#
Vibrating string

We shall take as working example
the experiment lineFree1.exp corresponding to the fundamental mode

####
Basic data

The basic data for this
experiment,
in adequate units, are:

- >Number of particles: n=21
- Mass of each particle: m=2,500
- Total length: L=2,000
- Springs default (rest) length: ls=10.0
- Spring constants: k=50,000

The rest distance between
neighboring particles is l=L/(n-1)= 100

This results in a mass per
unit
length: m = 2500/100 = 25

####
Period

An important quantity that
has to be measured is the Period of one oscillation. To accurately
measure
this quantity we have started the clock when the central particle
(monitored)
was at its highest point, with the Vz velocity just changing from plus
to minus (zero crossing velocity). Then we have measured the time for
reaching
exactly the same situation after 10 full oscillations. The result was
94.28125
and so, the Period:

####
Tension

We also measured the
Tension
of the string, t. To do this, we first set the string at its
equilibrium
state (straight line, all particles at rest).

Then we disconnected one
particle
(the central one from the right or left one) and measured the
unbalanced
force acting on it. The result was:

Note: to
disconect
the spring from one particle to a neighbour one, click the central
mouse
button on the first particle and drag to the next. To measure the
force,
activate "monitor" in the particle inspector panel and read the value
of
Fx.

####
Checking the results:

The value of the tension
is
coincident with the theoretical one given by:

For the fundamental mode,
the resonant frequency is given by: f

* 0* =v/(2L), v being the
velocity
of the wave in the string. This gives:

On the other hand, the
theoretical
value for this velocity is v = sqrt(t/m). This results in

It should be noticed that
the coincidence is extremely good, taking into account that we have
used
the equations corresponding to a string (distributed mass) and the
simulation
is made with a set of discrete localized particles.

####
Assignment:

Change the parameters of
the
system, measure the new period and check the results. For instance you
can load the following experiments:

with a mass=625 (k=50,000)

with a spring constant of
k=25000 (m=2500).

In both cases the central
particle
starts with zero vertical velocity.

Make a guess of the
resulting
period for each case.