Periodic Motion any repeated motion with regular time intervals Simple Harmonic Motion vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium ID: 579797
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Slide1
Chapter 11: Vibrations and WavesSlide2
Periodic Motion – any repeated motion with regular time intervalsSlide3
Simple Harmonic Motion – vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium Slide4
SHM
Speed is max at equilibriumSpeed is least at max displacementMax force is at max displacementMax acceleration is at max displacementSlide5
Simple Harmonic MotionSlide6
Hooke’s Law
F
elastic
= -
kx
Spring force= -(spring constant x displacement)
SI unit for k = N/m
(-) signifies the direction of the spring force is always opposite the direction of the mass’s displacement from equilibriumSlide7
ExampleIf a mass of 0.55kg attached to a vertical spring stretches the spring 2.0cm from its original equilibrium position, what is the spring constant?Slide8
Example Suppose the spring in the previous example is replaced with a spring that stretches 36 cm from its equilibrium position.
What is the spring constantIs this spring stiffer or less stiff than the one on the original example?Slide9
Simple Pendulum
Weight (force of gravity) is the restoring forceSHM for small angles onlySlide10Slide11Slide12
Bellringer 12/3
A 76N crate is hung from a spring (k=450N/m). How much displacement is caused by the weight of this crate?Slide13
Measuring Simple Harmonic Motion
Amplitude – maximum displacement from equilibriumPeriod – (T) – time it takes to complete one cycleFrequency (f) – number of cycles per unit of time (Hertz = s
-1
)
AmplitudeSlide14
Measuring SHMFrequency and period are inversely relatedSlide15Slide16
ExampleThe reading on a metronome indicates the number of oscillations per minute. What are the frequency and period of the metronome’s vibrations when the metronome is set at 180? Slide17
Pendulum in SHM Period of a simple pendulum depends on pendulum length and free-fall acceleration
Length is measured from center of mass of the bob and the pivot point
Period of a Simple Pendulum in SHM
T = 2
π
L
√
a
g
Period = 2 π x
sqrt
of (length divided by free fall)Slide18
ExampleYou need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12s. How all is the tower?Slide19
Mass/Spring in SHMPeriod of a mass spring system depends on mass and spring constant
Period of a Mass-Spring System in SHM
T = 2
π
m
√
k
Period = 2
π
x sqrt
of (mass divided by spring constant)Slide20
ExampleA child swings on a playground swing with a 2.5m long chain.
What is the period of the child’s motion?What is the frequency of vibration?Slide21
ExampleThe body of a 1275kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 153kg. When driven over a pothole in the road, the frame vibrates with a period of 0.840s. For the first seconds, the vibration approximates simple harmonic motion. Find the spring constant of a single spring.Slide22
Bellringer 12/4
A spring of spring constant 30.0N/m is attached to different masses, and the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes:2.3kg15kg1.9kgSlide23
Properties of WavesWave – motion of disturbance that transmits energy
Medium – physical environment through which a disturbance can travelMechanical waves – requires a mediumElectromagnetic waves – do not require a mediumSlide24
Pulse wave – single traveling pulsePeriodic wave – continuous pulse wavesSine wave – source vibrates with SHMSlide25
Wave TypesTransverse wave – particles vibrate perpendicularly to the direction the wave is traveling
Wavelength (λ)– distance the wave travels along its path during one cycleSlide26
Wave Types
Longitudinal wave – particles vibrate parallel to the direction the wave is travelingCrests – coils are compressed (high density and pressure)Troughs – coils are stretched (low density and pressure)Slide27Slide28
Period and FrequencyWave frequency – number of waves that pass a given point in a unit of time
Period – time needed for a complete wavelengthSlide29
Wave Speed
Speed of a mechanical wave is constant for any given medium
Wave Speed
v = f
λ
Speed of wave = frequency x wavelengthSlide30
ExampleA piano string tuned to middle C vibrates with a frequency of 262 Hz. Assuming the speed of sound in air is 343m/s, find the wavelength of the sound waves produced by the string.Slide31
ExampleA piano emits frequencies that range from a low of about 28Hz to a high of about 4200Hz. Find the range of wavelengths in air attained by this instrument when the speed of sound in air is 340m/s. Slide32
Bellringer 12/5
A tuning fork produces a sound with a frequency of 256 Hz and a wavelength in air of 1.35m. What value does this give for the speed of sound in air?What would be the wavelength of this same sound in water in which sound travels at 1500m/s Slide33
Constructive interference – two or more waves are added together to form a resultant waveSlide34Slide35
Destructive interference – two or more waves with displacements on opposite sides of equilibrium position are added together to make a resultant waveSlide36Slide37
Reflection
At a free boundary – waves are reflectedAt a fixed boundary – waves are reflected and invertedSlide38
Free End ReflectionSlide39
Fixed End ReflectionSlide40
Standing wavesStanding wave – results when two waves with same frequency, wavelength, and amplitude travel in opposite directions and interfere
Nodes – point in standing wave where it maintains zero displacementAntinodes – point in standing wave, halfway between two nodes, at which largest displacement occurs Slide41
Standing WavesSlide42
Standing WavesSlide43
ExampleA wave of amplitude 0.30m interferes with a second wave of amplitude 0.20m. What is the largest resultant displacement that
may occur?