Many oscillatory systems are subject to damping effects which are not exactly an

Question

Many oscillatory systems are subject to damping effects which are not exactly analogous to the frictional damping considered in Section 3.1. Nevertheless, such systems typically exhibit an exponential decrease in their average stored energy of the form (E) = E_0 exp(-vt). It is possible to define an effective quality factor for such oscillators as Q_f = omega_0/gamma, where omega_0 is the natural angular oscillation frequency. For example, when the note "middle C" on a piano is struck its oscillation energy decreases to one half of its initial value in about 1 second. The frequency of middle C is 256 Hz. What is the effective Q_f of the system?

Explanation / Answer

According to the question stored energy E=E0exp(-vt)

where v is the normal oscilllation frequency

again quality factor Qf= omega0/v

and omega0 is the angular frequency

now for the piano in 1sec E=1/2E0

again E0= <E>/exp(-vt)

now v=256hz

then omega0= 2*pie*v =2*3.14*256Hz = 1607. 68Hz

then Qf= 1607.68/256 = 6.28

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