# 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:
• T = 9.4281

#### 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:
• t = 4.500 e6
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:
• t = k(l-ls) = 4.500 e6
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:
• v = 2L/T = 424.27
On the other hand, the theoretical value for this velocity is v = sqrt(t/m). This results in
• v = sqrt(t/m) = 424.26
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.