Download - 1 13 Outline vibrations, waves, resonance Homework: 1, 2, 15, 30, 41, 45, 51, 64, 67, 101
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13 Outline
• vibrations, waves, resonance
• Homework:
• 1, 2, 15, 30, 41, 45, 51, 64, 67, 101.
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vibrations
• Examples:
• vibrating reed, mass on spring,
• drum, piano wire, string,…
• most vibrations are sinusoidal in time,
• and called “simple harmonic” motions (shm)
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terminology• x: displacement
• A: maximum displacement
• f: frequency (cycles/s)
• angular frequency : (rad/s)
• k: spring constant (N/m)
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sinusoidal nature of vibrations
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Kinematics of SHM
• position:
• frequency:
• angular frequency:
• maximums:
)cos( tAx
Av max2
max Aa
m
kf
2
1
m
k
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Dynamics of SHM
• F = -kx, a = -kx/m
• E = ½kx2 + ½mv2.
• Ex: k = 10N/m, m = 200grams, A = 10cm.
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Waves
• traveling disturbance
• Transverse Longitudinal
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wave phenomena
• interference of waves
• Examples, noise cancellation headphones, standing waves
• reflection, refraction, and diffraction.
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Periodic Waves
• continuous, well defined amplitude (A), frequency (f), wavelength (), and speed v = f.
• Example: f = 10 hertz, = 3 m.
• v = (10/s)(3m) = 30m/s.
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Waves on Strings
• Wave Velocity depends on:
• tension in string (F) and,
• the mass per unit length of string.
• Example: F=36N, m/L=0.010kg/m
Lm
Fv
/
smv /603600010.0
36
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Standing Waves
• Nodes (places with zero amplitude)
• Anti-nodes (places with maximum amplitude)
…waves in which the amplitude at a given location does not vary with time. Due to wave interference.
Features:
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Standing Wave: Both Ends Fixed
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Standing Wave: One End Fixed, One End Free
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summary
• many vibrations are simple harmonic
• one equation set describes all shm
• wave speed equations
• interference of waves & standing waves
• reflection, refraction, diffraction.
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Main ResultskxF
Av max
)cos( tAx T
2
2max Aa
m
k
2
f
221 kxPEelastic
2212
21 kxmvEmech
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Nature of Sound Waves
• Longitudinal
• Oscillations are:
• Condensations (higher pressure areas) and
• Rarefactions (lower pressure areas)
• Sound travels at about 343m/s at room temperature and normal atmospheric pressure
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Doppler Effect
• Frequency received is different than the Source frequency due to:
• Source Motion,
• Receiver Motion or,
• a combination of Source and Receiver motions.
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Example of wavelength distortion due to source motion:
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Height vs. Time
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25
Time (s)
Y (
m)
values of “A” and “f”?
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Decibels
• intensity level
where Io = 1.0x10-12 W/m2.
• Example: Intensity of sound is 4.0x10-5 W/m2. Intensity level is
oI
IdB log10
dBdBdB 76100.4log10100.1
100.4log10 7
12
5
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Sound Intensity (I)
• Intensity = power/area = P/A [watt/meter2]
• Spherical Radiation I = P/4r2.
• Example: Small speaker emits 1.0W of sound in all directions. Intensity 10m from the speaker is 1.0/(4102) W/m2.
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Frequency of Sound
• Audible Range: 20Hz to 20,000Hz
• Infrasonic: f < 20Hz
• Ultrasonic: f > 20,000Hz
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sinusoidal nature of shm
• position of blue mass moving on spring turns out to be same as the horizontal position of an object in uniform circular motion.
)cos( tAx
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L
gf 2 small angles
Simple Pendulum