Re: LC circuit/mass-spring analogy

["Prof. John P. Ertel (wizard)" <jpe@nadn.navy.mil>]

The 2nd Order Linear Dif-Eq which represents a simple spring-mass system
with v^1_damping together with the one for a simple SERIES LRC-circuit may
be thought of generally as "equations of motion".  Converting these into a
standard form, we have

  d^2x       dx                            d^2q       dq     1
m ----  +  b --  +  k x  =  Fdrv   <=>   L ----  +  R --  +  - q  =  Vdrv
  dt^2       dt                            dt^2       dt     C

In this regard, we would view the 2nd term as "inertia" where this
attribute is measured as either the mass (m) in a mechanical system or the
inductance (L) in an electrical system.  Thinking of inductance as the
"inertia" of a circuit is, I believe, a good pedagogical tool for
understanding the import of the di/dt part of the inductive reaction
voltage(NOT a potential) to signals passing through a circuit (- L di/dt).
As we know from Lenz's Law, the inductive part of a system "reacts so as
to oppose (or attempt to oppose) the change occurring within it."  What
simpler generic statement of inertia could we choose?

The 1st term is of course "loss" where this attribute is measured as
either the damping coefficient (b) or the resistance (R).  This gives us a
strong relationship between friction/damping and resistance thereby giving
insight into the microscopic causes resistance.

The zeroth term is "compliance" where this attribute is measured as either
the stiffness (k) or the reciprocal capacitance (1/C).  I find most
students very quickly get comfortable with thinking of 1/C as being
similar to "stiffness": the lower the capacitance, the easier it is to
"energize" a capacitor; and vice versa, the higher the capacitance, the
harder it is to energize a capacitor.

Over on the right side of each equation is the "inhomogeneity".  This is
the driving force (Fdrv) in a mechanical system and the driving
voltage(NOT a potential) Vdrv in an electrical system.  Each term in the
mechanical system is then a generalized (let's not forget the rotational
analog) force and each term in the electrical system is a voltage(NOT a
potential).

My answers to the questions:

1)  Strictly speaking, there is NOT a mechanical analogy of potential
difference.  But the mechanical analogy of voltage difference is the
generalized force.

2)  With the above explanations and some extensions, my students seem to
be less confused.  They can concentrate on the analogies/similarities and
not the lack thereof.

3)  I have not found any "limitations" to the LC<=>M-S analogy.  They each
seem perfectly analogous and fully supportive of physics of each other.

PS:  I cannot really take credit for most of the above (except that I
listened).  I was given these insights (if you want to call them that) by
two remarkable teachers of Physics while a student in the early 60's at
Emory University in Atlanta.  A belated ***THANKS*** to Drs. Robert Rohrer
and Jim Simmons!  Your lectures really were excellent!

=================================



On Wed, 9 Apr 1997, Jeff Marx wrote:

> Hi all,
>   Three questions related to the LC circuit/mass-spring analogy...
>
>   1)  According to Halliday, Resnick, and Walker's "Fundamentals of
> Physics" (Fourth Edition) the analogy between elements and variables in an
> LC circuit and a mass/spring system goes like this...
>
> LC <--> M-S
> -------------
>  q <--> x
>  i <--> v
>  C <--> 1/k
>  L <--> m
>
> My first question is, "What is the mechanical analogy of potential
> difference?"  (This is actually a question in the book.)  I feel that I
> have the answer they are looking for, but I was wondering what other people
> would come up with.
>   2)  My next question is how does one present the answer to the above
> question in a light that may not be confusing to the students? 
>   3)  What are some limitations of the LC <--> M-S analogy?
>
> Thanks in advance for any insights you may have on these questions.
>
> Jeff Marx
> RPI
> Troy, NY

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