Consider now the
mass
spring oscillator depicted physically in
Fig.
D.3, and in
equivalentcircuit form in
Fig.
D.4.
Figure D.3:
An ideal mass sliding on a
frictionless surface, attached via an ideal spring to a rigid
wall. The spring is at rest when the mass is centered at .

Figure D.4:
Equivalent circuit for the massspring oscillator.

By
Newton's second law of motion, the
force applied to a mass
equals its mass times its acceleration:
By
Hooke's law for ideal springs, the compression force
applied to a spring is equal to the spring constant
times the
displacement :
By Newton's third law of motion (``every action produces an equal and
opposite reaction''), we have
. That is, the compression
force
applied by the mass to the spring is equal and opposite to
the accelerating force
exerted in the negative
direction by
the spring on the mass. In other words, the forces at the massspring
contactpoint sum to zero:
We have thus derived a secondorder
differential equation governing
the motion of the mass and spring. (Note that
in
Fig.
D.3 is both the position of the mass and compression
of the spring at time
.)
Taking the
Laplace transform of both sides of this
differential
equation gives
To simplify notation, denote the initial position and
velocity by
and
, respectively. Solving for
gives
denoting the modulus and angle of the
pole residue
, respectively.
From §
D.1, the inverse Laplace transform of
is
, where
is the
Heaviside unit step function at time 0.
Then by linearity, the solution for
the motion of the mass is
If the initial velocity is zero (
), the above formula
reduces to
and the mass simply oscillates sinusoidally at frequency
, starting from its initial position
.
If instead the initial position is
, we obtain
Mechanical Equivalent of a Capacitor is a Spring
The mechanical analog of a capacitor is the
compliance of a
spring. The voltage
across a capacitor
corresponds to the
force used to
displace a spring. The charge
stored in
the capacitor corresponds to the
displacement of the spring.
Thus, Eq.
(
E.2) corresponds to
Hooke's law for ideal springs:
where
is called the
spring constant or
spring stiffness.
Note that
Hooke's law is usually written as
. The quantity
is called the
spring compliance.
Mechanical Equivalent of an Inductor is a Mass
The mechanical analog of an inductor is a
mass. The voltage
across an inductor
corresponds to the
force used to
accelerate a mass
. The current
through in the inductor
corresponds to the
velocity
of the mass. Thus,
Eq.
(
E.4) corresponds to
Newton's second law for an
ideal mass:
where
denotes the
acceleration of the mass
.
From the defining equation
for an inductor [Eq.
(
E.3)], we
see that the stored magnetic flux in an inductor is analogous to mass
times velocity, or
momentum. In other words, magnetic flux may
be regarded as electriccharge momentum.
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