Relationship between length of rubber beam and time to come to rest

[never4getthis]

Homework Statement

Imagine a rubber beam. One en is fixed and the other is pulled down 5cm and released. The beam wobbles until it comes to rest.

Independent (what I change): length of rubber beam
Dependent (what I measure): time to come to rest
Controlled (what stays the same): everything else

Homework Equations

I got a relationship of T=kL^2

The question is why this relationship. I can´t seem t find the answer

The Attempt at a Solution

One of the observations was that the shorter the beam, the faster it moved. When given potential energy (when ppulled down), then released, part of it gos to KE and the rest to termal (energy loss). If it moves faster, it looses energy faster so the time to come to rest should be less. This explains why it increases with length but not the x-squared relationship.

On the internet I have see things about stiffness, an iron beam, to support the same weight has to have a stiffness of the squared of the length. but does stiffness affect the time to come to rest?

This might also go into SHM, but I'm not sure

 

[PhanthomJay]

For a cantilever beam fixed at one end and free at the other, the effective spring constant , k, varies inversely with the cube of the length. The period of motion if undamped varies as the square root of the mass/k ratio, in simple harmonic motion. So if you halve the length of the beam, it's stiffness increases by a factor of 8, and its mass decreases by a factor of 2. In your equation for T = kx^2, the constant k is not the same as the beam's spring constant. So you might want to use a different letter to designate the constant, like T = cx^2, and show why (note, i have not included the complexity of energy loss during damped harmonic motion, but the principle still holds, I believe, by calculating the period of the motion as if undamped.).