Page 31 - Science at the nanoscale

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June 9, 2009
dependence, let us consider the simple case of a mass m attached
to a spring with spring constant k. From Hooke’s law, when a
spring is slightly displaced in the direction x from its equilibrium
position, it would undergo simple harmonic motion according to:
(2.1)
F = −kx
From Newton’s Second Law:
2
d x
(2.2)
= −kx
F = m
dt
2
k
d x
(2.3)
x = −ω x
= −
2
m
dt
1/2

k
, which is the frequency of the sinusoidal equa-
where ω =
m
tion that is a solution of equation (2.3):
(2.4)
x = A cos(ωt + ϕ)
where A is the amplitude of oscillation and φ is the phase of the
oscillation.
The frequency of oscillation f is the inverse of the period of os-
cillation T:
1
1
k
ω
(2.5)
f =
=
=
T
2π
Since the mass and spring are three-dimensional, the mass m
3
will vary as L and the spring constant k as L:
1 2 2π m 2 1/2 2.1. Mechanical Frequency 21 ch02
ω ∝ / L (2.6)
Hence the frequency is inversely proportional to the length
scale for a mechanical oscillator. Frequencies inversely propor-
tional to the length scale are typical of mechanical oscillators such
as string instruments like the violin or harp. In such oscillators
with two nodes at both ends (i.e. fixed at both ends), the length of
the string (oscillator) is related to its lowest order standing wave-
length by L = λ/2. If the oscillator has only one node at one end
(i.e. fixed at one end) as in a cantilever, then L = λ/4. Since λ = vt,
where t is the time for the wave to travel one oscillation and v the