1 13 outline vibrations, waves, resonance homework: 1, 2, 15, 30, 41, 45, 51, 64, 67, 101
TRANSCRIPT
1
13 Outline
• vibrations, waves, resonance
• Homework:
• 1, 2, 15, 30, 41, 45, 51, 64, 67, 101.
2
vibrations
• Examples:
• vibrating reed, mass on spring,
• drum, piano wire, string,…
• most vibrations are sinusoidal in time,
• and called “simple harmonic” motions (shm)
3
terminology• x: displacement
• A: maximum displacement
• f: frequency (cycles/s)
• angular frequency : (rad/s)
• k: spring constant (N/m)
4
sinusoidal nature of vibrations
5
Kinematics of SHM
• position:
• frequency:
• angular frequency:
• maximums:
)cos( tAx
Av max2
max Aa
m
kf
2
1
m
k
6
Dynamics of SHM
• F = -kx, a = -kx/m
• E = ½kx2 + ½mv2.
• Ex: k = 10N/m, m = 200grams, A = 10cm.
7
Waves
• traveling disturbance
• Transverse Longitudinal
8
wave phenomena
• interference of waves
• Examples, noise cancellation headphones, standing waves
• reflection, refraction, and diffraction.
9
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.
10
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
11
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:
12
Standing Wave: Both Ends Fixed
13
Standing Wave: One End Fixed, One End Free
14
summary
• many vibrations are simple harmonic
• one equation set describes all shm
• wave speed equations
• interference of waves & standing waves
• reflection, refraction, diffraction.
15
Main ResultskxF
Av max
)cos( tAx T
2
2max Aa
m
k
2
f
221 kxPEelastic
2212
21 kxmvEmech
16
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
17
Doppler Effect
• Frequency received is different than the Source frequency due to:
• Source Motion,
• Receiver Motion or,
• a combination of Source and Receiver motions.
18
Example of wavelength distortion due to source motion:
19
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”?
20
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
21
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.
22
Frequency of Sound
• Audible Range: 20Hz to 20,000Hz
• Infrasonic: f < 20Hz
• Ultrasonic: f > 20,000Hz
23
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
24
L
gf 2 small angles
Simple Pendulum