How Do You Calculate Energy Loss in a Damped Harmonic Oscillator?

AI Thread Summary
To calculate energy loss in a damped harmonic oscillator, the displacement amplitude decreases by 15% over five minutes, indicating a conversion of mechanical energy to other forms. The total energy (TE) is related to the amplitude squared, so a 15% decrease in amplitude results in approximately 72% of the initial mechanical energy remaining. The initial angular frequency (omega) is calculated as 160 rad/s, and the period is 0.04 seconds. The damping coefficient (gamma) can be derived from the amplitude decay, allowing for the calculation of the quality factor (Q) of the oscillator. Understanding the relationship between amplitude, energy, and damping is crucial for solving these types of problems.
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Homework Statement


The displacement amplitude of a lightly damped oscillator with m=0.250kg and k=6400N/m is observed to decrease by 15% in exactly five minutes
a) Calculate the fraction (in%0 of the initial mechanical energy of the oscillator that has been converted to other forms of energy (such as thermal energy) in the five-minute interval.
b)Calculate the Q value of this damped oscillator by first calculation omega initial and gama


Homework Equations


omega initial = sqrt(k/m)
gama = b/m
x = A initial exp^(gama t/2)
TE = 1/2kA initial exp^(-gamat)
A(t) = A inital exp^(bt/2m)
T = 2pisqrt(m/k)


The Attempt at a Solution


I have calculated omega inital = 160rads and the period to be .04s. I know there is the relationship between the displacement amplitude and the total energy but I cannot seem to figure out how to relate them. Any help is greatly appreciated.
 
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At maximum displacement, what are the kinetic and potential energies of the oscillator?
 
At maximum displacement, the kinetic energy should be zero. Should I be able to solve for the potential energy? I am so confused as to how to approach this. I know I have been given enough info but I feel like I don't have enough to find any additional values.
 
So at maximum displacement the entire energy is the potential energy. What is the potential energy of a spring?
 
1/2ka^2. How do i find this without being given a?
 
What is a? What is amplitude?
 
Ok, so I finally figured out I can find the % of TE in relation to the % of A lost. Which I can also use to find b which I can then use to find Q.
A=Ainital exp^bt/2m where A/Ainital will equal .85 then the only unknown is b.
And TE goes as A^2 so if A is 85% then TE is .85^2 or ~72%
 
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