2 Forces and elastic materials Capable of recovering shape after deformationRubber ball versus lump of claySpring forcesApplied force proportional to distance spring is compressed or stretchedInternal restoring force arises, returning spring to original shapeRestoring force also proportional to stretched or compressed distance
3 Forces and vibrations Vibration - repetitive back and forth motion At the equilibrium position, spring is not compressedWhen disturbed from equilibrium position, restoring force acts toward equilibriumCarried by inertia past equilibrium to other extremeExample of “simple harmonic motion”
4 Simple Harmonic Motion A spring exerts a restoring force that is proportional to the displacement from equilibrium:
5 Periodic Motion Period: time required for one cycle of periodic motion Frequency: number of oscillations per unit timeThis unit is called the Hertz:
6 Describing vibrations Amplitude - maximum extent of displacement from equilibriumCycle - one complete vibrationPeriod - time for one cycleFrequency - number of cycles per second (units = hertz, Hz)Period and frequency inversely related
7 Simple Harmonic Motion If we call the period of the motion T – this is the time to complete one full cycle – we can write the position as a function of time:It is then straightforward to show that the position at time t + T is the same as the position at time t, as we would expect.
8 Connections between Uniform Circular Motion and Simple Harmonic Motion An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:
10 Energy Conservation in Oscillatory Motion In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:
11 Energy Conservation in Oscillatory Motion This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.
12 The PendulumA simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).The angle it makes with the vertical varies with time as a sine or cosine.
13 The PendulumLooking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).
14 The PendulumHowever, for small angles, sin θ and θ are approximately equal.
15 The PendulumSubstituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:
16 Driven Oscillations and Resonance An oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.
17 Driven Oscillations and Resonance If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.
18 Waves Periodic (traveling) disturbances transporting energy Causes Periodic motion disturbing surroundingsPulse disturbance of short durationMechanical wavesRequire medium for propagationWaves move through mediumMedium remains in place
19 Kinds of waves Longitudinal waves Vibration direction parallel to wave propagation directionParticles in medium move closer together/farther apartExample: sound wavesGases and liquids - support only longitudinal waves
20 Kinds of waves, cont. Transverse waves Vibration direction perpendicular to wave propagation directionExample: plucked stringSolids - support both longitudinal and transverse wavesSurface water wavesCombination of bothParticle motion = circular
21 Waves in air Longitudinal waves only Large scale - swinging door creates macroscopic currentsSmall scale - tuning fork creates sound wavesSeries of condensations (overpressures) and rarefactions (underpressures)
22 Types of WavesWater waves are a combination of transverse and longitudinal waves.
23 Describing waves Graphical representation Wave terminology Pure harmonic waves = sines or cosinesWave terminologyWavelengthAmplitudeFrequencyPeriodWave propagation speed
24 Waves on a StringThe speed of a wave is determined by the properties of the material through which it propagates.For a string, the wave speed is determined by:the tension in the string, andthe mass of the string.As the tension in the string increases, the speed of waves on the string increases as well.
25 Waves on a StringThe total mass of the string depends on how long it is; what makes a difference in the speed is the mass per unit length. We expect that a larger mass per unit length results in a slower wave speed.
26 14-2 Waves on a StringAs we can see, the speed increases when the force increases, and decreases when the mass increases.
27 Waves on a StringWhen a wave reaches the end of a string, it will be reflected. If the end is fixed, the reflected wave will be inverted:
28 Waves on a StringIf the end of the string is free to move transversely, the wave will be reflected without inversion.
29 Sound WavesSound waves are longitudinal waves, similar to the waves on a Slinky:Here, the wave is a series of compressions and stretches.
30 Sound WavesIn a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.
31 Sound WavesThe speed of sound is different in different materials; in general, the denser the material, the faster sound travels through it.
32 Sound WavesSound waves can have any frequency; the human ear can hear sounds between about 20 Hz and 20,000 Hz.Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are called infrasonic.Ultrasonic waves are familiar from medical applications; elephants and whales communicate, in part, by infrasonic waves.
33 Sound IntensityThe intensity of a sound is the amount of energy that passes through a given area in a given time.
35 14-5 Sound IntensitySound intensity from a point source will decrease as the square of the distance.
36 Sound IntensityWhen you listen to a variety of sounds, a sound that seems twice as loud as another is ten times more intense. Therefore, we use a logarithmic scale to define intensity values.Here, I0 is the faintest sound that can be heard:
37 Sound IntensityThe quantity β is called a bel; a more common unit is the decibel, dB, which is a tenth of a bel.The intensity of a sound doubles with each increase in intensity level of 10 dB.
38 The Doppler EffectThe Doppler effect is the change in pitch of a sound when the source and observer are moving with respect to each other.When an observer moves toward a source, the wave speed appears to be higher, and the frequency appears to be higher as well.
39 The Doppler EffectThe Doppler effect from a moving source can be analyzed similarly; now it is the wavelength that appears to change:
40 The Doppler EffectCombining results gives us the case where both observer and source are moving:The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements.
41 Superposition and Interference Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition.
42 Superposition and Interference If two pulses combine to give a larger pulse, this is constructive interference (left). If they combine to give a smaller pulse, this is destructive interference (right).
43 Superposition and Interference Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.
44 Superposition and Interference Here is another example of an interference pattern, this one from two sources. If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.
45 Standing WavesA standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air.
46 Standing WavesThe fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics.
47 Standing WavesThere must be an integral number of half-wavelengths on the string; this means that only certain frequencies are possible.Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.
48 Standing WavesIn order for different strings to have different fundamental frequencies, they must differ in length and/or linear density.A guitar has strings that are all the same length, but the density varies.