Question

The quality factor of a sonometer wire of frequency 500 Hz is 5000 . The time in which its energy Reduce to 1/e of its value in the absence of damping

Answer 1

The energy of oscillation is

$$E(t)=E_0e^{\frac{-\omega_0t}{Q_0}}.$$

The phrase "The time in which its energy reduce to 1/e of its value in the absence of damping" means that the new energy must be

$$E(\tau)=\frac{E_0}{e}=E_0\cdot e^{-1}.$$

From the first and second equation, we notice that

$$\frac{\omega_0t}{Q_0}=1,\rightarrow\\ t=\tau=\frac{Q_0}{\omega_0}=\frac{Q_0}{2\pi f_0}=\frac{5000}{2\pi500}=1.6\text{ s}.$$