SlideShare a Scribd company logo
Beat Frequency
● Interference of two sound waves with slightly
different frequencies
 The sound of the beat is from the increase
and decrease in amplitude as a function of
time
What is a Beat?
What is Beat Frequency?
● the rate of beats or the number of beats per second
● difference between the frequency of two interacting
sound waves
fb = beat frequency
f2 = frequency of one wave
f1 = frequency of the other wave
Formula to Calculate Beat Frequency
fb = | f2 - f1 |
Equal to the difference between the absolute value of the frequencies of two
interacting waves
Units: Hertz (Hz)
- translates to beats per
second
Jessica and Simon both answer the following question: A tone
of 230 Hz and 235 Hz are played at the same time. What is the
beat frequency?
Their work is shown below:
Question 1:
How do you calculate the beat frequency?
Who’s calculations are correct?
Jessica’s, Simon’s or both? Explain why.
Jessica
fb = | f2 - f1 |
= |235 Hz - 230 Hz|
= 5 Hz
Simon
fb = | f1 - f2 |
= |230 Hz - 235 Hz|
= - 5 Hz
● Jessica is correct with her answer of 5 Hz
● Simon’s equation was still correct as he made the correct
substitutions for the f1 and f2
● His mistake was that he did not account for the absolute value in his
final answer.
o It is impossible for frequency to be negative it is considered to be
beats per second. Thus, the value can only be a positive quantity.
Solution
● You will hear beats when two or more sounds have a beat
frequency of 20-30 Hz
● You will hear a tone when two or more sounds have a beat
frequency greater than 30 Hz
● The greater the beat frequency, the easier it is to hear the
beats
● It is fairly easy to detect beat frequencies as low as 1 Hz
Detecting Beat Frequencies
If two waves have a beat frequency of 0.5 Hz and
two other waves have a beat frequency of 15 Hz,
which modulation would you detect first?
Question 2: Speed in Different Mediums
● You would detect the 15 Hz beat frequency first as it is greater than
0.5 Hz.
● A greater difference in the frequencies of two waves makes it easier
for humans to hear beats.
Note:
When there is a large difference between beat frequencies, you begin to
hear the sound of the beats. This is why string instruments are tuned so
that the frequency can be adjusted until the beating is minimized.
Solution
● Askill, John, Dr. "Physics of Music Chapter 3 by Dr. John Askill." Physics of Music Chapter 3 by Dr. John Askill.
N.p., n.d. Web. 16 Mar. 2015.
<http://fiziks.net/Music%20Sample%20Chapter%20Three/musicsamplechapter3.htm>.
● "Beats." Beat Frequencies. N.p., n.d. Web. 16 Mar. 2015.<http://hyperphysics.phy-
astr.gsu.edu/hbase/sound/beat.html>.
● "Beats." Beats. Michigan Tech, n.d. Web. 16 Mar. 2015. <http://www.phy.mtu.edu/~suits/beats.html>.
● Hawkes, Robert. Physics for Scientists and Engineers: An Interactive Approach. Toronto: Nelson Education,
2014. Print.
Citations

More Related Content

Viewers also liked

Groovy for Domain-Specific Languages
Groovy for Domain-Specific LanguagesGroovy for Domain-Specific Languages
Groovy for Domain-Specific Languages
Yaroslav Yermilov
 
Super Superpostion
Super SuperpostionSuper Superpostion
Super Superpostion
mrmeredith
 
Harley Learning Object 6
Harley Learning Object 6Harley Learning Object 6
Harley Learning Object 6
Harley Ma
 
7 superposition and standing waves
7 superposition and standing waves7 superposition and standing waves
7 superposition and standing waves
MissingWaldo
 
Standing waves and music
Standing waves and musicStanding waves and music
Standing waves and music
Alison Wu
 
Waves%20review%20practice%20questions%20with%20answers
Waves%20review%20practice%20questions%20with%20answersWaves%20review%20practice%20questions%20with%20answers
Waves%20review%20practice%20questions%20with%20answers
abu xaatim ibnu axmed
 
Standing Waves
Standing WavesStanding Waves
Standing Waves
snp96
 
Stationary Waves and Progressive Waves
Stationary Waves and Progressive WavesStationary Waves and Progressive Waves
Stationary Waves and Progressive Waves
Ayushmaan_02
 
Standing Waves
Standing  WavesStanding  Waves
Standing Waves
PhysicsJackson
 
Standing waves
Standing wavesStanding waves
Standing waves
annayys
 
Waves and Sound
Waves and SoundWaves and Sound
Waves and Sound
itutor
 
Standing waves final ppt
Standing waves final pptStanding waves final ppt
Standing waves final ppt
arbinddhaliwal
 
The characteristics of sound!
The characteristics of sound!The characteristics of sound!
The characteristics of sound!
tr00967
 
Characteristics of sound
Characteristics of soundCharacteristics of sound
Characteristics of sound
Department of Education - Philippines
 
Characteristic & Classification of Sound
Characteristic & Classification of SoundCharacteristic & Classification of Sound
Characteristic & Classification of Sound
Pratik Nakrani
 
CBSE Class IX Sciense Physics Sound
CBSE Class IX Sciense Physics SoundCBSE Class IX Sciense Physics Sound
CBSE Class IX Sciense Physics Sound
Pranav Ghildiyal
 
Characteristics of sound,teacherpres,ppt
Characteristics of sound,teacherpres,pptCharacteristics of sound,teacherpres,ppt
Characteristics of sound,teacherpres,ppt
Felix Bunagan
 
Sound Waves
Sound WavesSound Waves
Sound Waves
itutor
 
Sound - Physics
Sound - PhysicsSound - Physics
Sound - Physics
Disha Shahani
 
Properties of Sound
Properties of SoundProperties of Sound
Properties of Sound
I Wonder Why Science
 

Viewers also liked (20)

Groovy for Domain-Specific Languages
Groovy for Domain-Specific LanguagesGroovy for Domain-Specific Languages
Groovy for Domain-Specific Languages
 
Super Superpostion
Super SuperpostionSuper Superpostion
Super Superpostion
 
Harley Learning Object 6
Harley Learning Object 6Harley Learning Object 6
Harley Learning Object 6
 
7 superposition and standing waves
7 superposition and standing waves7 superposition and standing waves
7 superposition and standing waves
 
Standing waves and music
Standing waves and musicStanding waves and music
Standing waves and music
 
Waves%20review%20practice%20questions%20with%20answers
Waves%20review%20practice%20questions%20with%20answersWaves%20review%20practice%20questions%20with%20answers
Waves%20review%20practice%20questions%20with%20answers
 
Standing Waves
Standing WavesStanding Waves
Standing Waves
 
Stationary Waves and Progressive Waves
Stationary Waves and Progressive WavesStationary Waves and Progressive Waves
Stationary Waves and Progressive Waves
 
Standing Waves
Standing  WavesStanding  Waves
Standing Waves
 
Standing waves
Standing wavesStanding waves
Standing waves
 
Waves and Sound
Waves and SoundWaves and Sound
Waves and Sound
 
Standing waves final ppt
Standing waves final pptStanding waves final ppt
Standing waves final ppt
 
The characteristics of sound!
The characteristics of sound!The characteristics of sound!
The characteristics of sound!
 
Characteristics of sound
Characteristics of soundCharacteristics of sound
Characteristics of sound
 
Characteristic & Classification of Sound
Characteristic & Classification of SoundCharacteristic & Classification of Sound
Characteristic & Classification of Sound
 
CBSE Class IX Sciense Physics Sound
CBSE Class IX Sciense Physics SoundCBSE Class IX Sciense Physics Sound
CBSE Class IX Sciense Physics Sound
 
Characteristics of sound,teacherpres,ppt
Characteristics of sound,teacherpres,pptCharacteristics of sound,teacherpres,ppt
Characteristics of sound,teacherpres,ppt
 
Sound Waves
Sound WavesSound Waves
Sound Waves
 
Sound - Physics
Sound - PhysicsSound - Physics
Sound - Physics
 
Properties of Sound
Properties of SoundProperties of Sound
Properties of Sound
 

Similar to LO 7/8

Lo8 upload
Lo8 uploadLo8 upload
Lo8 upload
Julia Zhu
 
nLo2
nLo2nLo2
LD2
LD2LD2
LD2
LD2LD2
LD2
LD2LD2
Lo2
Lo2Lo2
Learning Objective 7 PHYS 101
Learning Objective 7 PHYS 101Learning Objective 7 PHYS 101
Learning Objective 7 PHYS 101
cindyshenn
 
Beats
BeatsBeats
Beats
Hedy Liang
 
Lo hedy liang(19403147)
Lo hedy liang(19403147)Lo hedy liang(19403147)
Lo hedy liang(19403147)
Hedy Liang
 
Beats
BeatsBeats
Beats
Edie Ye
 
Physics 101 lo8 (week 10)
Physics 101   lo8 (week 10)Physics 101   lo8 (week 10)
Physics 101 lo8 (week 10)
Leslie Liu
 
Sound&amp;resonance
Sound&amp;resonanceSound&amp;resonance
Sound&amp;resonance
Subas Nandy
 
foundation of Physics
foundation of Physics foundation of Physics
foundation of Physics
karangoyal972
 
Learning Objects 7 - Beats and Beat Frequency
Learning Objects 7 - Beats and Beat FrequencyLearning Objects 7 - Beats and Beat Frequency
Learning Objects 7 - Beats and Beat Frequency
Gillian Trotter
 
Sound waves
Sound wavesSound waves
Sound waves
shaffelder
 
Phys 101 lo #6
Phys 101 lo #6Phys 101 lo #6
Phys 101 lo #6
Tony Mo
 
Lo8
Lo8Lo8
Speech, hearing, noise, intelligibility.pptx
Speech, hearing, noise, intelligibility.pptxSpeech, hearing, noise, intelligibility.pptx
Speech, hearing, noise, intelligibility.pptx
priyankatabhane
 
Sound Waves
Sound WavesSound Waves
Sound Waves
Bruce Coulter
 
Beats
BeatsBeats

Similar to LO 7/8 (20)

Lo8 upload
Lo8 uploadLo8 upload
Lo8 upload
 
nLo2
nLo2nLo2
nLo2
 
LD2
LD2LD2
LD2
 
LD2
LD2LD2
LD2
 
LD2
LD2LD2
LD2
 
Lo2
Lo2Lo2
Lo2
 
Learning Objective 7 PHYS 101
Learning Objective 7 PHYS 101Learning Objective 7 PHYS 101
Learning Objective 7 PHYS 101
 
Beats
BeatsBeats
Beats
 
Lo hedy liang(19403147)
Lo hedy liang(19403147)Lo hedy liang(19403147)
Lo hedy liang(19403147)
 
Beats
BeatsBeats
Beats
 
Physics 101 lo8 (week 10)
Physics 101   lo8 (week 10)Physics 101   lo8 (week 10)
Physics 101 lo8 (week 10)
 
Sound&amp;resonance
Sound&amp;resonanceSound&amp;resonance
Sound&amp;resonance
 
foundation of Physics
foundation of Physics foundation of Physics
foundation of Physics
 
Learning Objects 7 - Beats and Beat Frequency
Learning Objects 7 - Beats and Beat FrequencyLearning Objects 7 - Beats and Beat Frequency
Learning Objects 7 - Beats and Beat Frequency
 
Sound waves
Sound wavesSound waves
Sound waves
 
Phys 101 lo #6
Phys 101 lo #6Phys 101 lo #6
Phys 101 lo #6
 
Lo8
Lo8Lo8
Lo8
 
Speech, hearing, noise, intelligibility.pptx
Speech, hearing, noise, intelligibility.pptxSpeech, hearing, noise, intelligibility.pptx
Speech, hearing, noise, intelligibility.pptx
 
Sound Waves
Sound WavesSound Waves
Sound Waves
 
Beats
BeatsBeats
Beats
 

Recently uploaded

Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
PsychoTech Services
 
Immersive Learning That Works: Research Grounding and Paths Forward
Immersive Learning That Works: Research Grounding and Paths ForwardImmersive Learning That Works: Research Grounding and Paths Forward
Immersive Learning That Works: Research Grounding and Paths Forward
Leonel Morgado
 
23PH301 - Optics - Optical Lenses.pptx
23PH301 - Optics  -  Optical Lenses.pptx23PH301 - Optics  -  Optical Lenses.pptx
23PH301 - Optics - Optical Lenses.pptx
RDhivya6
 
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Describing and Interpreting an Immersive Learning Case with the Immersion Cub...
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...
Leonel Morgado
 
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...
Sérgio Sacani
 
Microbiology of Central Nervous System INFECTIONS.pdf
Microbiology of Central Nervous System INFECTIONS.pdfMicrobiology of Central Nervous System INFECTIONS.pdf
Microbiology of Central Nervous System INFECTIONS.pdf
sammy700571
 
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)
eitps1506
 
Pests of Storage_Identification_Dr.UPR.pdf
Pests of Storage_Identification_Dr.UPR.pdfPests of Storage_Identification_Dr.UPR.pdf
Pests of Storage_Identification_Dr.UPR.pdf
PirithiRaju
 
Methods of grain storage Structures in India.pdf
Methods of grain storage Structures in India.pdfMethods of grain storage Structures in India.pdf
Methods of grain storage Structures in India.pdf
PirithiRaju
 
Physiology of Nervous System presentation.pptx
Physiology of Nervous System presentation.pptxPhysiology of Nervous System presentation.pptx
Physiology of Nervous System presentation.pptx
fatima132662
 
Gadgets for management of stored product pests_Dr.UPR.pdf
Gadgets for management of stored product pests_Dr.UPR.pdfGadgets for management of stored product pests_Dr.UPR.pdf
Gadgets for management of stored product pests_Dr.UPR.pdf
PirithiRaju
 
cathode ray oscilloscope and its applications
cathode ray oscilloscope and its applicationscathode ray oscilloscope and its applications
cathode ray oscilloscope and its applications
sandertein
 
TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx
TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptxTOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx
TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx
shubhijain836
 
Anti-Universe And Emergent Gravity and the Dark Universe
Anti-Universe And Emergent Gravity and the Dark UniverseAnti-Universe And Emergent Gravity and the Dark Universe
Anti-Universe And Emergent Gravity and the Dark Universe
Sérgio Sacani
 
Clinical periodontology and implant dentistry 2003.pdf
Clinical periodontology and implant dentistry 2003.pdfClinical periodontology and implant dentistry 2003.pdf
Clinical periodontology and implant dentistry 2003.pdf
RAYMUNDONAVARROCORON
 
Sustainable Land Management - Climate Smart Agriculture
Sustainable Land Management - Climate Smart AgricultureSustainable Land Management - Climate Smart Agriculture
Sustainable Land Management - Climate Smart Agriculture
International Food Policy Research Institute- South Asia Office
 
The cost of acquiring information by natural selection
The cost of acquiring information by natural selectionThe cost of acquiring information by natural selection
The cost of acquiring information by natural selection
Carl Bergstrom
 
Direct Seeded Rice - Climate Smart Agriculture
Direct Seeded Rice - Climate Smart AgricultureDirect Seeded Rice - Climate Smart Agriculture
Direct Seeded Rice - Climate Smart Agriculture
International Food Policy Research Institute- South Asia Office
 
GBSN - Biochemistry (Unit 6) Chemistry of Proteins
GBSN - Biochemistry (Unit 6) Chemistry of ProteinsGBSN - Biochemistry (Unit 6) Chemistry of Proteins
GBSN - Biochemistry (Unit 6) Chemistry of Proteins
Areesha Ahmad
 
Summary Of transcription and Translation.pdf
Summary Of transcription and Translation.pdfSummary Of transcription and Translation.pdf
Summary Of transcription and Translation.pdf
vadgavevedant86
 

Recently uploaded (20)

Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
 
Immersive Learning That Works: Research Grounding and Paths Forward
Immersive Learning That Works: Research Grounding and Paths ForwardImmersive Learning That Works: Research Grounding and Paths Forward
Immersive Learning That Works: Research Grounding and Paths Forward
 
23PH301 - Optics - Optical Lenses.pptx
23PH301 - Optics  -  Optical Lenses.pptx23PH301 - Optics  -  Optical Lenses.pptx
23PH301 - Optics - Optical Lenses.pptx
 
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Describing and Interpreting an Immersive Learning Case with the Immersion Cub...
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...
 
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...
 
Microbiology of Central Nervous System INFECTIONS.pdf
Microbiology of Central Nervous System INFECTIONS.pdfMicrobiology of Central Nervous System INFECTIONS.pdf
Microbiology of Central Nervous System INFECTIONS.pdf
 
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)
 
Pests of Storage_Identification_Dr.UPR.pdf
Pests of Storage_Identification_Dr.UPR.pdfPests of Storage_Identification_Dr.UPR.pdf
Pests of Storage_Identification_Dr.UPR.pdf
 
Methods of grain storage Structures in India.pdf
Methods of grain storage Structures in India.pdfMethods of grain storage Structures in India.pdf
Methods of grain storage Structures in India.pdf
 
Physiology of Nervous System presentation.pptx
Physiology of Nervous System presentation.pptxPhysiology of Nervous System presentation.pptx
Physiology of Nervous System presentation.pptx
 
Gadgets for management of stored product pests_Dr.UPR.pdf
Gadgets for management of stored product pests_Dr.UPR.pdfGadgets for management of stored product pests_Dr.UPR.pdf
Gadgets for management of stored product pests_Dr.UPR.pdf
 
cathode ray oscilloscope and its applications
cathode ray oscilloscope and its applicationscathode ray oscilloscope and its applications
cathode ray oscilloscope and its applications
 
TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx
TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptxTOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx
TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx
 
Anti-Universe And Emergent Gravity and the Dark Universe
Anti-Universe And Emergent Gravity and the Dark UniverseAnti-Universe And Emergent Gravity and the Dark Universe
Anti-Universe And Emergent Gravity and the Dark Universe
 
Clinical periodontology and implant dentistry 2003.pdf
Clinical periodontology and implant dentistry 2003.pdfClinical periodontology and implant dentistry 2003.pdf
Clinical periodontology and implant dentistry 2003.pdf
 
Sustainable Land Management - Climate Smart Agriculture
Sustainable Land Management - Climate Smart AgricultureSustainable Land Management - Climate Smart Agriculture
Sustainable Land Management - Climate Smart Agriculture
 
The cost of acquiring information by natural selection
The cost of acquiring information by natural selectionThe cost of acquiring information by natural selection
The cost of acquiring information by natural selection
 
Direct Seeded Rice - Climate Smart Agriculture
Direct Seeded Rice - Climate Smart AgricultureDirect Seeded Rice - Climate Smart Agriculture
Direct Seeded Rice - Climate Smart Agriculture
 
GBSN - Biochemistry (Unit 6) Chemistry of Proteins
GBSN - Biochemistry (Unit 6) Chemistry of ProteinsGBSN - Biochemistry (Unit 6) Chemistry of Proteins
GBSN - Biochemistry (Unit 6) Chemistry of Proteins
 
Summary Of transcription and Translation.pdf
Summary Of transcription and Translation.pdfSummary Of transcription and Translation.pdf
Summary Of transcription and Translation.pdf
 

LO 7/8

  • 2. ● Interference of two sound waves with slightly different frequencies  The sound of the beat is from the increase and decrease in amplitude as a function of time What is a Beat?
  • 3. What is Beat Frequency? ● the rate of beats or the number of beats per second ● difference between the frequency of two interacting sound waves
  • 4.
  • 5. fb = beat frequency f2 = frequency of one wave f1 = frequency of the other wave Formula to Calculate Beat Frequency fb = | f2 - f1 | Equal to the difference between the absolute value of the frequencies of two interacting waves Units: Hertz (Hz) - translates to beats per second
  • 6. Jessica and Simon both answer the following question: A tone of 230 Hz and 235 Hz are played at the same time. What is the beat frequency? Their work is shown below: Question 1: How do you calculate the beat frequency? Who’s calculations are correct? Jessica’s, Simon’s or both? Explain why. Jessica fb = | f2 - f1 | = |235 Hz - 230 Hz| = 5 Hz Simon fb = | f1 - f2 | = |230 Hz - 235 Hz| = - 5 Hz
  • 7. ● Jessica is correct with her answer of 5 Hz ● Simon’s equation was still correct as he made the correct substitutions for the f1 and f2 ● His mistake was that he did not account for the absolute value in his final answer. o It is impossible for frequency to be negative it is considered to be beats per second. Thus, the value can only be a positive quantity. Solution
  • 8. ● You will hear beats when two or more sounds have a beat frequency of 20-30 Hz ● You will hear a tone when two or more sounds have a beat frequency greater than 30 Hz ● The greater the beat frequency, the easier it is to hear the beats ● It is fairly easy to detect beat frequencies as low as 1 Hz Detecting Beat Frequencies
  • 9. If two waves have a beat frequency of 0.5 Hz and two other waves have a beat frequency of 15 Hz, which modulation would you detect first? Question 2: Speed in Different Mediums
  • 10. ● You would detect the 15 Hz beat frequency first as it is greater than 0.5 Hz. ● A greater difference in the frequencies of two waves makes it easier for humans to hear beats. Note: When there is a large difference between beat frequencies, you begin to hear the sound of the beats. This is why string instruments are tuned so that the frequency can be adjusted until the beating is minimized. Solution
  • 11. ● Askill, John, Dr. "Physics of Music Chapter 3 by Dr. John Askill." Physics of Music Chapter 3 by Dr. John Askill. N.p., n.d. Web. 16 Mar. 2015. <http://fiziks.net/Music%20Sample%20Chapter%20Three/musicsamplechapter3.htm>. ● "Beats." Beat Frequencies. N.p., n.d. Web. 16 Mar. 2015.<http://hyperphysics.phy- astr.gsu.edu/hbase/sound/beat.html>. ● "Beats." Beats. Michigan Tech, n.d. Web. 16 Mar. 2015. <http://www.phy.mtu.edu/~suits/beats.html>. ● Hawkes, Robert. Physics for Scientists and Engineers: An Interactive Approach. Toronto: Nelson Education, 2014. Print. Citations