Free SHM Superposition Superposition: the sum of solutions to an EOM is also a solution . . . . . . if the EOM is linear. EOM: Solutions: x and its derivatives appear only to first power. …or a linear combination: c 1 x 1 + c 2 x 2 x m kx 1 1 1 1 cos t A t x 2 2 2 2 cos t A t x
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Free SHM Superposition Superposition: the sum of solutions to an EOM is also a solution...... if the EOM is linear. EOM: Solutions: x and its derivatives.
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xmkx Free SHM
1111 cos tAtx
Superposition
Superposition: the sum of solutions to an EOM is also a solution . . . . . . if the EOM is linear.
EOM:
Solutions:
2222 cos tAtx
x and its derivatives appear only to first power.
…or a linear combination: c1x1 + c2x2
Is the linear combination useful?
initial displacement, begin at restxo , vo = 0
0111 cos0 xAx
0sin0 111 oAx
txtx oo cos1
101 ,0 Ax
initial velocity, begin at originvo , xo = 0
0cos0 222 Ax
oo vAx 222 sin0
22 cos
tv
tx oo
o
o
ovA
2
22
initial displacement xo and velocity vo
oo vAx 333 sin0
0333 cos0 xAx
Solve each for A3, equate, solve for :f
oo
o
xv
1
3 tan
Plug back into top equation to get A3:
ooo
o
xvx
A
13 tancos
Solution:
oooo
ooo
o xvtxv
xtx
1
13 tancostancos
-2
-1
0
1
2
0 5 10 15 20
X
Time
-2
-1
0
1
2
0 5 10 15 20X
Time
-2
-1
0
1
2
0 5 10 15 20
X
Time
-2
-1
0
1
2
0 5 10 15 20
X
Time
Superposition: The motion resulting from two simultaneous initial conditions is equal to the sum of the motions resulting from each initial condition. . . . . . if the EOM is linear.
x1
x2
x1+x2
x3
+x
Imagine SHM in 1D…
…as a projection of 2D motion.
The trig is getting complicated, let’s try something else…
Imagine SHM in 1D…
…as a projection of 2D motion.+x
“real” x-axis
“imaginary” y-axis
The trig is getting complicated, let’s try something else…
“real” x-axis
“imaginary” y-axis
Describe the position…
j -> “rotate 90 degrees”
…algebraically…geometrically OR
jbiar ˆˆ jbaz
j2b -> “go along original axis in opposite direction”