Consider now the
mass-
spring oscillator depicted physically in
Fig.
D.3, and in
equivalent-circuit 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 mass-spring 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 mass-spring
contact-point sum to zero:
We have thus derived a second-order
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 electric-charge momentum.
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