Extrapolating the Quality factor for decay in amplitudes

[PhysicsKid0123]

Homework Statement

By what factor does the amplitude decay in the time interval (which I had to find).

E: stored energy
γ:damping rate
a: amplitude
t: time
Q: quality factor
T:Period
ω: frequency

Homework Equations

Q = 2∏E/(ΔE)

The Attempt at a Solution

I found the interval to be t=QT
but "one cycle" indicates T=2∏?

So I'm not exactly sure if I got the right answer for the quality factor by which the quality factor decays? I feel like there is supposed to be a numerical answer. Any insights? Did I mess up anywhere or took the wrong approach? Anything will help!

Thanks!

 

[BvU]

Suppose the amplitude decays by a factor of 2 in a certain interval. What does that mean for the stored energy ?

Vice versa yields your answer without much effort !

[edit] In fact your answer can be made to look more attractive if you realize that $\omega T = 2\pi$

 

[PhysicsKid0123]

Suppose the amplitude decays by a factor of 2 in a certain interval. What does that mean for the stored energy ?

Vice versa yields your answer without much effort !

[edit] In fact your answer can be made to look more attractive if you realize that $\omega T = 2\pi$

It would lose half of it's stored energy?

 

[BvU]

Nope.

 

[HalfLostAndSearching]

I would first clarify the question with the person who provided it. It seems that the question is asking for the factor by which the amplitude decays in a given time interval, t, which is equal to QT. However, it is not clear what is meant by "one cycle" and how it relates to the period, T=2∏. It would be helpful to have more context or information about the system being studied.

Assuming that the question is asking for the factor by which the amplitude decays in one period, T=2∏, then the quality factor, Q, can be calculated using the equation Q = 2∏E/(ΔE). However, the given equation does not directly provide the factor by which the amplitude decays. It is a measure of the system's ability to store energy relative to the energy lost through damping. So, to answer the question, we would need to know the specific values of E and ΔE for the system in question.

In general, the quality factor is a dimensionless quantity, so it does not have a numerical value. It is used to compare the efficiency of different systems in storing energy and is often used in the design of mechanical and electrical systems. It is also important to note that the quality factor is not a constant value, but can vary depending on the frequency of the system's oscillations.

Overall, without more information about the specific system and values of E and ΔE, it is not possible to provide a numerical answer to the question. I would recommend discussing the question further with the person who provided it or seeking clarification on the context and values involved.