Problem statement

Let m be the mass of a structureless
body supported by a spring

with a uniform force constant k as shown in the diagram.

Set up the differential equation of motion that determines

the displacement of the mass from its equilibrium position

at time t when the intital conditions are x(0) = x_{0} and x'(0) = 0.

The net force on the mass is given by Hooke's law: *f = -kx*,
where *x* is the extension of the spring beyond its equilibrium length. According
to Newton's second law the force can be equated to mass times acceration giving us the
second order DE,

*mx" + kx = 0. *

To solve, define *w = (k/m) ^{1/2}* and
rewrite the DE as

The complete solution of the DE, with two arbitrary constants, is

*x(t) = A cos(w t) + B sin(w t).
*The initial conditions (

The constant *w*, characteristic of the system mass
and force constant, is the angular frequency of the motion. The frequency in cycles per
unit time is *n = w/2p*, and the period is *T = 2p/w*. Motion of this type, with amplitude given by the cosine (or
the sine) is called harmonic. When the initial conditions are different, the solution can
still be expressed as a cosine using the alternative parameterization *x(t) = x _{0}
cos(w t - f)*. This choice of
parameters includes the phase angle

A physically different system
with an equivalent DE and analogous "motion" consists of a series electric
circuit contaning capacitance *C* and inductance *L* as shown in the figure.
In this, the charge on the capacitor, *C*, varies in time in the
same way as the displacement of the mass on a spring.

Suppose that, with the switch in the open position, the capacitor has
charge *Q _{0}* and the current,

The potential differences across circuit elements must sum to zero
around the closed circuit. That across the inductance is *L dI/dt* and that across
the capacitor is *Q/C*. Since *I = dQ/dt* we find the DE: *LQ" + Q/C
= 0*. Make the substitution *w ^{2}*

This result is identical in form with that for the mass on a spring. Therefore the
electrical system and the mechanical system are said to be analogous. The equivalence
between them is exhibited in *x ~ Q, m ~ L, k ~ 1/C*. That is, displacement is
equivalent to quantity of electrical charge, mass to inductance, and spring constant to
the reciprocal of capacitance.

1. Show that the different parameterizations *x(t) = A cos(w
t) + B sin(w t)* and *x(t) = C cos(w
t - f)* are related by *C = (A ^{2} + B^{2})^{1/2}*
and

2. Deduce the form of the DE for the electrical LC oscillator if resistance, *R*,
is added to the series circuit. Only the ratios *R/L=2k*
and *1/LC=w ^{2}* are significant for the DE. (a)
To solve, introduce the integrating factor,

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Created or up-dated 08/03/99
by R.D. Poshusta