Coupled Oscillations. A single spring-mass system.
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Coupled Oscillations
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Transcript of Coupled Oscillations. A single spring-mass system.
m
k21 Let Natural freq. of each pendulum
122212
211211
xxk
xx
xxk
xx
m
m
c
c
21 2 1 1 2x x x x 0 Adding
Subtracting 21 2 1 1 2
2kx x x x 0c
m
1 1 2 10 1 1
2 1 2 20 2 2
x x cos
x x cos
q q t
q q t
Normal mode amplitudes : q10 and q20
Normal mode frequencies:
m
k21
mmck2k2
2
Mass displacements
2
cos2
cos2
coscos2
1x
2112
21211
tta
ttaqq
2
sin2
sin 2
coscos2
1x
2112
21212
tta
ttaqq
11 1 0
00 0 1
( )
( )
xmx mg k x x
lx
mx mg k x xl
Equation of motion
SHM Couplingterm term
With assumption
Compare model Coupled Pendula and Experiment
Check two pendula are identical
Determine inphase and out of phase mode
Check whether frequency of inphase mode (w1) is less than thatof out of phase mode (w2)