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Outline for Oscillations
• Simple Harmonic Motion– Position, Velocity, Acceleration– Force– Energy– Phase– Damping– Resonance
• Waves– Mechanical waves– Interference– Standing waves
Simple Harmonic Motion – Cheat Sheet
• Position, Velocity, and Acceleration
• Restoring Force gives ang. freq. dependence
AvfT
t(-AωatAωvtAx
max
2
,22
)sin ,)cos( ),sin(
m
kxkxF ,
dt
dm
2
2
Example Question• What is the period of a pendulum?
http://upload.wikimedia.org/wikipedia/en/thumb/a/a8/Pendulum.png/300px-Pendulum.png
Where is the restoring force the greatest? Where istension the greatest? Restoring force – At top of swing!
Tension – At bottom of swing
g
LT 2
Energy in SHM
2
22222
22
2
1
)(cos2
1)(sin
2
12
1
2
1
kAE
tAktAmE
kxmvE
http://www.farraguttn.com/science/milligan/APPhys/SHMOver_files/image022.jpg
Example Question
• A 0.5-kg mass hits a spring with a velocity of 2 m/s. The spring starts in its equilibrium position and has a spring constant of 0.5 N/m. What is the amplitude of the oscillation?
A = 2 m
Phase
• What happens if at t=0, x doesn’t equal 0?
)sin()0(
)sin(
0
0
Ax
tAx
http://www.learner.org/courses/mathilluminated/images/units/10/1858.png
Damping
• Damping – when a force (friction, air drag) causes the oscillator to lose energy
A
mbtmbt Aexm
btAex 2/
max2
2200
2/ ,4
),sin(
bvF
http://beltoforion.de/pendulum_revisited/Damped_oscillation_graph2.png
Resonance
• Resonance – when a driving force has the same frequency of oscillation as the oscillator
0ffdrive
Example Question 1
• A mass on a spring is driven at a driving frequency (red curve) and the resulting position vs. time of the mass is given (blue curve). Which case is closest to resonance?
Example Question 2
• When would you want to have resonance or damping? When would resonance or damping be unwanted?
Damping – when you don’t want oscillations: buildings, bridgesResonance – when you want to have oscillations: pushing a kid on a swing, making standing waves in cavity