Ch.10 Elasticity & Oscillations
• Problems: 3, 4, 27, 29.
• Elastic deformation
• Hooke’s Law
• Simple Harmonic Motion (SHM)
• period & frequency of SHM
• (sections 3, 4, 7, 8, 9, 10)
Hooke’s Law
• The restoring force exerted by an object is proportional to the amount the object has been deformed.
• F = -kx
• k = elastic constant (N/m)
• Ex. SHM-springs used in lab are about 8 N/m.
• /
Young’s Modulus: Object with length = L, and area = A.
L
LY
A
F
Y is Young’s Modulus which is a measure of stiffness.
k = YA/L.
Ex. Young’s Modulus. Area = 0.1m^2, L = 25m, Y = 5000 N/m^2. A pulling force of 100N applied at each end will cause the length to change by:
m 5N/m 5000
m 25
m 1.0
N 10022
Y
L
A
FL
Ex. An object with Area = 0.0001m^2, L = 1m, is pulled on both ends by a force of 10N, causing its length to increase to 1.4m. Young’s modulus is:
22
/000,2504.0
m 1
m 0001.0
N 10mN
m
L
L
A
FY
6
vibrations
• Examples:
• vibrating reed, mass on spring,
• drum, piano wire, string,…
• most vibrations are sinusoidal in time,
• and called “simple harmonic” motions (shm)
7
terminology• x: displacement
• A = amplitude = maximum displacement
• f: frequency (cycles/s) T = 1/f
• angular frequency f: (rad/s)
• k: spring constant (N/m)
8
sinusoidal nature of vibrations
m
kf
2
1
A 0.10kg mass is hung from a spring with spring constant 8N/m.
Also calculate for 0.2kg
mk
mgx 123.0
8
)8.9)(10.0(
Hzm
kf 43.1
1.0
8
2
1
2
1
Hzm
kf 01.1
2.0
8
2
1
2
1
10
SHM Formulas)2cos( ftAx
fAAv 2max AfAa 222max 4
m
kf
2
1
2max2
12212
212
21 mvkAmvkxE
Position:
frequency:
maximums:
11
SHM Formulas• position:
• frequency:
• maximums:
)2cos( ftAx
fAAv 2max AfAa 222max 4
m
kf
2
1
2max2
12212
212
21 mvkAmvkxE
Ex: SHM
2max2
12212
212
21 mvkAmvkxE
Ex: k = 10N/m, m = 200grams, A = 10cm.
JkAE 05.0)1.0)(10( 2212
21
Summary
•Young’s Modulus
•Hooke’s Law
•Simple Harmonic Motion
•SHM Examples: Mass-Spring System,
•Energy Conservation Applied to SHM
14
natural frequency
• lowest frequency an object vibrates with when struck
• also called “resonant frequency”
• Demo: Driving Tuning Fork
• /
15
resonance
• absorb max energy: frequency = resonant frequency
• example: 256 Hz guitar string resonates when exposed to 256 Hz.
• Chilandi plates video
16
Main ResultskxF
Av max
)cos( tAx T
2
2max Aa
m
k
2
f
221 kxPEelastic
2212
21 kxmvEmech
17
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
18
L
gf 2 small angles
Simple Pendulum
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