Physics - EMI/AC Question with Solution | TestHub

The amplitude of a damped oscillator decays as $A(t)=A_0{e}^{-\frac{bt}{2m}}$. Given that $A(T)=A_0{e}^{-\frac{bT}{2m}}=\frac{A_0}{e}$, we have $e^{-\frac{bT}{2m}}=\frac{1}{e}$, which implies $\frac{bT}{2m}=1$. The energy of a damped oscillator decays as $E(t)=E_0{e}^{-\frac{bt}{m}}$. At time $t=2T$, the energy will be $E(2T)=E_0{e}^{-\frac{b(2T)}{m}}=E_0{e}^{-2(\frac{bT}{m})}$. Since $\frac{bT}{2m}=1$, then $\frac{bT}{m}=2$. Substituting this, $E(2T)=E_0{e}^{-2(2)}=E_0{e}^{-4}=\frac{E_0}{e^4}$. Thus, the ratio $\frac{E(2T)}{E_0}=\frac{1}{e^4}$.