Oscillation problem with damping

[azagaros]

Homework Statement

A mass on a long elastic cord oscillates with an amplitude of 23 cm. If the energy of this system is reduced by fiction to half its value, what will be its amplitude?

Homework Equations

Looking though my book or resources on the internet have not revealed a path for this one: I assume I have A_i= 23 cm. E₀ = 1/2KA_i²e^t₀/T
is the assumption of the equation i am starting with and I am looking for .5E₀ so I can find A_f

The Attempt at a Solution

The starting logic is not coming together and any help would be appreciated.

 

[Simon Bridge]

You need the relationship between the total energy stored in the system and the amplitude of the oscillations.
The equation you have tells you how the stored energy varies with time for a strongly damped system... but the answer is in there. Maybe you can use it too - what is the time when the energy is reduced to half?
[edit: nah, easier the other way: how does energy depend on amplitude?]

pedantic aside: damping ... "dampening" would be the process of making it slightly wet.

 

[azagaros]

I not sure how to calculate the time it would be at half energy and I had that thought too. I do not have mass and I am starting with the amplitude of the oscillation. I have been reading my book back and forth trying to figure this one out. Energy does not depend on amplitude. Amplitude is the result of energy expenditure in the conservation of energy thought. It decreases over time due to the work of friction, or air resistance of the mass plus the mass of the cord acting on itself, is my supposition.

This is not getting me any closer to a direction of equations. If potential energy is the pull on the cord to produce the initial amplitude as described as mgy-1/2ky² = 1/2ky² + 1/2mv² would get me the first cycle but not the dampening section nor do I have time in the equation. I do not have a spring constant muchless any indication of time in this structure for a function of amplitude vs time. This all assumes this is a vertical structure. The masses do not cancel out. The oscillation dampening equation is y(t) = Ae ^bt/2mcos(ωt+φ) and ω = √(k/m - b²/4m²) = √(ω²-b²/4m²). These need some assumption of 0 in them for the start of the system.

 

[Simon Bridge]

Really? You should be able to solve for this sort of problem:
At $t=0$, $E_0 =\frac{1}{2}KA^2$ ... for $t>0$, $E(t) = \frac{1}{2}KA^2e^{t/\tau}$ So when $E(t)=\frac{1}{2}E_0$ - solve for $t$.

But if you look closely at the equation for $E_0$ - what sort of relationship is it between E and A? Is it a cubic relationship? A logarithmic relationship? What?
You do not need the cycles or anything. If you have the system oscillating with amplitude A0 - how would you find E0, the energy stored?

 

[Mark44]

pedantic aside: damping ... "dampening" would be the process of making it slightly wet.

Thread title now changed.

 

[azagaros]

the actual answer was A_f=A_i/√2 when I talked to my teacher.

 

[Simon Bridge]

Good - do you know how to get that answer?