cord between point P and the pulley is L = 2.00 m.

a) When the vibrator is set to a frequency of 150 Hz, a standing wave with six loops is formed. What must be the linear mass density of the cord?

b) How many loops (if any) will result if m is changed to 45.0 kg?



Problem 5.11: A standing wave is produced with a fixed length string, one end of which is attached to a vibrator, the other end of which is placed on a pulley and hung with a mass. Using the vibrator, the second harmonic standing wave is created (see figure below). The length between the vibrator and the pulley does not change.

If the frequency of the vibrator is doubled while everything else stays the same, a different harmonic standing wave is created. How would the wavelength of the new harmonic standing wave change?

a) Increase b) Decrease c) Stay the same

If the mass is increased by a factor of four while everything else stays the same, a

different harmonic standing wave is created. How would the wavelength of the new harmonic standing wave change?

a) Increase b) Decrease c) Stay the same

Physics 3 (SCI 32201) Chapter 3: Optics ___________________________________________________________________________________________


7สียงแล

Wave Optics

1. Double-slit interference 2. Single-slit diffraction 3. Diffraction gratings

Ray Optics

4. Plane mirrors 5. Concave mirrors, Convex mirrors, Converging lens, and Diverging lens 6. Defects of vision

- Nearsightedness - Farsightedness

7. Refraction of light - Refractive index and Law of refraction

Refractive index Law of refraction

- Critical angle and Total internal refraction Critical angle Total internal refraction

- Dispersion of light Prism Rainbows

- Real depth and Apparent depth 8. Light color and Pigment color

- Light color - Pigment color

Chapter 3: Optics



Wave Optics

1. Double-slit interference .

Note: All bright fringe has a similar width.

All bright fringe has a similar bright. The distance between any adjacent bright fringe is almost the same. The distance between any adjacent dark bright fringe is almost the same.

The key concept of double-slit interference is the ‘path difference’.

The general condition for interference:

|S1P - S2P| = nλ

- Constructive interference: n = 0, 1, 2, 3, … → bright fringe - Destructive interference: n = 0.5, 1.5, 2.5, … → dark fringe

Note: Constructive interference: Two waves are ‘in phase’ at point P.

Not just the crest - crest or the trough – trough. Destructive interference: Two waves are ‘out of phase’ at point P.

Not just the crest - trough.

N2 (n = 1.5)

d

A2 (n = 2)

N2 (n = 1.5)

A2 (n = 2)

A1 (n = 1)

A1 (n = 1)

N1 (n = 0.5)

N1 (n = 0.5)

A0 (n = 0)

x

L

θ

S2

S1



• The distance from the slits to the screen is so large in comparison to the distance between the slits (d ≪ L)

then |S1P - S2P| = d sin θ

Therefore, d sin θ = nλ

• The distances from the center of the central bright fringe to the center of the side bright fringe are often much smaller than the distance from the slits to the screen (x ≪ L). Hence the angle of line from slits to bright fringe on screen θ is very small, sin θ ≈ tan θ

then d sin θ = d tan θ = dx

L

Therefore, dx

L = nλ

Derivation |S

1P - S2P| = d sin θ = d

x

L



Example 1.1: Monochromatic light passes through two narrow slits and is projected onto a screen, creating a double-slit interference pattern. The first bright maximum occurs at a distance of 0.004 meters from the central maximum. The slit spacing is 3.0 x 10-6 meters, and the distance to the screen is 2.0 meters. Determine the wavelength of light used in this experiment.

Example 1.2: A viewing screen is separated from a double slit by 4.80 m. The distance between the two

slits is 0.0300 mm. Monochromatic light is directed toward the double slit and forms an interference pattern on the screen. The first dark fringe is 4.50 cm from the center line on the screen. a) Determine the wavelength of the light. b) Calculate the distance between adjacent bright fringes.

Example 1.3: Figure below shows a two-slit interference experiment in which the slits are 0.200 mm

apart and the screen is 1.00 m from the slits. The third bright fringe in the figure is 9.49 mm from the central fringe. Find the wavelength of the light.

L

n

n

n

n

n

n



Example 1.4: For the experimental setup we’ve been studying, assume that d = 1 mm, L = 6.0 m, and that the light used has a wavelength of 600 nm. a) How far above the center of the screen will the first bright fringe appear?

b) What would happen to the interference pattern if the slits were moved closer together?

Example 1.5: A light source emits visible light of two wavelengths: λ = 430 nm and λ' = 510 nm. The

source is used in a double-slit interference experiment in which L = 1.50 m and d = 0.0250 mm. Find the separation distance between the third-order bright fringes for the two wavelengths.



2. Single-slit diffraction .

Note: The central bright fringe is twice as wide as the other bright fringes.

The central bright fringe is brighter than the other bright fringes. The general condition for destructive interference (dark fringes):

d sin θ = nλ

dx

L = nλ

where n = 1, 2, 3, … Note: This equation provides the location of the dark fringes in the single-slit pattern,

which is similar to the equation for the locations of the bright fringes in a double-slit pattern. There is no equation for bright fringes in a single-slit pattern.

Stop to think How the interference patterns produced by several very narrow slits different from double-

slit or single-slit?

d L

N2 (n = 2)

N2 (n = 2)

N1 (n = 1)

N1 (n = 1)

θ x



Example 2.1: Light of wavelength 580 nm is incident on a slit having a width of 0.300 mm. The viewing screen is 2.00 m from the slit. Find the width of the central bright fringe. What if the slit width is increased by an order of magnitude to 3.00 mm? What happens to the diffraction pattern?

Example 2.2: You pass 633-nm laser light through a narrow slit and observe the diffraction pattern on a

screen 6.0 m away. The distance on the screen between the centers of the first minima on either side of the central bright fringe is 32 mm. How wide is the slit?

L



3. Diffraction gratings .

The diffraction grating consists of a large number of parallel slits, all with the same width and spaced equal distances d between centers.

Grating are often labeled with the number of slits per unit length. For example, a grating have about 5,000 slits per centimeter; the value of d is = __________ m.

ช่องคู่

The general condition for intensity maxima in the interference pattern:

d sin θ = nλ

dx

L = nλ

where n = 0, 1, 2, 3, …

Note: This equation provides the location of the bright fringes in the grating pattern,

which is similar to the equation for the locations of the bright fringes in a double-slit pattern. There is no equation for dark fringes in a grating pattern.

d

A2 2

A0 0

A1 1

A2 2

θ x

L d

d

A1 1

n



Derivation d sin θ = dx

L = nλ for intensity maxima

White light incident on a grating is dispersed into a spectrum.

- The central maximum is white because all wavelength interfere constructively. - __________ (short wavelength) is deviated from the central white at least. - __________ (long wavelength) is deviated from the central white the most.



Example 3.1: Monochromatic light from a helium–neon laser (λ = 632.8 nm) is incident normally on a diffraction grating containing 6,000 grooves per centimeter. Find the angles at which the first- and second-order maxima are observed.

Example 3.2: The wavelengths of the visible spectrum are approximately 380 nm (violet) to 750 nm

(red). a) Find the angular limits of the first-order visible spectrum produced by a plane grating

with 600 slits per millimeter when white light falls normally on the grating. b) Do the first-order and second-order spectra overlap? What about the second-order

and third-order spectra? Do your answers depend on the grating spacing?



Example 3.3: Light reflected from the surface of a video disc is multicolored as shown in figure below. The colors and their intensities depend on the orientation of the DVD relative to the eye and relative to the light source. Explain how that works.



Ray Optics

The light could be emitted by the object itself if it is self-luminous. Alternatively, the light could be emitted by another source (such as a lamp or the sun) and then reflected from the object.

Figure below shows light rays radiating in all directions from an object. For an observer to see this object directly, there must be the light rays from the object reach the observer’s eyes.

4. Plane mirrors .

Diverging light rays leave the source and are reflected from the mirror. Upon reflection, the rays continue to diverge. Image are located either at a point from which rays of light diverge.

The image is __________, __________, and __________. The image distance (s') always equals the object distance (s).



Example 4.1: Two mirrors are perpendicular to each other. A ray traveling in a plane perpendicular to both mirrors is reflected from one mirror at P, then the other at Q, as shown in figure below. What is the ray’s final direction relative to its original direction?

Example 4.2: Two mirrors make an angle of 120° with each other as illustrated in figure below. A ray is

incident on mirror M1 at an angle of 65° to the normal. Find the direction of the ray after it is reflected from mirror M2.

P

Q



5. Concave mirrors, Convex mirrors, Converging lens, and Diverging lens .

Reflection at a concave mirrors. Refraction at a converging lens.

principal axis

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________

Reflection at a convex mirrors. Refraction at a diverging lens.

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________

2F

C

C

F

F

2F

F

2F

F

F C C

C

F

F

F

2F

C

F

F

2F

F

2F

F

C

C

F

F

2F

2F

F

F

C

C

F

F

2F

2F

F

F

C

C

F

F

2F

2F

F

F

C

C

F

F

C

C

F

F



Image formation by concave mirrors. Image formation by converging lens.

_____________________________________ _____________________________________

The image is _________, _________, and _________. The image is _________, _________, and _________.

_____________________________________ _____________________________________


_____________________________________ _____________________________________

The image is _________, _________, and _________. The image is _________, _________, and _________. _____________________________________ _____________________________________



2F

C

C

F

F

2F

F

2F

F

F C C

C

F

F

C

F

C

2F

C

C

C

F

F

F

F

F

F

2F

2F

2F

F

F

F

2F

2F

2F

F

F

F



Image formation by convex mirrors. Image formation by diverging lens.

_____________________________________ _____________________________________






C

C

F

F

2F

2F

F

F

C

C

F

F

2F

2F

F

F

C

C

F

F

2F

2F

F

F

C

C

F

F

2F

2F

F

F

C

C

F

F

2F

2F

F

F



Conslusions Real image: - The outgoing rays don’t actually pass through the image point. - The real image is ______________.

- The real image is ______________ the lens and ______________ the mirror, - The image that are formed on a projection screen is real image.

Virtual image: - The outgoing rays really do pass through an image point. - The virtual image is ______________.

- The virtual image is ______________ the lens and ______________ the mirror.

Mirror equation and Thin lens equation.

• The object-image relationship: 1

f =

1

s +

1

s'

where s = object distance s' = image distance

• The focal length (f) of a spherical mirror: f = R

2

where R = radius of curvature

• Magnification (m) m ≡ I

O = -

s'

s

where I = image height

O = object height Sign conventions for spherical mirrors and thin lenses

Quantity Sign Keyword

f + Concave mirrors, Converging lens

- Convex mirrors, Diverging lens

s + Real object

- Virtual object

s' + Real image

- Virtual image

m + Upright

- Inverted



Derivation 1

f =

1

s +

1

s' for spherical mirrors.

Derivation 1

f =

1

s +

1

s' for thin lenses



Example 5.1: The figure below shows a concave mirror and an object (the bold arrow). Use a ray diagram to locate the image of the object.

Example 5.2: The figure below shows a convex mirror and an object (the arrow). Use a ray diagram to locate the image of the object.



Example 5.3: An object of height 4 cm is placed 30 cm in front of a concave mirror whose focal length is 10 cm. a) Where’s the image?

b) Is it real or virtual?

c) Is it upright or inverted?

d) What’s the height of the image? Example 5.4: An object of height 4 cm is placed 20 cm in front of a convex mirror whose focal length is

−30 cm. a) Where’s the image?

b) 16. Is it real or virtual? c) Is it upright or inverted?

d) What’s the height of the image?



Example 5.5: A concave mirror has a focal length of 10.0 cm. a) Locate and describe the image for an object distance of 25.0 cm. b) Locate and describe the image for an object distance of 5.0 cm.

Example 5.6: A concave mirror has a radius of curvature with absolute value 20 cm. Find graphically the

image of an object in the form of an arrow perpendicular to the axis of the mirror at object distances of ( a) 3 0 cm, (b) 2 0 cm, (c) 1 0 cm, and (d) 5 cm. Check the construction by computing the size and lateral magnification of each image.



Example 5.7: A concave mirror forms an image, on a wall 3.00 m in front of the mirror, of a headlamp filament 10.0 cm in front of the mirror. a) What are the radius of curvature and focal length of the mirror?

b) What is the lateral magnification?

c) What is the image height if the object height is 5.00 mm?

Example 5.8: A concave mirror with a focal length of 25 cm is used to create a real image that has twice the height of the object. How far is the image from the mirror?



Example 5.9: An automobile sideview mirror on the passenger side as shown in figure below shows an image of a truck located 50.0 m from the mirror. The focal length of the mirror is -0.60 m.

a) Find the position of the image of the truck. b) Find the magnification of the image.



Example 5.10: Santa checks himself for soot, using his reflection in a silvered Christmas tree ornament 0.750 m away. The diameter of the ornament is 7.20 cm. Standard reference texts state that he is a “right jolly old elf,” so we estimate his height to be 1.6 m.

a) Where is the image of Santa formed by the ornament? b) How tall is the image of Santa formed by the ornament? c) Is it erect or inverted?



Example 5.11: An object placed 60 cm in front of a spherical mirror forms a real image at a distance of 30 cm from the mirror. a) Is the mirror concave or convex?

b) What’s the mirror’s focal length?

c) Is the image taller or shorter than the object?



Example 5.12: The figure below shows a converging lens and an object (denoted by the bold arrow). Use a ray diagram to locate the image of the object.

Example 5.13: The figure below shows a diverging lens and an object. Use a ray diagram to locate the

image of the object.



Example 5.14: The object viewed by a convex lens is positioned outside of the focus, as shown in the diagram above.

Which of the following correctly describes the image? 1. No image is formed 2. Real and upright 3. Real and inverted 4. Virtual and upright 5. Virtual and inverted

Example 5.15: Which of the following is NOT true for a concave lens?

1. Concave lens are divergent. 2. The image is virtual. 3. The image is upright. 4. The image is larger than the object. 5. The image forms on the near side of the lens.



Example 5.16: An object of height 10 cm is placed 40 cm in front of a converging lens with a focal length of 20 cm. a) Where’s the image?



d) What’s the height of the image? Example 5.17: An object of height 9 cm is placed 48 cm in front of a diverging lens with a focal length of

−24 cm. a) Where’s the image?



d) What’s the height of the image?



Example 5.18: A converging lens has a focal length of 10.0 cm. a) An object is placed 30.0 cm from the lens. Construct a ray diagram, find the image

distance, and describe the image. b) An object is placed 5.0 cm from the lens. Construct a ray diagram, find the image

distance, and describe the image.

Example 5.19: Use ray diagrams to find the image position and magnification for an object at each of the following distances from a converging lens with a focal length of 20 cm: (a) 50 cm; (b) 20 cm; (c) 15 cm. Check your results by calculating the image position and lateral magnification.



Example 5.20: A diverging lens has a focal length of 10.0 cm. a) An object is placed 15.0 cm from the lens. Construct a ray diagram, find the image


b) An object is placed 5.0 cm from the lens. Construct a ray diagram, find the image


Example 5.21: A beam of parallel rays spreads out after passing through a thin diverging lens, as if the rays all came from a point 20.0 cm from the center of the lens. You want to use this lens

to form an erect, virtual image that is 1

3 the height of the object.

a) Where should the object be placed? Where will the image be?

b) Draw a principal ray diagram.

Example 5.22: A candle is placed a distance of 30 cm from lens of focal length 20 cm, forming an image

60 cm from the lens. Describe the image formed by the lens and calculate the magnification of the image.



Example 5.23: Two thin converging lenses of focal lengths f1 = 10.0 cm and f2 = 20.0 cm are separated by d = 20.0 cm as illustrated in figure below. An object is placed 30.0 cm to the left of lens 1. Find the position and the magnification of the final image.

Example 5.24: Converging lenses A and B, of focal lengths 8.0 cm and 6.0 cm, respectively, are placed

36.0 cm apart. Both lenses have the same optic axis. An object 8.0 cm high is placed 12.0 cm to the left of lens A. Find the position, size, and orientation of the image produced by the lenses in combination.



6. Defects of vision .

Nearsightedness or Myopia.

Example 6.1: The far point of a certain myopic eye is 50 cm in front of the eye. Find the focal length of

the eyeglass lens that will permit the wearer to see clearly an object at infinity. Assume that the lens is very close to the eye.

Farsightedness or Hyperopia.

Example 6.2: The near point of a certain hyperopic eye is 100 cm in front of the eye. Find the focal

length of the contact lens that will permit the wearer to see clearly an object that is 25 cm in front of the eye.



7. Refraction of light .

Refractive index and Law of refraction.

• Speed of light in vacuum (c) is approximately 3.0 × 108 m/s. The speed of light in any material is less than its speed in vacuum.

• Refractive index (n)

The refractive index (n) of a medium is defined to be the ratio of the speed of light in vacuum (c) to the speed of light in a medium (v). It is a dimensionless number without units.

n ≡

c

v

The value of the refractive index in anything other than vacuum is always greater than unity.

For vacuum, n ≈ 1. For water, n = 4

3 .

Example 7.1: The index of refraction for water is 1.3. What is the speed of light in water?



• Law of refraction 1. Snell’s law

For monochromatic light and for a given pair of materials, 1 and 2, on opposite sides of the interface, “the ratio of the sines of the angles of incident and the angle of refraction is equal to the inverse ratio of the two indexes of refraction.”

sin θ1

sin θ2=

n2

n1

or n1sin θ1 = n2sin θ2

n1v1 = n2v2

n1λ1 = n2λ2

2. The incident and refracted rays and the normal to the surface all lie in the same plane.

n1 > n2 n1 < n2

A ray entering a material of smaller index of refraction A ray entering a material of larger index of refraction bends away from the normal. bends toward the normal.

n1

n2

n1

n2



Example 7.2: A beam of light in air is incident upon a piece of glass, striking the surface at an angle of 60°. If the index of refraction of the glass is 1.5, what are the angles of reflection and refraction?

Example 7.3: In figure below, material a is water and material b is glass with index of refraction 1.52. The

incident ray makes an angle of 60.0o with the normal; find the directions of the reflected and refracted rays.



Example 7.4: A beam of light enters the flat surface of a diamond from the air at an angle of 30o from the normal. The angle of refraction in the diamond is measured to be 12o from the normal. What is the index of refraction of this diamond? Interpret your answer physically. (sin 12o = 0.2)

Example 7.5: A beam of light moving in air strikes the surface of a lake at an angle of 45°, as shown in

the diagram below.

Which statement below is true? 1. nwater < nair and θ < 45° 2. nwater = nair and θ = 45° 3. nwater > nair and θ > 45° 4. nwater < nair and θ > 45° 5. nwater > nair and θ < 45°



Example 7.6: A light ray of wavelength 589 nm traveling through air is incident on a smooth, flat slab of crown glass with index of reflection 1.52 at an angle of 30.0° to the normal. a) Find the angle of refraction. b) Find the speed of this light once it enters the glass. c) What is the wavelength of this light in the glass?

Example 7.7: The wavelength of the red light from a helium-neon laser is 633 nm in air but 474 nm in

the aqueous humor inside your eyeball. Calculate the index of refraction of the aqueous humor and the speed and frequency of the light in it.



Example 7.8: A light beam passes from medium 1 to medium 2, with the latter medium being a thick slab of material whose index of refraction is n2 (figure below). Show that the beam emerging into medium 1 from the other side is parallel to the incident beam.



Critical angle and Total internal reflection.

• Critical angle (θc) The critical angle occurs when a ray passes from one material into another material having a

smaller index of refraction. As the angle of incident θ1 is increased, the angle of refraction θ2 also become larger and the refracted ray bends away from the normal so much that it approaches a direction parallel to the interface (θ2 = 90°). The angle of incidence for which the refracted ray emerges tangent to the surface is called the ‘critical angle’ (θc).

n1 > n2

We can use Snell’s law to find the critical angle. n1sin θ1 = n2sin θ2 When θ1 = θc, θ2 = 90° and we then have n1 sin θc = n2 sin 90°

• Total internal reflection Total internal refraction occurs when the angles of incidence greater than the critical angle

(θ1 > θc). The ray cannot pass into another the material and is entirely reflected at the boundary. Note: The critical angle and total internal reflection occurs only when light is directed from a medium of

a given index of refraction toward a medium of lower index of refraction.

n1

n2



Example 7.9: When a light ray is shined from a dense medium into air and the incident angle is slowly increased, total internal reflection is first noticed when the incident angle reaches 53°. Determine the index of refraction of the initial dense medium.

Example 7.10: Find the critical angle for an air–water boundary. (Assume the index of refraction of water

is 1.33.) Example 7.11: The critical angle for total internal reflection between air and water is known to be 49°.

a) If a beam of light striking an air/water boundary undergoes total internal reflection, will it stay in the air or in the water?

b) Describe what happens if a beam of light in the air strikes the surface of a calm body of water at an angle of 50° to the normal.



Dispersion of light.

• Prism Now suppose a beam of white light (a combination of all visible wavelengths) is incident on a

prism. The rays that emerge spread out in a series of colors known as the ‘spectrum’.

All colors of light incident on the prism with the same angle of incidence, but refract with different angles of refraction, so they are disperse into a spectrum.

- __________ is deviated least. - __________ is deviated most.

Therefore, the index of refraction of a material depends on wavelength. - __________ has the most refractive index. - __________ has the smallest refractive index.

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• Rainbows A ray of sunlight (which is white light) strikes a drop of water in the atmosphere. It is first

refracted at the front surface of the drop, with the violet light deviating the most and the red light the least. At the back surface of the drop, the light is reflected and returns to the front surface, where it again undergoes refraction as it leaves the frop.



Real depth and Apparent depth.

You look straight down into a swimming pool.

The fish appears to be closer to the surface than it actually is.

n object

n eye

= real depth

apparent depth

Derivation n object

n eye =

real depth

apparent depth

n object

n eye

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https://www.google.co.th/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwjU2riv2ILaAhVFUrwKHQnqChwQjRx6BAgAEAU&url=https://www.vexels.com/png-svg/preview/143192/orange-fish-cartoon&psig=AOvVaw0j3F-WVRxrFiIHlH1--yxR&ust=1521902552707997



Example 7.12: If you look straight down into a swimming pool where it is 2.00 m deep, how deep does it appear to be?

Example 7.13: A small fish is swimming at a depth d below the surface of a pond (figure below).

a) What is the apparent depth of the fish as viewed from directly overhead?

b) If your face is a distance d above the water surface, at what apparent distance above

the surface does the fish see your face?

What if you look more carefully at the fish and measure its apparent height from its upper

fin to its lower fin? Is the apparent height h' of the fish different from the actual height h?



8. Light color and Pigment color .

Light color. Pigment color.

Concept: All the light colors other than the pigment colors are absorbed. All the light colors similar to the pigment colors are reflected.

→ White reflects all the light colors. → Black absorbs all the light colors.

Example 8.1:

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Example 8.2:





Summary .

1. Double-slit interference The general condition for interference:

|S1P - S2P| = nλ

d sin θ = nλ

dx

L = nλ

- Constructive interference: n = 0, 1, 2, 3, … → bright fringe - Destructive interference: n = 0.5, 1.5, 2.5, … → dark fringe

2. Single-slit diffraction

The general condition for destructive interference (dark fringes):

d sin θ = nλ

dx

L = nλ

3. Diffraction gratings

The general condition for intensity maxima in the interference pattern:

d sin θ = nλ

dx

L = nλ

4. Plane mirrors

The image is virsual, upright, and same size as the object. The image distance always equals the object distance.

5. Concave mirrors, Convex mirrors, Converging lens, and Diverging lens

The object-image relationship: 1

f =

1

s +

1

s'

The focal length of a spherical mirror: f = R

2

Magnification: m ≡ I

O = -

s'

s

6.



7. Refraction of light - Refractive index and Law of refraction

Refractive index: n ≡ c

v

Law of refraction: n1sin θ1 = n2sin θ2

n1v1 = n2v2

n1λ1 = n2λ2

- Critical angle and Total internal refraction Critical angle is the angle of incidence for which the refracted ray emerges tangent to

the surface. n1 sin θc = n2 sin 90°

Total internal refraction Total internal refraction occurs when the angles of incidence

greater than the critical angle - Dispersion of light

The index of refraction of a material depends on wavelength. (Violet has the most refractive index. Red has the smallest refractive index.)

- Real depth and Apparent depth The fish appears to be closer to the surface than it actually is.

n object

n eye

= real depth

apparent depth

8. Light color and Pigment color

All the light colors other than the pigment colors are absorbed. All the light colors similar to the pigment colors are reflected.



Assignment .

1. Double-slit interference .

Problem 1.1: In a Young’s double-slit experiment, two parallel slits with a slit separation of 0.100 mm are illuminated by light of wavelength 589 nm, and the interference pattern is observed on a screen located 4.00 m from the slits. a) What is the difference in path lengths from each of the slits to the location of the

center of a third-order bright fringe on the screen? b) What is the difference in path lengths from the two slits to the location of the center

of the third dark fringe away from the center of the pattern?

Problem 1.2: Light of wavelength 620 nm falls on a double slit, and the first bright fringe of the interference pattern is seen at an angle of 15.0° with the horizontal. Find the separation between the slits.

Problem 1.3: A laser beam is incident on two slits with a separation of 0.200 mm, and a screen is placed

5.00 m from the slits. An interference pattern appears on the screen. If the angle from the center fringe to the first bright fringe to the side is 0.181°, what is the wavelength of the laser light?



Problem 1.4: In the double-slit arrangement of Figure below, d = 0.150 mm, L = 140 cm, λ = 643 nm, and x = 1.80 cm.

a) What is the path difference for the rays from the two slits arriving at P? b) Express this path difference in terms of λ. c) Does P correspond to a maximum, a minimum, or an intermediate condition? Give

evidence for your answer.

x

A



Problem 1.5: Monochromatic light of wavelength λ is incident on a pair of slits separated by 2.40 × 10-4 m and forms an interference pattern on a screen placed 1.80 m from the slits. The first-order bright fringe is at a position xbright = 4.52 mm measured from the center of the central maximum. Calculate the wavelength of the light.

Problem 1.6: Young’s experiment is performed with light of wavelength 502 nm. Fringes are measured

carefully on a screen 1.20 m away from the double slit, and the center of the 2th fringe (not counting the central bright fringe) is found to be 1.06 mm from the center of the central bright fringe. What is the separation of the two slits?

Problem 1.7: Coherent light with wavelength 450 nm falls on a pair of slits. On a screen 1.80 m away,

the distance between dark fringes is 3.90 mm. What is the slit separation?



Problem 1.8: Two slits spaced 0.450 mm apart are placed 78.5 cm from a screen. What is the distance between the second and third dark lines of the interference pattern on the screen when the slits are illuminated with coherent light with a wavelength of 490 nm?

Problem 1.9: Coherent light with wavelength 400 nm passes through two very narrow slits that are

separated by 0.200 mm, and the interference pattern is observed on a screen 4.00 m from the slits. a) What is the width (in mm) of the central interference maximum? b) What is the width of the first-order bright fringe?

Problem 1.10: Two thin parallel slits that are 0.0116 mm apart are illuminated by a laser beam of

wavelength 585 nm. On a very large distant screen, what is the total number of bright fringes, including the central fringe and those on both sides of it?



2. Single-slit diffraction .

Problem 2.1: Monochromatic light from a distant source is incident on a slit 0.765 mm wide. On a screen 2.05 m away, the distance from the central maximum of the diffraction pattern to the first minimum is measured to be 1.51 mm. Calculate the wavelength of the light.

Problem 2.2: Monochromatic electromagnetic radiation with wavelength λ from a distant source passes

through a slit. The diffraction pattern is observed on a screen 2.50 m from the slit. If the width of the central maximum is 6.00 mm, what is the slit width if the wavelength is 500 nm (visible light).



Problem 2.3: Light of wavelength 633 nm from a distant source is incident on a slit 0.750 mm wide, and the resulting diffraction pattern is observed on a screen 3.50 m away. What is the distance between the two dark fringes on either side of the central bright fringe?

Problem 2.4: Red light of wavelength 633 nm from a helium–neon laser passes through a slit 0.350 mm

wide. The diffraction pattern is observed on a screen 3.10 m away. a) What is the width of the central bright fringe? b) What is the width of the first bright fringe on either side of the central one?

Additional Problem 2.4: Helium–neon laser light (λ = 632.8 nm) is sent through a 0.300-mm-wide single slit. What is the width of the central maximum on a screen 1.00 m from the slit?



Problem 2.5: Light of wavelength 540 nm passes through a slit of width 0.200 mm. a) The width of the central maximum on a screen is 8.10 mm. How far is the screen from

the slit? b) Determine the width of the first bright fringe to the side of the central maximum.

Problem 2.6: Light of wavelength 585 nm falls on a slit 0.0666 mm wide. On a very large and distant

screen, how many totally dark fringes (indicating complete cancellation) will there be, including both sides of the central bright spot?



Problem 2.7: Laser light with a wavelength of 650 nm is directed through one slit or two slits and allowed to fall on a screen 2.31 m beyond. Figure below shows the pattern on the screen, with a centimeter ruler below it.

a) Did the light pass through one slit or two slits? Explain how you can determine the answer.

b) If one slit, find its width. If two slits, find the distance between their centers.



3. Diffraction gratings .

Problem 3.1: What is the wavelength of light that is deviated in the first order through an angle of 14.6o by a transmission grating having 5000 slits/cm?

Problem 3.2: A beam of 541-nm light is incident on a diffraction grating that has 400 grooves/mm.

Determine the angle of the second-order ray. Problem 3.3: Light from an argon laser strikes a diffraction grating that has 5,310 grooves per centimeter.

The central and first-order principal maxima are separated by 0.488 m on a wall 1.72 m from the grating. Determine the wavelength of the laser light.



Problem 3.4: When laser light of wavelength 632.8 nm passes through a diffraction grating, the first bright spots occur a 17.8o from the central maximum. a) What is the line density (in lines/cm) of this grating? b) How many additional bright spots are there beyond the first bright spots, and at what

angles do they occur?

Problem 3.5: If a diffraction grating produces its third-order bright band at an angle of 77.4o for light of wavelength 681 nm, find a) the number of slits per centimeter for the grating b) Will there be a fourth-order bright band?



5. Concave mirrors, Convex mirrors, Converging lens, and Diverging lens .

Convex mirrors and Concave mirrors.

Problem 5.1: A candle 4.50 cm tall is 40.0 cm to the left of a plane mirror. a) Where is the image formed by the mirror? b) What is the height of this image?

Problem 5.2: The image of a tree just covers the length of a plane mirror 3.70 cm tall when the mirror is held 34.0 cm from the eye. The tree is 20.0 m from the mirror. What is its height?

Problem 5.3: A pencil that is 9.0 cm long is held perpendicular to the surface of a plane mirror with the

tip of the pencil lead 12.0 cm from the mirror surface and the end of the eraser 21.0 cm from the mirror surface. a) What is the length of the image of the pencil that is formed by the mirror? b) Which end of the image is closer to the mirror surface: the tip of the lead or the end

of the eraser?



Problem 5.4: An object is placed 50.0 cm from a concave spherical mirror with focal length of magnitude 20.0 cm. a) Draw a principal-ray diagram showing the formation of the image. b) Find the location of the image. c) What is the magnification of the image? d) Is the image real or virtual? e) Is the image upright or inverted?



Problem 5.5: An object of height 2.00 cm is placed 30.0 cm from a convex spherical mirror of focal length of magnitude 10.0 cm. a) Draw a principal-ray diagram showing the formation of the image. b) Find the location of the image. c) Indicate whether the image is upright or inverted. d) Determine the height of the image.



Problem 5.6: A concave spherical mirror has a radius of curvature of magnitude 24.0 cm. a) Determine the object position for which the resulting image is upright and larger than

the object by a factor of 3.00. b) Draw a ray diagram to determine the position of the image. c) Is the image real or virtual?

Problem 5.7: A dentist uses a curved mirror to view teeth on the upper side of the mouth. Suppose she

wants an erect image with a magnification of 2.00 when the mirror is 1.25 cm from a tooth. a) What kind of mirror (concave or convex) is needed? b) What must be the focal length and radius of curvature of this mirror?



Problem 5.8: For a concave spherical mirror that has focal length of 18 cm, what is the distance of an object from the mirror’s vertex if the image is real and has the same height as the object?

Problem 5.9: For a convex spherical mirror that has focal length of 12 cm, what is the distance of an object from the mirror’s vertex if the height of the image is half the height of the object?



Converging lens and Civerging lens.

Problem 5.10: A converging lens has a focal length of 10.0 cm. Construct accurate ray diagrams for object distances of 20.0 cm.

a) Determine the location of each image. b) Is the image real or virtual? c) Is the image upright or inverted? d) What is the magnification of the image?



Problem 5.11: A converging lens has a focal length of 10.0 cm. Construct accurate ray diagrams for object distances of 5.00 cm.

a) Determine the location of each image. b) Is the image real or virtual? c) Is the image upright or inverted? d) What is the magnification of the image?

Problem 5.12: An object is placed 10.0 cm from a diverging lens of focal length 10 cm.

a) Find the location of the image. b) Find the magnification of the image.



Problem 5.13: A converging lens with a focal length of 70.0 cm forms an image of a 3.20-cm-tall real object that is to the left of the lens. The image is 4.50 cm tall and inverted. a) Where are the object located? b) Where are the image located? Is the image real or virtual?

Additional Problem 5.13: A converging lens with a focal length of 7.00 cm forms an image of a 3.20-mm-tall real object that is to the left of the lens. The image is 1.70 cm tall and erect. a) Where are the object located b) Where are the image located? Is the image real or virtual?



Problem 5.14: A converging lens with a focal length of 12.0 cm forms a virtual image 0 . 8 0 cm tall, 17.0 cm to the left of the lens. a) Determine the position and size of the object. b) Is the image erect or inverted? c) Are the object and image on the same side or opposite sides of the lens? d) Draw a principal-ray diagram for this situation.

Additional Problem 5.14: A diverging lens with a focal length of 48.0 cm forms a virtual image 8.00 mm tall, 17.0 cm to the right of the lens. a) Determine the position and size of the object. b) Is the image erect or inverted? c) Are the object and image on the same side or opposite sides of the lens? d) Draw a principal-ray diagram for this situation.



Problem 5.15: A converging lens forms an image of an 0.75-cm-tall real object. The image is 12.5 cm to the left of the lens, 3.30 cm tall, and erect. a) What is the focal length of the lens? b) Where is the object located?

Problem 5.16: A lens forms an image of an object. The object is 16.0 cm from the lens. The image is 12.0

cm from the lens on the same side as the object. a) What is the focal length of the lens? Is the lens converging or diverging? b) If the object is 0.85 cm tall, how tall is the image? Is it erect or inverted? c) Draw a principal-ray diagram.



Problem 5.17: An object is 16.0 cm to the left of a lens. The lens forms an image 36.0 cm to the right of the lens. a) What is the focal length of the lens? Is the lens converging or diverging? b) If the object is 0.80 cm tall, how tall is the image? Is it erect or inverted? c) Draw a principal-ray diagram.

Additional Problem 5.17: An object located 32.0 cm in front of a lens forms an image on a screen 8.00 cm behind the lens. a) Find the focal length of the lens. b) Determine the magnification. c) Is the lens converging or diverging?



Problem 5.18: A 1.20-cm-tall object is 50.0 cm to the left of a converging lens of focal length 40.0 cm. A second converging lens, this one having a focal length of 60.0 cm, is located 300.0 cm to the right of the first lens along the same optic axis. a) Find the location and height of the image (call it I1) formed by the lens with a focal

length of 40.0 cm. b) I1 is now the object for the second lens. Find the location and height of the image

produced by the second lens. This is the final image produced by the combination of lenses.

Problem 5.19: A 1.20-cm-tall object is 50.0 cm to the left of a converging lens of focal length 40.0 cm. A second diverging lens, this one having a focal length of 60.0 cm, is located 300.0 cm to the right of the first lens along the same optic axis. a) Find the location and height of the image (call it I1) formed by the lens with a focal





Problem 5.20: A 1.20-cm-tall object is 50.0 cm to the left of a diverging lens of focal length 40.0 cm. A second converging lens, this one having a focal length of 60.0 cm, is located 300.0 cm to the right of the first lens along the same optic axis. a) Find the location and height of the image (call it I1) formed by the lens with a focal



Problem 5.21: A 1.20-cm-tall object is 50.0 cm to the left of a diverging lens of focal length 40.0 cm. A second diverging lens, this one having a focal length of 60.0 cm, is located 300.0 cm to the right of the first lens along the same optic axis. a) Find the location and height of the image (call it I1) formed by the first lens. b) I1 is now the object for the second lens. Find the location and height of the image




Problem 5.22: Two thin lenses with a focal length of magnitude 15.0 cm, the first diverging and the second converging, are located 11.3 cm apart. An object 0.16 cm tall is placed 25.0 cm to the left of the first (diverging) lens. a) How far from this first lens is the final image formed? b) Is the final image real or virtual? c) What is the height of the final image? Is it erect or inverted?



6. Defects of vision .

Problem 6.1: A certain person can see distant objects well, but his near point is 45.0 cm from his eyes instead of the usual 25.0 cm. a) Is this person nearsighted or farsighted? b) What type of lens (converging or diverging) is needed to correct his vision? c) What focal length lens is needed to clearly an object that is 25 cm in front of the eye?

Problem 6.2: A person can see clearly up close but cannot focus on objects beyond 75.0 cm. She opts

for contact lenses to correct her vision. a) Is she nearsighted or farsighted? b) What type of lens (converging or diverging) is needed to correct her vision? c) What focal length lens is needed?



7. Refraction of light .

Problem 7.1: The index of refraction of diamond is 2.417. Find the speed of light in diamond.

Problem 7.2: A light beam travels at 1.94 × 108 m/s in quartz. What is the index of refraction of quartz?

Problem 7.3: A ray of light travels from air into another medium, making an angle of θ1 = 45.0° with the

normal as in Figure below. Find the angle of refraction θ2 if the second medium is water (nwater = 1.33).

Problem 7.4: A ray of light traveling in water is incident on an interface with a flat piece of glass. The

wavelength of the light in the water is 727 nm and its wavelength in the glass is 542 nm. If the ray in water makes an angle of 41.5o with respect to the normal to the interface, what angle does the refracted ray in the glass make with respect to the normal?

(sin 41.5°= 0.66 and sin 48.5°= 0.75)



Problem 7.5: Light traveling in air is incident on the surface of a block of plastic at an angle of 60.0o to the normal and is bent so that it makes a 45.0o angle with the normal in the plastic. Find the speed of light in the plastic.

Problem 7.6: A parallel beam of light in air makes an angle of 41.0o with the surface of a glass plate

having a refractive index of 1.57. (sin 41.0°= 0.66 and sin 49.0°= 0.75) a) What is the angle between the reflected part of the beam and the surface of the

glass? b) What is the angle between the refracted beam and the surface of the glass?



Problem 7.7: A beam of light both reflects and refracts at the surface between air and glass as shown in Figure below. If the refractive index of the glass is ng , find the angle of incidence θ1 in the air that would result in the reflected ray and the refracted ray being perpendicular to each other.

Problem 7.8: As shown in Fig. below, a layer of water (nwater = 1.33) covers a slab of material X in a

beaker. A ray of light traveling upward follows the path indicated.

(sin 65°= 0.91, sin 25°= 0.42, sin 48°= 0.74 and sin 52°= 0.79)

Using the information on the figure, find a) the index of refraction of material X. b) the angle the light makes with the normal in the air.



Problem 7.9: A ray of light strikes a flat block of glass (n = 1.50) of thickness 2.00 cm at an angle of 30.0° with the normal. Trace the light beam through the glass and find the angles of incidence and refraction at each surface.

Problem 7.10: A ray of light strikes the midpoint of one face of an equiangular (60°–60°–60°) glass prism

(n = 1.5) at an angle of incidence of 30°. a) Trace the path of the light ray through the glass and find the angles of incidence and

refraction at each surface. b) If a small fraction of light is also reflected at each surface, what are the angles of

reflection at the surfaces?



Critical angle and Total internal reflection.

Problem 7.11: A beam of light is traveling inside a solid glass cube that has index of refraction 1 . 5 3 . It strikes the surface of the cube from the inside. If the cube is in air, at what minimum angle with the normal inside the glass will this light not enter the air at this surface?

Problem 7.12: A ray of light is traveling in a glass cube that is totally immersed in water (nwater = 1.33).

You find that if the ray is incident on the glass–water interface at an angle to the normal larger than 45.0o no light is refracted into the water. What is the refractive index of the glass?

Problem 7.13: The critical angle for total internal reflection at a liquid–air interface is 45.0o. If a ray of

light traveling in the liquid has an angle of incidence at the interface of 32.5o, what angle

does the refracted ray in the air make with the normal? (sin 32.5°= 0.54 and sin 57.5°= 0.84)



Problem 7.14: Light is incident along the normal on face AB of a glass prism of refractive index 1.52, as

shown in Fig. below. Find the largest value the angle α can have without any light refracted out of the prism at face AC if the prism is immersed in air.

Problem 7.15: A ray of light is incident in air on a block of a transparent solid whose index of refraction

is n. If n = 1.37, what is the largest angle of incidence θa for which total internal reflection will occur at the vertical face (point shown in Fig. below)?



Problem 7.16: A ray of light traveling in air is incident at angle θa on one face of a 90.0o prism made of glass. Part of the light refracts into the prism and strikes the opposite face at point A (Fig

below). If the ray at A is at the critical angle, what is the value of θa? (sin 40.0° = 0.64 and

sin 50.0° = 0.77)

Problem 7.17: Figure below shows the path of a light beam through several slabs with different indices

of refraction.

a) If θ1 = 30.0o, what is the angle θ2 of the emerging beam? b) What must the incident angle θ1 be to have total internal reflection at the surface

between the medium with n = 1.20 and the medium with n = 1.00?



8. Light color and Pigment color .

Problem 8.1: What color is seen when white light shines on the blue pigment?

Problem 8.2: What color is seen when white light shines on the yellow pigment?

Problem 8.3: What color is seen when white light shines on the red pigment? Problem 8.4: What color is seen when white light shines on a cyan pigment?

Problem 8.5: What color is seen when white light shines on a green pigment?

Problem 8.6: What color is seen when red light shines on a green pigment?

Problem 8.7: What color is seen when magenta light shines on a red pigment?

Problem 8.8: in the diagrams below, several sheets of paper are illuminated by different primary colors

of light (R for red, B for blue, and G for green). Indicate what primary colors of light will be reflected and the appearance of the sheet of paper.

Physics 3 (SCI 32201) Chapter 4: Sound ___________________________________________________________________________________________


7สียงแล

1. Speed of sound waves 2. Reflection, Refraction, Interference, and Diffraction of sound waves

- Reflection - Refraction - Interference - Diffraction

3. Power of a sound, Sound intensity, and Sound level - Power - Sound intensity - Sound level

4. Beats 5. Resonance

- String fixed at both ends - Pipe closed at one end and Pipe open at both ends

Closed at one end Open at both ends

6. Shock waves 7. Doppler effect

Chapter 4: Sound



Sound waves are generated by the vibration of an object.

Tuning fork Air particle Direction of travel of sound a.

b.

c.

d.

e.

Direction of travel of sound

Types of sound waves.

• Audible range: frequency 20 – 20,000 Hertz

• Infrasonic waves: frequency < 20 Hertz

• Ultrasonic waves: frequency > 20,000 Hertz For example, bat sound

compression

compression

compression compression

compression compression rarefaction rarefaction

rarefaction

rarefaction

rarefaction rarefaction compression compression



1. Speed of sound waves .

The speed of sound in air depends on the temperature of the air:

vsound = 331 + 0.6T

where T = air temperature (oC)

Using this equation, one finds that at 25 oC, the speed of sound in air is approximately

vsound = 331 + 0.6 (25)

vsound ≈ 350 m/s

*** This equation works best at air temperatures -50 oC < T < 50 oC *** Example 1.1: A sound wave traveling through water has a frequency of 500 Hz and a wavelength of 3

m. How fast does sound travel through water? Example 1.2: A sound wave with a frequency of 343 Hz travels through the air at 20 oC.

a) What is its wavelength?

b) If its frequency increased to 686 Hz, what is its wave speed and the wavelength? ___________________________________________________________________________________________ Additional Problem 1.2: What is the wavelength of a sound wave in air at 20 oC if the frequency is 262 Hz.



2. Reflection, Refraction, Interference, and Diffraction of sound waves .

Reflection.

The human ear can distinguish ‘echo’ from the original direct sound if the delay is more than 0.1 of a second.

Sound waves are reflected only from objects larger than their wavelength. The advantage to bats of using high-pitched sound (ultrasonic) to echolate is that these sounds are carried on smaller waves and consequently permit the localization of smaller object.

Refraction.

Lightning without thunder We hear thunder when the lightning is close to us, but we often do not hear the thunder for distant

lightning because of refraction – Sound waves propagating through air are bent and undergo refraction when the air temperature varies. The air near the ground is warmer than the upper air, so the speed of sound near the ground is greater. The sound will be refracted and bent away from the ground.

Interference.

Two identical loudspeakers emit sound waves toward each other. When they overlap, identical waves traveling in opposite directions will combine to form standing waves.

Diffraction.

Sound waves bend around the corner of building.



Example 2.1: Two identical loudspeakers placed 3 .00 m apart are driven by the same oscillator (figure below) . A listener is originally at point O, located 8 . 0 0 m from the center of the line connecting the two speakers. The listener then moves to point P, which is a perpendicular distance 0.350 m from O, and she experiences the first minimum in sound intensity. What is the frequency of the oscillator? The speed of sound in air is 343 m/s.

Example 2.2: Two small loud speaker, A and B are driven by the same amplifier and emit pure sinusoidal

waves in phase. The speed of sound is 350 m/s.

a) For what frequencies does constructive interference occur?

b) For what frequencies does destructive interference occur?



3. Power of a sound, Sound intensity, and Sound level .

Power of a sound (P).

Power of a sound is defined as the rate at which the energy transported by the wave or the energy that the sound source emits per unit time. The unit of power is Joule per second (J/s) or Watt (W).

Sound intensity (I).

Sound intensity is defined as the power of a sound per unit area.

I ≡ P

A

The SI unit of sound intensity is Watt per square meter (w/m2).

For a point sound source, the sound will radiate in all directions equally as a spherical surface (A = 4πr2) with the sound source at the center of the sphere. Hence, the intensity of a sound wave at a distance r from the source is

I = P

4πr2

The intensity decreases with increasing distance r from the source according to the inverse-square

law.

Note: Threshold of hearing I0 = 1.0 × 10–12 w/m2



Example 3.1: A point source emits sound waves with an average power output of 80.0 W. a) Find the intensity 3.00 m from the source.

b) Find the distance at which the intensity of the sound is 1.00 × 10-8 W/m2. Example 3.2: A siren on a tall pole radiates sound waves uniformly in all directions. At a distance of 15.0

m from the siren, the sound intensity is 0.250 W/m2. At what distance is the intensity 0.010 W/m2.



Sound level (β).

Sound level is defined as the logarithm measure of sound intensity.

β = 10 logI

I0

The unit of sound level is decibel (dB)

Example 3.3: Find the sound level of the sound wave with sound intensity 3.2 × 10-7 W/m2.

Example 3.4: What sound intensities correspond to 48 dB?

Sound intensity (I) Sound level (β)

10-12 d

10-11 d

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1

Audible waves: f = 20 – 20,000 Hz I = 10–12 – 1 w/m2 β = 0 – 120 dB



• The different between any two sound levels is related to the corresponding intensity by

β1 - β

2 = 10 log (

I1

I2

)

β1 - β

2 = 10 log (

P1

4πr12

P2

4πr22

)

If the power of a sound does not change P1 = P2, then

β1 - β

2 = 10 log (

r2

r1

)2

β1 - β

2 = 20 log (

r2

r1

)

Example 3.5: How many times more intense is an 82 dB sound than a 52 dB sound? Example 3.6: Consider an idealized bird (treated as a point source) that emits constant sound power,

with intensity obeying the inverse-square law. If you move twice the distance from the bird, by how many decibels does the sound level drop?



Example 3.7: Two identical machines are positioned the same distance from a worker. The intensity of

sound delivered by each operating machine at the worker’s location is 2.0 × 10-7 W/m2. a) Find the sound level heard by the worker when one machine is operating. b) Find the sound level heard by the worker when two machines are operating.

Loudness is a psychological response to a sound. It depends on both the intensity and the frequency of the sound. As a rule of thumb, a doubling in loudness is approximately associated with an increase in sound level of 10 dB. If the loudness of the machines in this example is to be doubled, how many machines at the same distance from the worker must be running?



• An audiogram of the human hearing The audiogram shows the values of the sound level threshold as a function of frequency. It

shows how sound levels need to be at different frequencies for you to hear them.

According to the diagram, the sound level threshold is dependent on frequency.



Pitch.

• Low pitch (low frequency) → Bass

• High pitch (high frequency) → Treble

Quality of sound.

The quality of the sound is part of what allows you to identify instruments playing the same note. For example, you can differentiate between a guitar, piano, or a flute playing the same note.

Musical sounds have wave functions that are more complicated than a simple sine function. The pattern is so complex because the musical instruments generates many frequency at the same time. These frequencies are called ‘harmonics’. The lowest frequency is called the ‘fundamental frequency’ or ‘first harmonic’. The frequencies of the remaining harmonics are integer multiples of the fundamental frequency.

The wave patterns produced by a musical instrument are the result of the superposition of harnonics. Two tones produced by different instruments might have the same fundamental frequency but sound different because of different harmonic content.

Fundamental frequency or first harmonic

second harmonic

third harmonic

resultant wave

time

time

time

time



4. Beats .

Beating is the periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies. A listener would hear a sound of periodically varying loudness.

Beat frequency (fB).

Beat frequency is the number of amplitude maxima one hears per second. It equals the difference in frequency between the two sources:

fb = |f1 - f2|

Note: At frequency differences greater than about 7 Hz, we no longer hear individual beats. Average frequency.

The resultant wave has a frequency equal to the average frequency:

f average = f1 + f2

2

For instance, if one tuning fork vibrates at 438 Hz and a second one vibrates at 442 Hz, a listener

would hear a 440-Hz sound wave go through an intensity maximum four times every second. Example 4.1: Two identical piano strings of length 0.750 m are each tuned exactly to 440 Hz. The tension

in one of the strings is then increased by 1.0%. If they are now struck, what is the beat frequency between the fundamentals of the two strings?

Dis

pla

cem

ent

time (s)

0 1 2 3 4



5. Resonance .

The frequency at which a system tends to oscillate in the absence of any driving or damping force is called ‘natural frequency’.

Resonance is the peaking of the amplitude when the frequency of the driving force is near the natural frequency of oscillation. For example, building up the oscillations of a child on a swing by pushing with a frequency equal to the swing’s natural frequency.

• Fundamental frequency: Lowest frequency

• Harmonic: All integer multiples of the fundamental frequency

• Overtone: Any frequency greater than the fundamental frequency

String fixed at both ends.

first harmonic fundamental frequency

second harmonic

ก third harmonic



Example 5.1: The middle C string on a piano has a fundamental frequency of 262 Hz, and the string for the first A above middle C has a fundamental frequency of 440 Hz. a) Calculate the frequencies of the next two harmonics of the C string.

b) If the A and C strings have the same linear mass density µ and length L, determine the ratio of tensions in the two strings.

Example 5.2: A string of length 12 m that’s fixed at both ends supports a standing wave with a total of

5 nodes. What are the harmonic number and wavelength of this standing wave?



Example 5.3: In an attempt to get your name in Guinness World Records, you build a bass viol with strings of length 5.00 m between fixed points. One string, with linear mass density 40.0 g/m, is tuned to a 20.0-Hz fundamental frequency (the lowest frequency that the human ear can hear). a) Calculate the tension of this string. b) Calculate the frequency and wavelength on the string of the second harmonic. c) Calculate the frequency and wavelength on the string of the second overtone. d) What are the frequency and wavelength of the sound waves produced in the air when

the string is vibrating at its fundamental frequency? The speed of sound in air at 20 oC is 344 m/s.



Example 5.4: A piano tuner uses a tuning fork to adjust the key that plays the A note above middle C (whose frequency should be 440 Hz). The tuning fork emits a perfect 440 Hz tone. When the tuning fork and the piano key are struck, beats of frequency 3 Hz are heard. a) What is the frequency of the piano key?

b) If it’s known that the piano key’s frequency is too high, should the piano tuner tighten or loosen the wire inside the piano to tune it?

Example 5.5: A piano tuner causes a piano string to vibrate. He then loosens the string a little, decreasing its tension, without changing the length of the string. a) What happens to the fundamental frequency?

b) What happens to the fundamental wavelength?



Pipe closed at one end and Pipe open at both ends.

Resonance occurs when the air in the pipe is stimulated by sound waves that have the same frequency as the natural frequency of the air in the pipe.

• A ‘tuning fork’ vibrating at an unknown frequency is placed near the top of the pipe. The length of the air column can be adjusted. The experiment shows that there are many positions of water levels in the pipes that cause resonance.

• A tuning fork is vibrated near the top of the pipe. There are many forks frequencies that can cause the resonance of the air in one pipe.

Pipe closed at one end Pipe open at both ends

fundamental f fundamental f first harmonic first harmonic

first overtone first overtone third harmonic second harmonic

second overtone second overtone fifth harmonic third harmonic

B

E

A

D

C

F

Displacement

Position

Displacement

Position

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Example 5.6: A closed-end tube resonates at a fundamental frequency of 343 Hz. The air in the tube is at a temperature of 20°C, and it conducts sound at a speed of 343 m/s. a) What is the length of the tube?

b) What is the next higher harmonic frequency?

c) Answer the questions posed in questions a.) and b.) assuming that the tube was open at its far end.



Example 5.7: A simple apparatus for demonstrating resonance in an air column is depicted in figure below. A vertical pipe open at both ends is partially submerged in water, and a tuning fork vibrating at an unknown frequency is placed near the top of the pipe. The length L of the air column can be adjusted by moving the pipe vertically. The sound waves generated by the fork are reinforced when L corresponds to one of the resonance frequencies of the pipe. For a certain pipe, the smallest value of L for which a peak occurs in the sound intensity is 9.00 cm. The speed of sound in air is 343 m/s.

a) What is the frequency of the tuning fork? b) What are the values of L for the next two resonance conditions?



Example 5.8: A section of drainage culvert 1.23 m in length makes a howling noise when the wind blows across its open ends. Determine the frequencies of the first three harmonics of the culvert if it is cylindrical in shape and open at both ends. Take v = 343 m/s as the speed of sound in air.

Example 5.9: On a day when the speed of sound is 345 m/s, the fundamental frequency of a particular

stopped organ pipe is 220 Hz. a) How long is this pipe?

b) The second overtone of this pipe has the same wavelength as the third harmonic of

an open pipe. How long is the open pipe?

Example 5.10: A stopped organ pipe is sounded near a guitar, causing one of the strings to vibrate with large amplitude. We vary the string tension until we find the maximum amplitude. The string is 80% as long as the pipe. If both pipe and string vibrate at their fundamental frequency, calculate the ratio of the wave speed on the string to the speed of sound in air.



6. Shock waves .

vsource = 0.

vsource < vsound.

- If the source is moving toward the listener, the frequency heard by the listener is __________. - If the source is moving away from the listener, the frequency heard by the listener is __________.

λ behind = λ in front =

λ =



Example 6.1: a) A sound source emits the wave fronts shown in the diagram below. In what direction is the sound

source traveling compared to a stationary observer?

b) In the diagram below, where does an observer need to be positioned to hear a higher-pitch sound?

Example 6.2: A police car’s siren emits a sinusoidal wave with frequency f0 = 300 Hz. The speed of sound

is 340 m/s and the air is still. a) Find the wavelength of the waves if the siren is at rest. b) Find the wavelengths of the waves in front of and behind the siren if it is moving at

30 m/s.



vsoundt

θ = Mach angle

vsourcet

vsource = vsound.

v source > v sound.

Shock waves occur when the speed of source exceeds the wave speed.

• Mach number: Mach No. ≡ Vsource

Vsound =

1

sin θ

• The circular wave fronts interfere constructively at a point along the surface of the cone, leading to a very-large-amplitude wave front. The conical wave front is called a ‘shock wave’. The arrival of this shock wave causes the ‘sonic boom’ you hear after an airplane has passed by.

Example 6.3: An airplane is flying at Mach 1.75 at an altitude of 8,000 m, where the speed of sound is

320 m/s. How long after the plane passes directly overhead will you hear the sonic boom?

sin θ = vsound

vsource



7. Doppler effect .

Doppler effect is a phenomenon that occurs when a source of sound and a listener are in motion relative to each other causes the frequency of the sound heard by the listener is not the same as the source frequency.

where v = velocity of sound wave

vs = velocity of source vL = velocity of listener f0 = frequency of sound wave emitted from the source

fL = frequency heard by the listener

vs

vs

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vL

vs

vL

vs

vL

vL

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∴ fL =

vL vs

vs

vL vs






Example 7.1: You are standing on the street not moving, and an ambulance with its siren on is approaching you at the speed of vs. What is the frequency of the siren that you detect?

Once the ambulance passes you, what is the frequency of the siren that you detect? Example 7.2: If a listener L is at rest and the siren in Example 6.2 is moving away from L at 30 m/s, what

frequency does the listener hear?



Example 7.3: If the siren is at rest and the listener is moving away from it at 30 m/s, what frequency does the listener hear?

Example 7.4: The siren is moving away from the listener with a speed of 45 m/s relative to the air, and

the listener is moving toward the siren with a speed of 15 m/s relative to the air. What frequency does the listener hear?



Example 7.5: A source of 4 kHz sound waves travels at 1

9 the speed of sound toward a detector that’s

moving at 1

9 the speed of sound, toward the source.

a) What is the frequency of the waves as they’re received by the detector?

b) A train sounds its whistle as it travels at a constant speed by a train station. Describe the pitch of the train’s whistle as heard by someone standing on the station platform.

Example 7.6: A submarine (sub A) travels through water at a speed of 8.00 m/s, emitting a sonar wave

at a frequency of 1,400 Hz. The speed of sound in the water is 1,533 m/s. A second submarine (sub B) is located such that both submarines are traveling directly toward each other. The second submarine is moving at 9.00 m/s. a) What frequency is detected by an observer riding on sub B as the subs approach each

other?

b) The subs barely miss each other and pass. What frequency is detected by an observer riding on sub B as the subs recede from each other?



Summary .

1. Speed of sound waves vsound = 331 + 0.6T

2. Reflection, Refraction, Interference, and Diffraction of sound waves - Reflection

The human ear can distinguish ‘echo’ from the original direct sound if the delay is more than 0.1 of a second.

Sound waves are reflected only from objects larger than their wavelength. - Refraction

We hear thunder when the lightning is close to us, but we often do not hear the thunder for distant lightning because of refraction.

- Interference Two identical loudspeakers emit sound waves toward each other. When they overlap,

identical waves traveling in opposite directions will combine to form standing waves. - Diffraction

Sound waves bend around the corner of building. 3. Power of a sound, Sound intensity, and Sound level

- Sound intensity: I ≡ P

A

- Sound level: β = 10 logI

I0

4. Beats Beating is the periodic variation in amplitude at a given point due to the superposition of two

waves having slightly different frequencies. A listener would hear a sound of periodically varying loudness.

fb = |f1 - f2|

5. Resonance Resonance is the peaking of the amplitude when the frequency of the driving force is near the

natural frequency of oscillation. 6. Shock waves

Mach number: Mach No. ≡ Vsource

Vsound =

1

sin θ

7. Doppler effect Doppler effect is a phenomenon that occurs when a source of sound and a listener are in

motion relative to each other causes the frequency of the sound heard by the listener is not the same as the source frequency.



Assignment .

1. Speed of sound waves .

Problem 1.1: To find the height of the cliff, you drop a rock from the top; 9.60 s later you hear the sound of the rock hitting the ground at the foot of the cliff. If you ignore air resistance, how high is the cliff if the speed of sound is 330 m/s?

Problem 1.2: The speed of sound in air is 344 m/s. What is the wavelength of a sound wave with a

frequency of 784 Hz? Additional Problem 1.2: The sound source of a ship’s sonar system operates at a frequency of 30.0 kHz. The speed of sound in water is 1,482 m/s. What is the wavelength of the waves emitted by the source?



Problem 1.3: A sound wave propagates in air at 27oC with frequency 4.00 kHz. It passes through a region where the temperature gradually changes and then moves through air at 0oC. Give numerical answers to the following questions to the extent possible and state your reasoning about what happens to the wave physically. a) What happens to the speed of the wave?

b) What happens to its frequency?

c) What happens to its wavelength?



2. Reflection, Refraction, Interference, and Diffraction of sound waves .

Interference.

Unless identicated otherwise, assume the speed of sound in air to be v = 344 m/s.

Problem 2.1: Two identical loudspeakers 10.0 m apart are driven by the same oscillator with a frequency of f = 21.5 Hz (Fig. below) in an area where the speed of sound is 344 m/s.

Show that a receiver at point A records a minimum in sound intensity from the two speakers.



Problem 2.2: Two loudspeakers, A and B (see Fig. below), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. The frequency of the sound waves produced by the loudspeakers is 206 Hz. Consider a point P between the speakers and along the line connecting them, a distance x to the right of A. Both speakers emit sound waves that travel directly from the speaker to point P.

For what values of x will a) destructive interference occur at P?

b) constructive interference occur at P?



Problem 2.3: Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 235.0 Hz in a room where the speed of sound is 340.0 m/s. A woman starts out at the midpoint between the two speakers. The room’s walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision. a) What does she hear: constructive or destructive interference? Why?

b) She now walks slowly toward one of the speakers. How far from the center must she

walk before she first hears the sound reach a minimum intensity?

c) How far from the center must she walk before she first hears the sound maximally enhanced?

Problem 2.4: Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in

phase. Speaker B is 12.0 m to the right of speaker A. The frequency of the waves emitted by each speaker is 688 Hz. You are standing between the speakers, along the line connecting them, and are at a point of constructive interference. How far must you walk toward speaker B to move to a point of destructive interference?



Problem 2.5: Small speakers A and B are driven in phase at 725 Hz by the same audio oscillator. Both speakers start out 4.50 m from the listener, but speaker A is slowly moved away (Figure below)

a) At what distance d will the sound from the speakers first produce destructive interference at the listener’s location?

b) If A is moved even farther away than in part (a), at what distance d will the speakers next produce destructive interference at the listener’s location

c) After A starts moving away from its original spot, at what distance d will the speakers first produce constructive interference at the listener’s location?



Problem 2.6: Two loudspeakers, A and B (Fig. below), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speakers emit sound waves that travel directly from the speaker to point Q.

What is the lowest frequency for which a) constructive interference occurs at point Q?

b) destructive interference occurs at point Q?



Problem 2.7: Two speakers, emitting identical sound waves of wavelength 2.0 m in phase with each other, and an observer are located as shown in Fig. below.

a) At the observer’s location, what is the path difference for waves from the two speakers?

b) Will the sound waves interfere constructively or destructively at the observer’s location—or something in between constructive and destructive?

c) Suppose the observer now increases her distance from the closest speaker to 17.0 m, staying directly in front of the same speaker as initially. Answer the questions of parts (a) and (b) for this new situation.



Problem 2.8: The ship in Figure below travels along a straight line parallel to the shore and a distance d = 600 m from it. The ship’s radio receives simultaneous signals of the same frequency from antennas A and B, separated by a distance L = 800 m. The signals interfere constructively at point C, which is equidistant from A and B. The signal goes through the first minimum at point D, which is directly outward from the shore from point B. Determine the wavelength of the radio waves.



Problem 2.9: Two identical loudspeakers are located at points A and B, 2.00 m apart. The loudspeakers are driven by the same amplifier and produce sound waves with a frequency of 784 Hz. Take the speed of sound in air to be 344 m/s. A small microphone is moved out from point B along a line perpendicular to the line connecting A and B (line BC in Fig. below).

a) At what distances from B will there be destructive interference?

b) At what distances from B will there be constructive interference?

Additional Problem 2.9: Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone arranged as shown in Fig. below. For what frequencies does their sound at the speakers produce

a) constructive interference. b) destructive interference.



Problem 2.10: Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is 172 Hz. You are 8.00 m from A. What is the closest you can be to B and be at a point of destructive interference?



3. Power of a sound, Sound intensity, and Sound level .

Problem 3.1: A fireworks rocket explodes at a height of 100 m above the ground. An observer on the

ground directly under the explosion experiences an average sound intensity of 7.00 × 10-2 W/m2. What is the sound level (in decibels) heard by the observer?

Problem 3.2: At point A, 4.0 m from a small source of sound that is emitting uniformly in all directions,

the sound level is 52 dB. What is the intensity of the sound at A?



Problem 3.3: You live on a busy street, but as a music lover, you want to reduce the traffic noise. If you reduce the intensity by half, what change (in dB) do you make in the sound level?

Problem 3.4: If you increase the volume of your stereo so that the intensity doubles, by how much

does the sound level increase? Problem 3.5: If one sound is 190 times as intense as another, by how much do they differ in sound level

(in decibels)?



Problem 3.6: The Sacramento City Council adopted a law to reduce the allowed sound level of the much-despised leaf blowers from their current level of about 95 dB to 70 dB. With the new law, what is the ratio of the new allowed intensity to the previously allowed intensity?

Problem 3.7: You live on a busy street, but as a music lover, you want to reduce the traffic noise. If you

install special sound-reflecting windows that reduce the sound level (in dB) by 26 dB, by what fraction have you lowered the sound intensity (in W/m2)?

Problem 3.8: By what factor must the sound intensity be increased to raise the sound level by 13.0 dB? Problem 3.9: If two sounds differ by 15.00 dB, find the ratio of the intensity of the louder sound to that

of the softer one.



Problem 3.10: The intensity due to a number of independent sound sources is the sum of the individual intensities. a) When four quadruplets cry simultaneously, how many decibels greater is the sound

level than when a single one cries? b) To increase the sound level again by the same number of decibels as in part (a), how

many more crying babies are required?



Problem 3.11: Show that the difference between decibel levels β1 and β

2 of a sound is related to the

ratio of the distances r1 and r2 from the sound source by

β2 - β

1 = = 20 log (

r1

r2

)

Problem 3.12: A baby’s mouth is 25 cm from her father’s ear and 1.55 m from her mother’s ear. What is

the difference between the sound levels heard by the father and by the mother? Problem 3.13: You are trying to overhear a juicy conversation, but from your distance of 15.0 m, it sounds

like only an average whisper of 20.0 dB. How close should you move to the chatterboxes for the sound level to be 60.0 dB?

Additional Problem 3.13: The sound from a trumpet radiates uniformly in all directions in air. At a distance of 5.08 m from the trumpet the sound level is 53.0 dB. At what distance is the sound level 30.0 dB?



Problem 3.14: At point A, 4.0 m from a small source of sound that is emitting uniformly in all directions, the sound level is 52 dB. a) How far from the source must you go so that the intensity is one-fourth of what it was

at A? b) How far must you go so that the sound level is one-fourth of what it was at A?

Problem 3.15: The power output of a certain public-address speaker is 6.00 W. Suppose it broadcasts

equally in all directions. a) Within what distance from the speaker would the sound be painful to the ear? b) At what distance from the speaker would the sound be barely audible?



4. Beats .

Problem 4.1: Two strings are vibrating at the same frequency of 150 Hz. After the tension in one of the strings is decreased, an observer hears four beats each second when the strings vibrate together. Find the new frequency in the adjusted string.

Problem 4.2: While attempting to tune the note C at 523 Hz, a piano tuner hears 2.00 beats/s between

a reference oscillator and the string. a) What are the possible frequencies of the string?

b) When she tightens the string slightly, she hears 3.00 beats/s. What is the frequency of

the string now?

Problem 4.3: A violinist is tuning her instrument to concert A (440 Hz). She plays the note while listening

to an electronically generated tone of exactly that frequency and hears a beat frequency of 5 Hz, which increases to 6 Hz when she tightens her violin string slightly. a) What was the frequency of the note played by her violin when she heard the 5-Hz

beats?

b) To get her violin perfectly tuned to concert A, should she tighten or loosen her string from what it was when she heard the 5-Hz beat?

Problem 4.4: Two guitarists attempt to play the same note of wavelength 6.50 cm at the same time,

but one of the instruments is slightly out of tune and plays a note of wavelength 6.52 cm instead. What is the frequency of the beats these musicians hear when they play together? Assume the speed of sound in the air to be v = 344 m/s.



5. Resonance .

String fixed at both ends.

Problem 5.1: One of the 63.5-cm-long strings of an ordinary guitar is tuned to produce the note B3 (frequency 245 Hz) when vibrating in its fundamental mode. a) Find the speed of transverse waves on this string.

b) If the speed of sound in the surrounding air is 344 m/s, find the frequency and wavelength of the sound wave produced in the air by the vibration of the B3 string. How do these compare to the frequency and wavelength of the standing wave on the string?

Problem 5.2: A piano tuner stretches a steel piano wire with a tension of 800 N. The steel wire is 0.400

m long and has a mass of 3.00 g. What is the frequency of its fundamental mode of vibration?

Problem 5.3: A 75.0-cm-long wire of mass 5.625 g is tied at both ends and adjusted to a tension of 35.0

N. When it is vibrating in its second overtone, find a) the frequency and wavelength at which it is vibrating.

b) the frequency and wavelength of the sound waves it is vibrating.



Problem 5.4: One string of a certain musical instrument is 80.0 cm long and has a mass of 8.71 g. It is being played in a room where the speed of sound is 344 m/s. a) To what tension must you adjust the string so that, when vibrating in its second

overtone, it produces sound of wavelength 0.765 m?

b) What frequency sound does this string produce in its fundamental mode of vibration?

Problem 5.5: A musician tunes the C-string of her instrument to a fundamental frequency of 65.4 Hz. The vibrating portion of the string is 0.625 m long and has a mass of 14.7 g. With what tension must the musician stretch it?



Problem 5.6: The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is 60.0 cm long, and this length of the string has mass 2.00 g. The string sounds an A4 note (440 Hz) when played.

a) Where must the player put a finger (what distance x from the bridge) to play a note

D5 (587 Hz)? (See Fig. above) For both the A4 and D5 notes, the string vibrates in its fundamental mode.

b) Without retuning, is it possible to play a note G4 (392 Hz) on this string? Why or why not?

Additional Problem 5.6: A violin string has a length of 0.350 m and is tuned to concert G, with fG = 392 Hz. How far from the end of the string must the violinist place her finger to play concert A, with fA = 440 Hz?



Pipe closed at one end and Pipe open at both ends.

Problem 5.7: An air column in a glass tube is open at one end and closed at the other by a movable piston. The air in the tube is warmed above room temperature, and a 384-Hz tuning fork is held at the open end. Resonance is heard when the piston is at a distance d1 = 22.8 cm from the open end and again when it is at a distance d2 = 68.3 cm from the open end. a) What speed of sound is implied by these data?

b) How far from the open end will the piston be when the next resonance is heard?

Additional Problem 5.7: A long tube contains air. The tube is open at one end and closed at the other by a movable piston. A tuning fork that vibrates with a frequency of 500 Hz is placed near the open end. Resonance is produced when the piston is at a distances 18.0 cm, 55.5 cm, and 93.0 cm from the open end. From these values, what is the speed of sound in the air? ___________________________________________________________________________________________ Problem 5.8: You blow across the open mouth of an empty test tube and produce the fundamental

standing wave of the air column inside the test tube. The speed of sound in air is 344 m/s and the test tube acts as a stopped pipe. a) If the length of the air column in the test tube is 14.0 cm, what is the frequency of

this standing wave?

b) What is the frequency of the fundamental standing wave in the air column if the test tube is half filled with water?



Problem 5.9: The frequency of the note F4 is 349 Hz. If an organ pipe is open at one end and closed at the other, what length must it have for its fundamental mode to produce this note at 20.0 oC.

Problem 5.10: Two organ pipes, open at one end but closed at the other, are each 1.10 m long. One is now lengthened by 1.90 cm. Find the beat frequency they produce when playing together in their fundamental. Assume the speed of sound in air to be v = 344 m/s.



Problem 5.11: The fundamental frequency of a pipe that is open at both ends is 588 Hz. Assume the speed of sound wave in air to be v = 344 m/s. a) How long is this pipe?

b) If one end is now closed, find the wavelength and the frequency of the new fundamental.

Additional Problem 5.11: A glass tube (open at both ends) of length L is positioned near an audio speaker of frequency f = 680 Hz. For what values of L will the tube resonate with the speaker? Assume the speed of sound wave in air is 343 m/s.



Problem 5.12: The longest pipe found in most medium-size pipe organs is 4.88 m long. Assume the speed of sound in air to be v = 344 m/s. What is the frequency of the note corresponding to the fundamental mode if the pipe is a) open at both ends?

b) open at one end and closed at the other?

Problem 5.13: Standing sound waves are produced in a pipe that is 1.20 m long. Assume the speed of sound wave in air to be v = 344 m/s. For the fundamental and first two overtones, determine the locations along the pipe (measured from the open end) of the nodes if a) the pipe is open at both ends.

b) the pipe is closed at the left end and open at the right end.



Problem 5.14: The fundamental frequency of an open organ pipe corresponds to middle C (261.6 Hz on the chromatic musical scale). The third resonance of a closed organ pipe has the same frequency. Assume the speed of sound in air to be v = 344 m/s. What is the length of a) the open pipe?

b) the closed pipe?

Problem 5.15: An organ pipe has two successive harmonics with frequencies 1372 and 1764 Hz. Assume the speed of sound in air to be v = 344 m/s. a) Is this an open or a stopped pipe? b) What two harmonics are these?

c) What is the length of the pipe?



6. Shock waves .

Problem 6.1: A jet plane flies overhead at Mach 1.67 and at a constant altitude of 949 m. Assume the speed of sound in air to be v = 344 m/s. a) What is the angle of the shock-wave cone?

b) How much time after the plane passes directly overhead do you hear the sonic boom?

Problem 6.2: The shock-wave cone created by the space shuttle at one instant during its reentry into the atmosphere makes an angle of 61.0o with its direction of motion. The speed of sound at this altitude is 329 m/s. a) What is the Mach number of the shuttle at this instant?

b) how fast (in m/s and in km/h) is it traveling relative to the atmosphere?



7. Doppler effect .

Unless otherwise specified, the speed of sound in air is 344 m/s.

Problem 7.1: A railroad train is traveling at 24.0 m/s in still air. The frequency of the note emitted by the locomotive whistle is 390 Hz. Assume the speed of sound in air to be v = 344 m/s. What is the wavelength of the sound waves a) in front of the locomotive?

b) behind the locomotive?

What is the frequency of the sound heard by a stationary listener c) in front of the locomotive?

d) behind the locomotive?



Problem 7.2: a) A sound source producing 1.00-kHz waves moves toward a stationary listener at one-half the speed of

sound. What frequency will the listener hear?

b) Suppose instead that the source is stationary and the listener moves toward the source at one-half the speed of sound. What frequency does the listener hear?

Problem 7.3: A car alarm is emitting sound waves of frequency 520 Hz. You are on a motorcycle,

traveling directly away from the parked car. How fast must you be traveling if you detect a frequency of 490 Hz?



Problem 7.4: Two swift canaries fly toward each other, each moving at 16.0 m/s relative to the ground, each warbling a note of frequency 1780 Hz. a) What wavelength will each canary measure for the note from the other one?

b) What frequency note does each bird hear from the other one?

Problem 7.5: A railroad train is traveling at 30.0 m/s in still air. The frequency of the note emitted by the train whistle is 352 Hz. What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 m/s and a) approaching the first?

b) receding from the first?



Problem 7.6: While sitting in your car by the side of a country road, you are approached by your friend, who happens to be in an identical car. You blow your car’s horn, which has a frequency of 260 Hz. Your friend blows his car’s horn, which is identical to yours, and you hear a beat frequency of 6.0 Hz. How fast is your friend approaching you?

Problem 7.7: Two train whistles, A and B, each have a frequency of 392 Hz. A is stationary and B is

moving toward the right (away from A) at a speed of 35.0 m/s. A listener is between the two whistles and is moving toward the right with a speed of 15.0 m/s (Fig. below). No wind is blowing.

a) What is the frequency from A as heard by the listener?

b) What is the frequency from B as heard by the listener?

c) What is the beat frequency detected by the listener?

References

1. Young, H. D., Freedman, R. A. (2016). Sears and Zemansky’s University Physics with Modern Physics. (14th ed). Pearson.

2. Serway, R. A., Jewett, Jr., J. W. (2018). Physics for Scientists and Engineers with Modern Physics. (10th ed). Cengage learning.

3. Henderson, H. (2017). SAT Subject Test Physics. (10th ed). Grace Freedson’s Publishing Network.

4. The Staff of the Princeton Review. (2017). Cracking the SAT Subject Test in Physics. (16th ed). TPR Educational IP Holdings.

5. Jansen, R. and Young, G. (2016). BARRON’S SAT Subject Test in Physics. (2nd ed). Barron’s Educational Series.

6. ขวัญ อารยะธนิตกุล, นฤมล เอมะรัตต์, รัชภาคย์ จิตต์อารี และ เชิญโชค ศรขวัญ. (2558). ฟิสิกส์ 1. (ฉบับปรับปรุงครั้งที่ 9). ภาควิชาฟิสิกส์ คณะวิทยาศาสตร์ มหาวิทยาลัยมหิดล.

7. สถาบันส่งเสริมการสอนวิทยาศาสตร์และเทคโนโลยี. (2562). หนังสือเรียนรายวิชาเพิ ่มเติมวิทยาศาสตร์ ฟิสิกส์ เล่ม 3. (พิมพ์ครั้งที่ 1). กรุงเทพฯ: โรงพิมพ์ สกสค. ลาดพร้าว.

8. สถาบันส่งเสริมการสอนวิทยาศาสตร์และเทคโนโลยี. (2562). หนังสือเรียนรายวิชาเพิ ่มเติมวิทยาศาสตร์ ฟิสิกส์ เล่ม 4. (พิมพ์ครั้งที่ 1). กรุงเทพฯ: โรงพิมพ์ สกสค. ลาดพร้าว.

Biography

Name Date of Birth Place of Birth

Ariyaphol Jiwalak 4 July 1992 Chanthaburi, Thailand

Educational Background Mahidol University, 2011-2014 Bachelor of Science (Physics)

Mahidol University, 2015-2018 Master of Science (Physics, International Program) Royal Government of Thailand Scholarship

Development and Promotion of Science and Technology Talent Project (DPST) by the Institute for the Promotion of Teaching Science and Technology (IPST)

Home Address 5/371 Ideo Mobi Bangsue Grand Interchange Prachachuen Road, Bangsue Sub-district, Bangsue District, Bangkok, 10800

Tel. 089-7525223 E-mail: [email protected] Publication / Presentation Jiwalak A., Emarat N. and Arayathanitkul K., Students' physics laboratory skill in

measurement and uncertainty. Siam Physics Congress, 20-22 May 2015, Krabi, Thailand.

Jiwalak A., Emarat N. and Arayathanitkul K., An activity sheet for teaching double-slit interference. Siam Physics Congress, 21-23 May 2018, Phitsanulok, Thailand.

Demonstration School of Suan Sunandha Rajabhat University,

1 U-Thong nok Road, Dusit, Bangkok 10300 Thailand.

Documents

Physics 3 (SCI 32201 - SSRU