• There are many objects and things that by nature oscillate, vibrate, wiggle, etc• In fact, this motion is the reason why we can SEE and HEAR things around us. As a result, it is essential to know this stuff!…can you think of any examples of objects that oscillate, vibrate, or wiggle??
Simple Harmonic Motion• Simple Harmonic Motion (SHM) is back and forth (oscillatory) motion caused by a restoring force that is directly proportional to its displacement. – Restoring force: the force that acts in the direction of returning an object to the equilibrium position – The displacement centers around an equilibrium point – Ex: springs, pendulums Fsα x
Simple Harmonic Motion• If an object vibrates or oscillates back and forth over the same amount of time each cycle, it is called periodic motion.• SHM consists of a period, T, and an amplitude.• Amplitude (A) is the maximum displacement of an object from rest.• In SHM, Period (T) is the time to perform one complete cycle of motion (oscillation)• Frequency (f) is the # of cycles per unit of time – f = 1/T [units: cycles per second = Hertz]
At rest SHM of Springs Let’s say that a spring that is originally at rest is pushed in by a distance of –A, and then is released so that it oscillates. •What do we notice about the force that the spring exerts on the mass, and about the displacement of the object at each point in its oscillation?
Springs – Hooke’s LawOne of the simplest type of simple harmonic motion is called Hookes Law. This is primarily in reference to SPRINGS. The negative sign only tells usFs α x that “F” is what is called a RESTORING FORCE (force ofk = Spring Constant(Unit:N/m) spring on mass), in that it works in the OPPOSITE direction of the displacement.Fs = −kx
ExampleA load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount? Fs = kx Fs = kx 50 = k (0.05) Fs = (1000)(0.11) Fs = 110 N k = 1000 N/m
Example 11-1When a family of four people with a total mass of 200 kg step into their 1200 kg car,the car’s springs compress 3.0 cm.(b)What is the spring constant of the car’s springs, assuming they act as a singlespring?(c)How far would the car lower if loaded with 300 kg?
Hooke’s Law from a Graphical Point of ViewSuppose we had the following data: Fs = kx There is a linearx(m) Force(N) Fs relationship between F k= and x! 0 0 x 0.1 12 k = Slope of a F vs. x graph 0.2 24 Force vs. Displacement y = 120x + 1E-14 R2 = 1 0.3 36 80 70 0.4 48 60 0.5 60 Force(Newtons) 50 40 0.6 72 30 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Displacement(Meters) k =120 N/m
At rest SHM of Springs Let’s say that a spring that is originally at rest is pushed in by a distance of –A, and then ist0 released so that it oscillates. •What would our position vs. time graph look like for the mass?t1t2t3 t4
Springs The amplitude, A, of a wave is the CREST same as the displacement ,x, of a spring. Both are in meters. Equilibrium Line Period, T, is the time for one revolution or Trough in the case of springs the time for ONE COMPLETE oscillation (One crest andTs=sec/cycle. Let’s assume that trough). Oscillations could also be calledthe wave crosses the equilibrium vibrations and cycles. In the wave aboveline in one second intervals. T we have 1.75 cycles or waves or=3.5 seconds/1.75 cycles. T = 2 vibrations or oscillations.sec.
Period of a springThe period of a spring dependson mass in the spring constant- NOT on the amplitude ofoscillation! m Ts = 2π If the spring constants are all the k same in the above picture, which spring is carrying the largest mass?
Pendulums as SHOsPendulums, like springs, oscillate back and forth exhibiting simple harmonic behavior. A shadow projector would show a pendulum moving in synchronization with a circle. Here, the angular amplitude is equal to the radius of a circle.
Period of a PendulumWe know that mass and the spring constant are the only factors that effect the period of a spring… But what factors effect the period of a pendulum?
The Period of a PendulumThe period of a pendulum dependson the length of the pendulumswing, and gravity! l T pendulum = 2π g 4π 2lT = 2 gT 2α l Like a spring, the4π 2 = Constant of Proportionality restoring force will g always oppose the direction of displacement
Galileo’s at it again!Galileo credited for discovering thiswhile attending church at Cathedral ofPisa.-He timed its frequency using hispulse-Noted that larger amplitude = largervelocity, but also larger distance tocover
Example 11-8(a) What would be the period of a grandfather clock with a 1.0-m pendulum?(b) Estimate the length of the pendulum in a grandfather clock that ticks once per second (ticks once per cycle). 2.0 s, 0.25 m
Example to determine theA visitor to a lighthouse wishes 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? lTP = 2π → l = height g 4π 2l TP2 g 9.4 2 (9.8)TP2 = →l = = = g 4π 2 4(3.141592) 2 L = Height = 21.93 m
SHM and Uniform Circular MotionSprings and Waves behave very similar to objects that move in circles.The radius of the circle is symbolic of the displacement, x, of a spring or the amplitude, A, of a wave.xspring = Awave = rcircle
Damped Harmonic Motion What does damping do to a wave? (recall from string on a wave simulation)Damping decreases the amplitude of oscillation over time – This naturally occurs due to air resistance – energy is dissipated to thermal energy – There are three types of damping: overdamping, underdamping (large curve shown), and critical damping – Critical damping reaches equilibrium the quickest • Can you think of any cases where we would want a spring to give critical damping?
WavesA wave: a rhythmicdisturbance that carriesENERGY
Wave motion• Mechanical waves (what we will focus on!) – Requires a medium to travel through – Local particle motion vs. overall wave motion – Ex: SOUND!• Non-mechanical waves – Requires no medium (can travel through a vacuum) – Ex: electromagnetic waves, light!, radio waves, microwaves, X-rays
Wave PropertiesWavelength (λ) length or size of one oscillationAmplitude (A) strength of disturbance (intensity)Frequency (f) oscillation per secondWave velocity (v) velocity of wave through a mediumCrest maximum positive ATrough min negative A
What is the Wave length?• Measure from any identical two successive points 5 10 15 20 25 30 35 40
What is the Wave length?• Measure from any identical two successive points 5 10 15 20 25 30 35 40 30 - 10 = 20
What is the Wave length?• Measure from any identical two successive points 5 10 15 20 25 30 35 40 22.5 - 2.5 = 20• There are 4 complete oscillations depicted here• ONE WAVE = 1 COMPLETE OSCILLATION
Wave PropertiesWaves are oscillations and they transport energy.The energy of a wave is proportional to its frequency. Fast oscillation = high frequency = high energy Slow oscillation = low frequency = low energyThe amplitude is a measure of the wave intensity. SOUND: amplitude corresponds to loudness LIGHT: amplitude corresponds to brightness
Wave SpeedYou can find the speed of a wave by multiplying the wave’swavelength in meters by the frequency (cycles per second).Since a “cycle” is not a standard unit this gives youmeters/second.
How waves travel• Pulse waves – a single disturbance• Periodic waves – repeated pulse waves
Two types of WavesThe first type of wave is called Longitudinal. Longitudinal Wave - A fixed point on medium will move parallel with the wave motion 2 areas Compression- an area of high molecular density and pressure Rarefaction - an area of low molecular density and pressure (expansion)
Two types of WavesThe second type of wave is called Transverse. Transverse Wave - A fixed point on medium will move perpendicular with the wave motion. Wave parts - crest, trough, wavelength, amplitude, frequency, period
ExampleA harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum travels 425 cm along a rope in 10.0 s .(b)What is the frequency?(c)What is the velocity?(d)What is the wavelength? cycles 40f = = = 1.33 Hz sec 30 ∆x 4.25v= = = 0.425 m/s ∆t 10 vwavevwave = λf → λ = = 0.319 m f
ExampleA piano string that produces the middle C note vibrates with afrequency of 264 Hz. If the sound waves have a wavelength in airof 1.30 m, what is the speed of sound?343 m/s**note that for mechanical waves, wave speed is the same for a given medium.Consider a concert – the sound from all instruments reaches your ears at thesame time, even though they are different frequencies. Frequencies andwavelengths may differ, but the produce of the two is the same!
Waves Activity with a SlinkyFollow directions and answer questionsto learn more about waves!Be careful with the springs. They can bedangerous if overstretched, and alsomust be replaced. Also, please do not leta stretched spring release quickly EVER!A. Transverse & Longitudinal WavesB. The speed of all waves of thesame kind in a given mediumC. Wavelength and frequencyD. The interference of wavesE. Reflected wavesF. Wave Transferral
Wave interactions• Reflection: when object strikes an obstacle or a different medium, at least part of wave is reflected – Law of reflection or angle of incidence
Interference• Principle of superposition – algebraic sum of their separate displacements – Constructive interference – Destructive interference
Two Equal Waves• Upon arriving in the same place, they add constructively
Constructive Interference• Waves combine without any phase difference• When they oscillate together (“in phase”)
Wave AdditionAmplitude ~ Intensity
Two Opposite Waves• Upon arriving in the same place, they cancel, destructively
Destructive Interference• Waves combine differing by multiples of 1/2 wavelength• They oscillate “out-of-phase”
Standing Waves• Standing wave patterns are produced as the result of the repeated interference of two waves of identical frequency while moving in opposite directions.• A= Antinode• N= Node Blue wave moving right, green wave moving left… black is the wave formed
Soundtype of standing wave as theySound Waves are a common Waves are caused by RESONANCE.Resonance – when a FORCED vibration matches an object’s natural frequency thus producing vibration, sound, or even damage.One example of this involves shattering awine glass by hitting a musical note that ison the same frequency as the naturalfrequency of the glass. (Natural frequencydepends on the size, shape, andcomposition of the object in question.)Because the frequencies resonate, or arein sync with one another, maximum energytransfer is possible.