# Simple Harmonic Motion. Simple harmonic motion (SHM) refers an oscillatory, or wave-like motion that describes the behavior of many physical phenomena:

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Simple Harmonic Motion

Simple harmonic motion (SHM) refers an oscillatory, or wave-like motion that describes the behavior of many physical phenomena: –a pendulum –a bob attached to a spring –low amp. waves in air, water, the ground –vibration of a plucked guitar string

Objects undergoing SHM trace out sine waves where the d is pos and neg with time.

Simple Pendulum

displacement– time graph w worksheet

Velocity and acceleration in SHM The position of an object undergoing SHM changes with time, thus it has a velocity The velocity of an object is the slope of its graph of position vs. time. Thus, we can see that velocity in SHM also changes with time, and so object is accelerating:

Vibrations & Waves

Waves are an energy disturbance propagates through material or empty space. Energy Transfer by Waves

Waves Transfer Energy Matter is not transferred Ex: Cork on water or buoy

Some Types of Energy that travel as Waves Sound – vibrating tuning fork, string, wood etc. Light (EM) – vibrating charges. Earthquake – vibration of Earth’s crust

How can we prove that waves transfer energy? -Can waves do work? -Give examples.

Mechanical waves need medium through which to travel. Mediums include: Gasses - air liquids/water Solids - wood Ex: Sound/Earthquake waves

Non mechanical – no medium required! Electromagnetic Waves (EM) need no medium *EM waves can also propagate through a medium

Two Main Types of Waves Transverse (all EM waves), seismic S waves Longitudinal (Compressional) Sound, seismic P waves

Transverse Wave Pulse One disturbance

Transverse Periodic Wave Pulses Pass at Regular Intervals Particles vibrate perpendicular to energy transport. Trace out sine wave.

Particle motion transverse wave.

Longitudinal/Compressional Wave Particles compressed and expanded parallel to energy propagation.

Another View

Sound Waves ex of mechanical wave. Need medium to propagate. Vibrations in air molecules from vibrating tuning fork or vibrating string.

Eardrum sound waves do work on eardrum

Water Waves Combination of 2 Types

Parts of a Wave Wavelength  distance btw Crests or Troughs Midpoint = Equilibrium Position

Crests d d

Wave Pulse Single disturbance Periodic Wave Many pulses with regular  and period

1. State the difference between a mechanical and non-mechanical wave.

Longitudinal Wave Parts

Transverse & Longitudinal Waves can be represented by sine waves. Longitudinal Waves can be graphed as density of particles vs time. Then will graph as sine wave.

Period (T) & Frequency (f) Period = time to complete one cycle of wave crests or troughs. Time for disturbance to travel 1. Usually measured in seconds. T = 0.5 s/cycle.

Frequency = Number of cycles in unit time. Inverse of period. Usually number per second called Hertz (Hz) Ex: 3 crests or cycles per second = 3s -1 or 3 hz

f = how often T = how long f = a rate T = a time T & f are inverse f = 1/Tor T = 1/f.

2. A wave has a period T of 5.0 seconds. What is its frequency? 0.2 hz

3. A wave has a frequency of 100 Hz. What is its period? 0.01 s.

4. The wave below shows a “snapshot” that lasted 4.0 seconds. What is the frequency of the wave? 4.0 seconds 2 cycles/4 s=0.5 Hz

Wave Speed Speed/Velocity = d/t If a crest (or any point on a wave) moves 20m in 5 sec,v = 20m/5s = 4 m/s.

Relationship of wave speed to wavelength( ) and frequency(f). v = d/tbut for waves d = 1 occurs in time T (1period) so v = /T since freqf =1/T v = f

5. A piano emits from 28 Hz to 4200 Hz. Find the range of wavelengths in air attained by this instrument when the speed of sound in air is 340 m/s. = 0.081 m to 12 m

Wave speed is constant if medium is uniform. Air at constant T and P. Homogenous solids. Water with constant T. Only the medium through which it travels! What determines wave speed?

7. A tuning fork produces a sound with f = 256 Hz and in air of 1.35 m. What is the speed of sound in air? What would be the wavelength of this tuning fork is sound travels through water at 1500 m/s? 346 m/s 5.86 m

Velocity depends on medium’s properties: -EM waves all travel at c in a vacuum. - EM waves slower through materials. -Vibrations travel faster on tighter strings - slower on loose strings. -v sound constant in air but depends on temp/density of air.

8. What determines the wave’s frequency? Vibrational Rate

Example Problems & Hwk. Read Text 12 - 3 Read Text Chap 12-3 Do pg 470 #23- 32, 35, 36. Write all out will collect.

Do Now Text Pg 457 #2

Quiz 1. What is only factor that determines wave speed. 2. Give a real life example of: –A longitudinal wave –A transverse wave. Sketch a transverse wave. Label the –Wavelength –Amplitude –Equilibrium position

What is the motion of points on a wave?

Up & down motion of particle on wave.

Another View

Given a wave moving to the left as below, what will be the motion of the red beach ball just after the time shown? Up Down Right Left Down

Wave Behaviors & Interactions Boundary Behavior & Wave Superposition

When a wave enters a material with new properties it: Goes through it without noticing Slows down Speeds up Accelerates

Wave Behaviors and Interactions

Reflection- a wave incident on a boundary (new material), part bounces off, part transmitted.

Example Echo: A sound wave is traveling in air at STP. The echo is heard 2.6 second later. How far away is the reflecting object? Time to object = 1.3 seconds. Speed sound STP = 331 m/s v = d/t tv = d (1.3s)(331 m/s) 430.3 m

Reflection off Fixed Boundary – pulse inverts

Pulse Passing into new medium from less dense material. What happens to pulse?

Changes in: Velocity and Amplitude change with medium. No Change - frequency.

More dense into less dense

How do multiple waves combine? Waves can overlap and occupy the same space at the same time. How they do it depends on the position or phase of the crests and troughs. Superposition – constructive destructive interference.

Phase of Particles in Wave “in phase” = points in identical position. Whole number of apart.(A,F B,G E,J C,H) 180 o out of phase = equal displacement fr equilibrium but moving opposite directions. Odd number of ½ apart. (A,D)

Superposition /Interference– 2 or more waves or pulses interact/superimpose & combine. Their amplitudes add or subtract. The resultant wave is the sum of the two.

Constructive Interference – waves superimpose with displacement in same direction + or -, amplitude increases.

Constructive Interference.

Destructive Interference- waves or pulses meet with opposite displacement. Waves partially or totally cancel.

Points on waves that meet “in phase” interfere constructively.

Points that meet “out of phase” interfere destructively. Below is total destructive interference.

Standing Waves Wave pattern that results when 2 waves, of same f,, & v travel in opposite directions. Often formed from pulses reflected off a boundary. Waves interfere constructively & destructively at fixed points.

Standing Wave – wave appears to be standing still. No net transfer of energy.

Standing wave formed from wave pulses in same medium.

Nodes are points of max. destructive interference. Antinodes = points of max. constructive interference.

Standing Wave Formation from 2 waves.

Standing waves have no net transfer of energy – no propagation of energy.

Sketch Resultant on wksht from RB pg 271. Hwk Text pg 471 #33 - 44, 48-49

Hwk Text pg 471 #36 - 44, 48-50

Relation of Wavelength to String Length for Standing Waves

Standing waves form only when the string length allows a whole number of half wavelengths to fit.

½ = L or 2L =.

= L.

= L

General expression relating wavelength to string length for standing waves: n ( ½ ) = L n is a whole number

Mechanical Universe “Waves”

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