Lecture 14 1/38 Phys 220. Final Exam. Wednesday, August 6 th 10:30 am 12:30 pm Phys multiple choice problems (15 points each 300 total)

Transcription

1 Lecture 14 1/38 Phys 220 Final Exam Wednesday, August 6 th 10:30 am 12:30 pm Phys multiple choice problems (15 points each 300 total) 75% will be from Chapters % from Chapters 1-9 Students needing accommodations must contact me by July 30 th Equation Sheet will be provided Practice Exam and Equation Sheet on course website

2 Lecture 14 2/38 Phys 220 Physics Topics Kinematics Forces, Torque, and Work Conservation of Energy Conservation of Momentum (Linear & Angular) Density and Pressure Harmonic Motion Traveling Waves Standing Waves

3 Lecture 14 3/38 Phys 220 Standing Waves Standing waves are formed by interference of two waves traveling in opposite directions Waves travel along the string and are reflected from the ends Points where the string displacement is zero are called nodes largest are called antinodes Many standing waves may fit into n = 1 the length of the string n = 2 n = 3 n 2L n n 1, 2,3,

4 Lecture 14 4/38 Phys 220 Standing Waves Fixed endpoints Fundamental frequency means n=1 v = f n = 1 n = 2 n 2L n n = 3 nv f n 2L f n = n f 1 No energy is transmitted by a standing wave n = 4

5 Lecture 14 5/38 Phys 220 Harmonics Longest wavelength has smallest frequency fundamental frequency (f 1 ) which means n=1 Next longest wavelength second harmonic (f 2 ) n=2 Pattern continues n = 1 n = 2 n 2L n nv f n 2L n = 3 Note that these above equations are only valid for standing waves on a string fixed at both ends n = 4 f n n f 1

6 Musical Tones Lecture 14 6/38 Phys 220 Many musical instruments use strings as a vibrating element Your fingers press down on the string and changes its length The string vibrates with all the possible standing wave pattern frequencies The pitch of note is determined by its fundamental frequency Two notes whose fundamental frequencies differ by a factor of 2 are said to be separated by an octave Combination of harmonics determine sound of instruments Each instrument have different frequencies that are loud and others that are quiet, etc.

7 Lecture 14 7/38 Phys 220 Seismic Waves Seismic waves propagate through the Earth Source can be any large mechanical disturbance such as an earthquake Three types of seismic waves S waves (S for shear) transverse waves P waves (P for pressure) longitudinal sound waves Surface waves similar to water waves Seismic waves can be detected by a seismograph

8 Lecture 14 8/38 Phys 220 Sound Waves Sound waves are an important example of wave motion Differences in pressure can be modeled by a wave Formed by the vibration of the speaker Particles move back and forth to form areas of low and high pressure Overall particles move away from the source Most sounds are a combination of many frequencies

9 Lecture 14 9/38 Phys 220 Speed of Sound Adjacent regions of high pressure are separated by a distance equal to the wavelength Adjacent regions of low pressure are also separated by the wavelength The speed of the sound wave is given by v f The velocity of the wave and the velocity of the particles within the medium are not the same The speed of sound depends on the medium the wave Medium Speed (m/s) is traveling through Air 343 In a gas, the speed of sound Helium 1000 also depends on temperature of the gas: Water 1480 v sound ( T 20 C) m/s Steel 5790

10 Lecture 14 10/38 Phys 220 Intensity and Pitch For Sound Waves I = p 2 0 / (2 r v) (p o is the pressure amplitude) Proportional to p 2 0 Loudness Loudness perception is logarithmic Threshold for hearing I 0 = W/m 2 b = (10 db) log 10 ( I / I 0 ) b 2 b 1 = (10 db) log 10 (I 2 /I 1 ) Pitch Perception of frequency

11 Lecture 14 11/38 Phys 220 Decibels Sound intensity level b b = (10 db) log 10 ( I / I 0 ) Units: Bels (Named after A.G. Bell) I I W/m W/m A ratio of 10 7 indicates a sound intensity of 7 bels or 70 decibels (db) An intensity of 0 db corresponds to the hearing threshold Intensity increase by factor 10 intensity level adds 10 db Adding 3 db to the intensity level doubles the intensity (log 10 2=0.3) Alexander Graham Bell ( ) b (10 db)log 10 I I 0

12 Lecture 14 12/38 Phys 220 Human Perception of Sound The lowest line plots the intensity of the minimum detectable sound as a function of the frequency Doubling the perceived loudness corresponds to moving from one contour line to the next The highest contour curve has a loudness at which the ear is damaged Occurs at β 120 db The loudness contours depend on frequency Frequency range of normal human hearing is 20 Hz to 20 khz Sound waves below 20 Hz are called infrasonic Sound waves above 20 khz are called ultrasonic

13 Lecture 14 13/38 Phys 220 Question If 1 person can shout with loudness 50 db. How loud will it be when 100 people shout? A) 52 db B) 70 db C) 150 db b 100 b 1 = (10 db) log 10 (I 100 /I 1 ) b 100 = b 1 +(10 db) log 10 (I 100 /I 1 ) b 100 = 50 + (10 db) log 10 (100/1) b 100 = 50 + (10 db) log 10 (100) b 100 =

14 Lecture 14 14/38 Phys 220 Standing Sound Waves Pipe open (or closed) at both ends 2 n (n = 1, 2, 3, ) n L f n nv 2L sound

15 Lecture 14 15/38 Phys 220 Standing Sound Waves Pipe open at one end 4L (n = 1, 3, 5, ) n n f n nv 4L sound

16 Lecture 14 16/38 Phys 220 Demo Standing Waves

17 Lecture 14 17/38 Phys 220 Standing Waves in Pipes Open at both ends: Pressure node at end = (2L)/n (n=1,2,3,...) Open at one end: Pressure antinode at closed end = (4L)/n (n=1,3,5,...)

18 Lecture 14 18/38 Phys 220 Demo Standing Waves in a Gas

19 Lecture 14 19/38 Phys 220 Organ Pipe Example A 0.9 m organ pipe (open at both ends) is measured to have it s n=2 harmonic at a frequency of 382 Hz. What is the speed of sound in the pipe? Pressure node at each end. = 2 L / n n=1,2,3.. = L for n=2 harmonic f = v / v = f = (382 s -1 ) (0.9 m) = 343 m/s

20 Lecture 14 20/38 Phys 220 Question What happens to the fundamental frequency of a closed pipe, if the air (v=343 m/s) is replaced by helium (v=972 m/s)? A) Increases B) Same C) Decreases

21 Lecture 14 21/38 Phys 220 Demo Monochord

22 Lecture 14 22/38 Phys 220 Timbre Timbre is the combination of standing waves of different frequencies in an the instrument depends on the mix of frequencies different for different instruments and how the instrument is played Fourier decomposition Fundamental + Overtones

23 Lecture 14 23/38 Phys 220 Superposition & Interference Consider two harmonic waves A and B meeting at x=0. Same amplitudes, but 2 = 1.15 x 1. The displacement versus time for each is shown below: A( 1 t) B( 2 t) What does C(t) = A(t) + B(t) look like?

24 Lecture 14 24/38 Phys 220 Superposition & Interference Consider two harmonic waves A and B meeting at x=0. Same amplitudes, but 2 = 1.15 x 1. The displacement versus time for each is shown below: A( 1 t) B( 2 t) C(t) = A(t) + B(t) DESTRUCTIVE INTERFERENCE CONSTRUCTIVE INTERFERENCE

25 Lecture 14 25/38 Phys 220 Beats Add two cosines and remember the identity: Acos( 1t) Acos( 2t) 2Acos( L t) Acos( H t) 1 1 where L ( 1 2 ) and H ( 1 2) 2 2 cos( L t) Beat Frequency: beat 2 L or f beat f

26 Lecture 14 26/38 Phys 220 Beat Frequency Two different waves with close frequencies Amplitudes must be similar The frequency of the oscillations is the beat frequency, ƒ beat f beat f 1 f 2 Beats can be used to tune musical instruments

27 Lecture 14 27/38 Phys 220 Question What is the beat frequency given these two frequencies? A) 6 Hz B) 7 Hz C) -2 Hz D) 3 Hz Two sound waves are shown at the right. The first frequency is 250 Hz. The second frequency is 253 Hz.

28 Lecture 14 28/38 Phys 220 Doppler Effect Moving Source When source is coming toward you (v s > 0) Distance between waves decreases Frequency increases When source is going away from you (v s < 0) Distance between waves increases Frequency decreases f obs f source 1 1 v v source sound source moving towards + source moving away

29 Lecture 14 29/38 Phys 220 Question A police car passes you with its siren on. The frequency of the sound you hear from its siren after the car passes A) Increases B) Decreases C) Same

30 Lecture 14 30/38 Phys 220 Doppler Effect Moving Observer When moving toward source (v o < 0) Distance between waves decreases Frequency increases When moving away from source (v o > 0) Distance between waves increases Frequency decreases v f obs fsource 1 v obs sound + moving towards source moving away from source

31 Lecture 14 31/38 Phys 220 Moving Observer and Source Combine moving observer and moving source equations Frequency is increased when moving toward each other Frequency is decreased when moving away from each other f obs f source v 1 v v 1 v obs sound source sound moving towards + on top, on bottom (1 + v obs /v sound ) on top (1 v source /v sound ) on bottom moving away on top, + on bottom (1 v obs /v sound ) on top (1 + v source /v sound ) on bottom

32 Lecture 14 32/38 Phys 220 Demo Doppler Effect

33 Lecture 14 33/38 Phys 220 Question A: You are driving along the highway at 65 mph, and behind you a police car, also traveling at 65 mph, has its siren turned on. B: You and the police car have both pulled over to the side of the road, but the siren is still turned on. In which case, is the observed frequency of the siren higher? A) Case A B) Case B C) same f v f v s v o

34 Lecture 14 34/38 Phys 220 Doppler Effect Example: Speed Gun A speed gun uses the Doppler effect with reflected electromagnetic waves to measure the speed of a moving object As the initial waves hit the object, it acts like a moving observer For the scattered rays, the object acts like a moving source The speed gun uses the Dopplershifted frequency to deduce the speed of the object

35 Lecture 14 35/38 Phys 220 Doppler Effect Example: Bats Bats employ an approach similar to the speed gun Bats generate pulses of ultrasonic waves These waves are reflected and the bat uses the reflected sound to judge its environment

36 Lecture 14 36/38 Phys 220 Shock Waves The speed of the wave s source can be greater than the speed of the wave When the source moves at or faster than the speed of the wave, the pressure in front of the object piles up along a conic envelope As many different sound waves arrive at a point on the envelope, they produce a very loud sound intensity and form a shock wave

37 Lecture 14 37/38 Phys 220 Ultrasound Images Ultrasonic imaging uses sound waves to obtain images inside a material The most familiar use is to produce views inside the human body The depth of objects inside the body is determined by the time it takes the reflected ray to return to the detector Some medical ultrasounds also use the Doppler effect

38 Lecture 14 38/38 Phys 220 If a car is not moving, the sound waves from the horn move out evenly, as shown in the figure below. Question Where is the frequency higher for a stationary observer if the car is moving to the right? A) at the blue side B) at the red side C) both sides see the same frequency The car is moving to the right, and the sound waves from the horn move out from the car, as shown above.