Calculating Qf: Piano Key Frequency 256Hz Energy Dissipated in 1s

[Eric_meyers]

Homework Statement

"Systems typically exhibit an exponential decrease in their average stored energy of the form <E> = E$$_{0}$$ e^(-vt) ---- If a piano key of frequency 256 Hz is struck and its oscillation energy decreases to one half of its initial value in about 1 second what is the effective Qf of the system?

Homework Equations

Qf = w/v
<E> = E$$_{0}$$ e^(-vt)

The Attempt at a Solution

I need to find the energy dissipated and from the equation they gave me I put 1/2 (the energy the system has from its initial value after 1 second)

1/2 = e^(-v)

v = .693

Qf = 256/.693 = 369

I'm not quite sure if I did the dissipation energy equation correctly.

 

[Winzer]

$$Q=\frac{\omega}{v}=\frac{2 \pi f}{v}$$. Your numerator needs to be in angular frequency. Looks fine though.

 

[tomcue]

But assuming that I did, the effective Qf of the system would be 369. This value represents the efficiency of the system in converting the initial energy into sound. A higher Qf would indicate a more efficient system, while a lower Qf would indicate a less efficient system. In this case, the system has a relatively high Qf, which means that it can sustain its oscillation and produce sound for a longer period of time before dissipating all of its energy. This information can be useful for designing and improving piano keys to produce longer and clearer sounds. Additionally, understanding the Qf of a system can also help in predicting its behavior and making adjustments to optimize its performance.