PHYS 302
: Unit 3 Viewing Notes –
Athabasca University
1
PHYS 302: Vibrations and Waves  Unit 3 Viewing notes
As in Unit 2, Lewin considers a mass on a spring, with
b/m=
γ
and
ω
0
2
=
k/m
. Initially, a force
F
0
cos
ω
t
is considered
to be applied to the object. Summing the forces in Newton’s second law,
0
cos
mx
kx
bx
F
t
; and much
like near the central, mathematicallyoriented “18 minutes” of Unit 2, in the complex plane, reorganized, and with
the RHS nonzero, one gets
0
2
0
F
j
t
m
z
z
z
e
(3:30). We propose a solution
(
)
i
t
z
Ae
, which
represents, in the long term, an imposition of vibration at frequency
ω
. This is the
steady state
solution: although it
changes in time because of the vibration, the overall state does not change; it just keeps vibrating. So
steady state
does not mean
static
. Doing the derivatives and substituting,
0
2
2
(
)
0
(
)
F
j
t
j
t
m
j
Ae
e
(6:30).
Equivalently,
0
0
2
2
0
(
)
(cos
sin
)
F
F
j
m
m
j
A
e
j
. This is really two equations, one for the real
part, and one for the imaginary part, which
both
need to be true at the same time. Separating them,
0
2
2
0
(
)
cos
F
m
A
and
0
sin
F
m
j
A
j
or
0
sin
F
m
A
. Whenever we have equations like this
with sin and cos, we always think of the identity
2
2
cos
sin
1
and of the tangent. In squaring and adding
these equations we have
0
2
2
2
2
2
2
0
(
)
(
) cos
F
m
A
and
0
2
2
2
(
)
(
) sin
F
m
A
, giving
0
0
2
2
2
2
2
2
2
2
2
2
0
(
)
(
)
(
) (cos
sin
)
(
)
F
F
m
m
A
A
. Thus,
0
2
2
2
2
2
2
0
((
)
(
) )
(
)
F
m
A
.
Dividing through and taking the square root gives us
0
2
2
2
2
0
/
(
)
(
)
F
m
A
as the amplitude of driven,
damped motion (9:40). Although this function is complicated, it should be immediately clear that the amplitude will
be highest when the driving frequency
ω
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 Spring '10
 martinconner
 Force, Mass, Cos, Radio spectrum

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