CLASS 2 CLASS 2 Section 13.5
Simple Pendulum The simple pendulum is another example of a system that exhibits simple harmonic motion The force is the component of the weight tangent to the path of motion F t = - mg sin θ Section 13.5
Sound Waves Sound waves are longitudinal waves traveling through a medium Sound waves are produced from vibrating objects Introduction
Simple Pendulum In general, the motion of a pendulum is not simple harmonic However, for small angles, it becomes simple harmonic In general, angles < 15 are small enough sin θ θ F t = - mg θ = -mg (s/l) This force obeys Hooke s La: so simple harmonic motion
Period of Simple Pendulum This shows that the period is independent of the amplitude and the mass The period depends on the length of the pendulum and the acceleration of gravity at the location of the pendulum
Simple Pendulum Compared to a Spring-Mass System Section 13.5
Example A visitor to a lighthouse wishes to determine the height of the tower. She ties a spool of thread to a small rock to make a simple pendulum, which she hangs down the center of a spiral staircase of the tower. The period of oscillation is 9.40 s. What is the height of the tower? T T P 2 P 2 2 4 l g l g l l height 2 TP g 2 4 2 9.4 (9.8) 4(3.141592) 2 L = Height = 21.93 m
Damped Oscillations Only ideal systems oscillate indefinitely In real systems, friction retards the motion Friction reduces the total energy of the system and the oscillation is said to be damped
Damped Oscillations, cont. Damped motion varies depending on the fluid used With a low viscosity fluid, the vibrating motion is preserved, but the amplitude of vibration decreases in time and the motion ultimately ceases This is known as underdamped oscillation Section 13.6
Graphs of Damped Oscillators Curve a shows an underdamped oscillator Curve b shows a critically damped oscillator Curve c shows an overdamped oscillator Section 13.6
Wave Motion A wave is the motion of a disturbance Mechanical waves require Some source of disturbance A medium that can be disturbed Some physical connection or mechanism though which adjacent portions of the medium influence each other All waves carry energy and momentum
Types of Waves Traveling Waves Flip one end of a long rope that is under tension and fixed at the other end The pulse travels to the right with a definite speed A disturbance of this type is called a traveling wave Section 13.7
Types of Waves Transverse In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion Section 13.7
Types of Waves Longitudinal In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave A longitudinal wave is also called a compression wave
Other Types of Waves Waves may be a combination of transverse and longitudinal A soliton consists of a solitary wave front that propagates in isolation First studied by John Scott Russell in 1849 Now used widely to model physical phenomena
Waveform A Picture of a Wave The brown curve is a snapshot of the wave at some instant in time The blue curve is later in time The high points are crests of the wave The low points are troughs of the wave Section 13.7
Description of a Wave A steady stream of pulses on a very long string produces a continuous wave The blade oscillates in simple harmonic motion Each small segment of the string, such as P, oscillates with simple harmonic motion
Amplitude and Wavelength Amplitude is the maximum displacement of string above the equilibrium position Wavelength, λ, is the distance between two successive points that behave identically Section 13.8
Speed of a Wave v = ƒλ Is derived from the basic speed equation of distance/time This is a general equation that can be applied to many types of waves
Speed of a Wave on a String The speed on a wave stretched under some tension, F m is called the linear density, mass per unit length of string The speed depends only upon the properties of the medium through which the disturbance travels Section 13.9
Interference of Waves Two traveling waves can meet and pass through each other without being destroyed or even altered Waves obey the Superposition Principle When two or more traveling waves encounter each other while moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point Actually only true for waves with small amplitudes Section 13.10
Constructive Interference Two waves, a and b, have the same frequency and amplitude Are in phase The combined wave, c, has the same frequency and a greater amplitude Section 13.10
Constructive Interference in a String Two pulses are traveling in opposite directions The net displacement when they overlap is the sum of the displacements of the pulses Note that the pulses are unchanged after the interference Section 13.10
Destructive Interference Two waves, a and b, have the same amplitude and frequency One wave is inverted relative to the other They are 180 out of phase When they combine, the waveforms cancel Section 13.10
Destructive Interference in a String Two pulses are traveling in opposite directions The net displacement when they overlap is decreased since the displacements of the pulses subtract Note that the pulses are unchanged after the interference Section 13.10
Reflection of Waves Whenever a traveling wave reaches a boundary, some or all of the wave is reflected When it is reflected from a fixed end, the wave is inverted The shape remains the same Fixed End Section 13.11
Reflected Wave Free End When a traveling wave reaches a boundary, all or part of it is reflected When reflected from a free end, the pulse is not inverted Section 13.11
Sound Waves: Molecular View When sound travels through a medium, there are alternating regions of high and low pressure. Compressions are high pressure regions where the molecules are crowded together. Rarefactions are low pressure regions where the molecules are more spread out. An individual molecule moves side to side with each compression. The speed at which a compression propagates through the medium is the wave speed, but this is different than the speed of the molecules themselves. wavelength,
Using a Tuning Fork to Produce a Sound A tuning fork will produce a pure musical note As the tines vibrate, they disturb the air near them As the tine swings to the right, it forces the air molecules near it closer together This produces a high density area in the air This is an area of compression Wave Section 14.1
Using a Tuning Fork, cont. As the tine moves toward the left, the air molecules to the right of the tine spread out This produces an area of low density This area is called a rarefaction Section 14.1
Using a Tuning Fork, final λ As the tuning fork continues to vibrate, a succession of compressions and rarefactions spread out from the fork A sinusoidal curve can be used to represent the longitudinal wave Crests correspond to compressions and troughs to rarefactions Section 14.1
Classification of Sound Waves Audible waves Lay within the normal range of hearing of the human ear Normally between 20 Hz to 20 000 Hz Infrasonic waves Frequencies are below the audible range Earthquakes are an example Ultrasonic waves Frequencies are above the audible range Dog whistles are an example Section 14.2
Speed of Sound, General The speed of sound is higher in solids than in gases The molecules in a solid interact more strongly The speed is slower in liquids than in solids Liquids are more compressible Speed of sound only depends on the medium not on frequency or wavelength Speed is higher in more elastic medium Section 14.3
Speed of Sound in a Liquid In a fluid, the speed depends on the fluid s compressibility and inertia B is the Bulk Modulus of the liquid ρ is the density of the liquid Compares with the equation for a transverse wave on a string Speed of sound is less in more compressible medium Water, Air, Solid,? What order? Section 14.3
Speed of sound depends on medium: Quiz 3. Two waves are traveling through the same container of nitrogen gas. Wave A has a wavelength of 1.5 m. Wave B has a wavelength of 4.5 m. The speed of wave B must be the speed of wave A. a. one-ninth b. one-third c. the same as d. three times larger than Answer: C
Speed of sound depends on medium: Quiz 3. Doubling the frequency of a wave in the same medium doubles the speed of the waves. TRUE or FALSE: FALSE! What Ted by about the medium! v = f*λ? Wavelength becomes half when the frequency is doubled
Speed of Sound in a Solid Rod The speed depends on the rod s compressibility and inertial properties Y is the Young s Modulus of the material ρ is the density of the material Section 14.3
Speed of sound depends on medium In which solid speed of sound is higher? Lead or Steel? Why? Steel, because it has higher elastic modulus
Speed of sound in solid rods If the speed of sound in steel rod is V, what is the speed of sound in copper rod? Density of Steel = 8,050 kg/m 3 Density of Copper = 8960 kg/m 3 Youngs modulus of Steel = 2.00 x 10 11 N/m 2 Youngs modulus of Copper = 1.17 x 10 11 N/m 2
Speed of Sound in Air 331 m/s is the speed of sound at 0 C T is the absolute temperature Higher the temperature of air, higher is the speed of sound Section 14.3
Speed of sound in air What is the speed of sound in air at 27 C? Here T = 27 C = (27 + 273) = 300 K Must be converted into Kelvin Hence, 300 V = 346.8 m/s
Intensity of Sound Waves The average intensity I of a wave on a given surface is defined as the rate at which the energy flows through the surface, ΔE /Δt divided by the surface area, A The direction of energy flow is perpendicular to the surface at every point The rate of energy transfer is the power SI unit: W/m 2 Section 14.4
Various Intensities of Sound Threshold of hearing Faintest sound most humans can hear About 1 x 10-12 W/m 2 Threshold of pain Loudest sound most humans can tolerate About 1 W/m 2 The ear is a very sensitive detector of sound waves It can detect pressure fluctuations as small as about 3 parts in 10 10 Section 14.4
Frequency Response Curves Bottom curve is the threshold of hearing Threshold of hearing is strongly dependent on frequency Easiest frequency to hear is about 3300 Hz When the sound is loud (top curve, threshold of pain) all frequencies can be heard equally well Section 14.13
Intensity Level of Sound Waves The sensation of loudness is logarithmic in the human ear β is the intensity level or the decibel level of the sound I o is the threshold of hearing Section 14.4
Intensity Ratio problem What is the intensity ratio for 50 and 60 decibels?
Various Intensity Levels Threshold of hearing is 0 db Threshold of pain is 120 db Jet airplanes are about 150 db Multiplying a given intensity by 10 adds 10 db to the intensity level Section 14.4
Spherical Waves A spherical wave propagates radially outward from the oscillating sphere The energy propagates equally in all directions The intensity is Section 14.5
Intensity from a point source If 100J energy is given off by a point source in space in 30 seconds, what is its intensity at 5m? Intensity is energy received by a unit surface area (1 m 2 ) in unit time (1 second) I = E A t = 100 4πr 2 t = 100 4π 5 5 30 W/m2
Intensity of a Point Source Since the intensity varies as 1/r 2, this is an inverse square relationship The average power is the same through any spherical surface centered on the source To compare intensities at two locations, the inverse square relationship can be used Section 14.5
Representations of Waves Wave fronts are the concentric arcs The distance between successive wave fronts is the wavelength Rays are the radial lines pointing out from the source and perpendicular to the wave fronts Section 14.5
Wavefront Distance between two wave fronts is the wavelength
Plane Wave Far away from the source, the wave fronts are nearly parallel planes The rays are nearly parallel lines A small segment of the wave front is approximately a plane wave Section 14.5
Plane Waves, cont Any small portion of a spherical wave that is far from the source can be considered a plane wave This shows a plane wave moving in the positive x direction The wave fronts are parallel to the plane containing the y- and z- axes Section 14.5
Doppler Effect A Doppler effect is experienced whenever there is relative motion between a source of waves and an observer. When the source and the observer are moving toward each other, the observer hears a higher frequency When the source and the observer are moving away from each other, the observer hears a lower frequency Section 14.6
Doppler Effect, cont. Although the Doppler Effect is commonly experienced with sound waves, it is a phenomena common to all waves Section 14.6
Doppler Effect, Case 1 (Observer Toward Source) An observer is moving toward a stationary source Due to his movement, the observer detects an additional number of wave fronts The frequency heard is increased Section 14.6
Doppler Effect, Case 1 (Observer Away from Source) An observer is moving away from a stationary source The observer detects fewer wave fronts per second The frequency appears lower Section 14.6
Doppler Effect, Case 1 Equation Observer in motion Towards the source Away from the source Section 14.6
Doppler Effect, Case 2 (Source in Motion) As the source moves toward the observer (A), the wavelength appears shorter and the frequency increases As the source moves away from the observer (B), the wavelength appears longer and the frequency appears to be lower Section 14.6
Doppler Effect, Source Moving Equation Towards the observer Away from the observer Section 14.6
Doppler Effect, General Case source and observer both in motion Both the source and the observer could be moving Use positive values of v o and v s if the motion is towards the source or observer Frequency appears higher Use negative values of v o and v s if the motion is away from the source or observer Frequency appears lower Section 14.6
Example At rest, a car s horn sounds the note A (440 Hz). The horn is sounded while the car moves down the street. A bicyclist moving in the same direction at 10 m/s hears a frequency of 415 Hz. DATA: v sound = 343 m/s. What is the speed of the car? (Assume the cyclist is behind the car)
Group Task A student measures the frequency of a car s honk 400 Hz when both were at rest. (a) what will be his measurement when the car moves with a constant speed of 20 m/s away from him? The student now starts following the car at 10 m\s on his bike. (b) What will be his measurement now? In a third experiment, the student moves exactly opposite to the moving car at the same speed (10 m/s) and makes the measurement (c) what should be his reading now? Use speed of sound = 340 m\s
Shock Waves A shock wave is generated when the source velocity exceeds the speed of the wave itself The circles represent the wave fronts emitted by the source Section 14.6
Shock Waves, cont Tangent lines are drawn from S n to the wave front centered on S o The angle between one of these tangent lines and the direction of travel is given by sin θ = v / v s The ratio v s /v is called the Mach Number The conical wave front is the shock wave Section 14.6
Shock Waves, final Shock waves carry energy concentrated on the surface of the cone, with correspondingly great pressure variations A jet produces a shock wave seen as a fog of water vapor Section 14.6
Interference of Sound Waves Sound waves interfere Constructive interference occurs when the path difference between two waves motion is zero or some integer multiple of wavelengths Path difference = nλ (n = 0, 1, 2, ) Destructive interference occurs when the path difference between two waves motion is an odd half wavelength Path difference = (n + ½)λ (n = 0, 1, 2, ) Section 14.7
Standing Waves When a traveling wave reflects back on itself, it creates traveling waves in both directions The wave and its reflection interfere according to the superposition principle With exactly the right frequency, the wave will appear to stand still This is called a standing wave Section 14.8
Standing Waves on a String Nodes must occur at the ends of the string because these points are fixed Section 14.8
Standing Waves, cont A node occurs where the two traveling waves have the same magnitude of displacement, but the displacements are in opposite directions Net displacement is zero at that point The distance between two nodes is ½λ An antinode occurs where the standing wave vibrates at maximum amplitude Section 14.8
Standing Waves on a String, final The lowest frequency of vibration (b) is called the fundamental frequency (ƒ 1 ) Higher harmonics are positive integer multiples of the fundamental λ n = 1, 2, 3,. Section 14.8
Problem, classwork graded, 10 minutes Find the frequency of first and third harmonics on a 1 m long string tightly fixed at two ends with a tension of 100N. Diameter of the string is 10mm and the density of the string material is 8500 kg/m 3 Speed of the wave on the string is 5000 m/s Hint: calculate the mass of the string from its volume and density. Consider the string as a solid cylinder.
Standing Waves on a String Frequencies ƒ 1, ƒ 2, ƒ 3 form a harmonic series ƒ 1 is the fundamental and also the first harmonic ƒ 2 is the second harmonic or the first overtone Waves in the string that are not in the harmonic series are quickly damped out In effect, when the string is disturbed, it selects the standing wave frequencies Section 14.8
Forced Vibrations and Resonance A system with a driving force will cause a vibration at the frequency of the driving force When the frequency of the driving force equals the natural frequency of the system, the system is said to be in resonance Section 14.9
An Example of Resonance Pendulum A is set in motion The others begin to vibrate due to the vibrations in the flexible beam Pendulum C oscillates at the greatest amplitude since its length, and therefore its natural frequency, matches that of A Section 14.9
Standing Waves in Air Columns If one end of the air column is closed, a node must exist at this end since the movement of the air is restricted If the end is open, the elements of the air have complete freedom of movement and an antinode exists Section 14.10
Tube Open at Both Ends Tube open at both ends should have antinodes at both ends and node at the middle for the fundamental frequency Both end open tube, f 1 = V/2L f n = nf 1 = nv/2l for n = 1, 2, 3, 4, 5, Section 14.10
Tube Closed at One End Tube closed at end should have node at the closed end and antinode at the other end for the fundamental frequency Both end open tube, f 1 = V/4L f n = nf 1 = nv/4l for n = 1, 3, 5, Only odd harmonics are possible. Even harmonics are not allowed Section 14.10
Beats Beats are alternations in loudness, due to interference of two sound waves of slightly different frequencies. The beat frequency equals the difference in frequency between the two sources: Section 14.11
Beats, cont. Red arrows: destructive, Black arrows: Constructive Section 14.11
Quality of Sound Tuning Fork Tuning fork produces only the fundamental frequency Produces single frequency Section 14.12
Quality of Sound Flute The same note played on a flute sounds differently The second harmonic is very strong The fourth harmonic is close in strength to the first Section 14.12
Quality of Sound Clarinet The fifth harmonic is very strong The first and fourth harmonics are very similar, with the third being close to them Section 14.12
Timbre In music, the characteristic sound of any instrument is referred to as the quality of sound, or the timbre, of the sound The quality depends on the mixture of harmonics in the sound What is the difference between noise and music? Section 14.12
Pitch Pitch is related mainly, although not completely, to the frequency of the sound. Higher frequency has higher pitch Section 14.12
Summary Longitudinal Waves Velocity of sound in medium Doppler effect Intensity level Waves in String Waves in Pipes Musical notes