Simple Harmonic Motion What is an Oscillation? Vibration Goes back and forth without any resulting...
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Transcript of Simple Harmonic Motion What is an Oscillation? Vibration Goes back and forth without any resulting...
Simple Harmonic Motion
What is an Oscillation?
VibrationGoes back and forth without any resulting movement
SHM - Simple Harmonic Motion
An object in SHM oscillates about a fixed point.
This fixed point is called mean position, or equilibrium position
This is the point where the object would come to rest if no external forces acted on it
Describe restoring force
Restoring force, and therefore acceleration, is proportional to the displacement from mean position and directed toward it
Examples of SHM: Simple pendulum Mass on a spring Bungee jumping Diving board Object bobbing in the water Earthquakes Musical instruments
Simple Pendululm Equation:
€
T = 2ΠL
g
Time is independent of amplitude or mass
Assumptions: 1. Mass of string is negligible compared to mass of load
2. Friction is negligible 3. Angle of swing is small 4. Gravitational acceleration is constant
5. Length is constant
Mass on a Spring Equation:
€
T = 2Πm
kTime is independent gravitational acceleration
Assumptions: 1. Mass of spring is negligible compared to mass of load
2. Friction is negligible 3. Spring obeys Hooke’s Law at all times
4. Gravitational acceleration is constant
5. Fixed end of spring can’t move
Restoring Force is proportional to (-) displacement
Meanpositiondisplacementrestoring force
Sketch:
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F ∝−x
€
F = −kxNegative sign means force is in the opposite direction of the displacement
Variables for SHM: x displacement from mean position A maximum displacement (amplitude) Ø phase angle (initial displacement at t = 0)
T period (time for one oscillation) f frequency (number of oscillations per unit time)
angular frequency
€
=k
m
€
2 =k
m
Relationships between variables
€
F = −kx
€
a =−kx
m
€
=ma
€
=− 2x
€
T = 2Πm
k
€
=2Π(1
ω)
€
=2π
T
Other relationships:
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x = xo sinωt
€
x = xo cosωt
€
v = vo cosωt
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v = vo sinωt
€
v = ±ω x02 − x 2
Diagrams
QuickTime™ and a decompressor
are needed to see this picture.
Graphs:http://physics.bu.edu/~duffy/semester1/c18_SHM_graphs.html
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
Kinetic and Potential Energies in SHM
€
Ek =1
2mv 2
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v = ±ω x02 − x 2since
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Ek =1
2mω2(x0
2 − x 2)
€
Ekmax =1
2mω2(x0
2)
€
ET =1
2mω2(x0
2)€
E p =1
2mω2x 2
Damping
Energy losses (energy dissipation) due to friction - removes energy from system
For an oscillating object with no damping, total energy is constant - depends on mass, square of initial amplitude, angular frequency
Damping (continued)
Amplitude decreases exponentially - all energy is eventually converted to heat
Critical damping (controlled)- oscillations die out in shortest time possible
QuickTime™ and a decompressor
are needed to see this picture.
Resonance
System displaced from equilibrium position will vibrate at its natural frequency
System can be forced to vibrate with a driving force at the natural frequency
Examples: musical instruments, machinery, glass, microwave, tuning a radio