Question

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5\text{s}$. In another $10\text{s}$, it will decrease to $\alpha$ times its original magnitude, where $\alpha$ equals

• A. $0.7$
• B. $0.81$
• C. $0.729$
• D. $0.6$

Answer 1

The correct option is C $0.729$

Amplitude of damped oscillator is given by
$A=A_0{e}^{-\frac{bt}{2m}}$
After $5\text{s}$, $0.9A_0=A_0{e}^{-\frac{b\left(5\right)}{2m}}$
$\Rightarrow 0.9=e^{-\frac{b\left(5\right)}{2m}}$ ...(i)
After next $10\text{s}$, $A=A_0{e}^{-\frac{b\left(15\right)}{2m}}$
$\Rightarrow A=A_0{\left(e^{-\frac{5b}{2m}}\right)}^3$ ...(ii)
From Eqs. (i) and (ii), we get
$A={0.9}^3\times A_0=0.729A_0$
Hence, $\alpha =0.729$

Why this question? Note: In damped oscillator, the initial amplitude decays exponentially. Hence it is not necessary to know the exact value of initial amplitude if all one needs to find out is the time in which the initial amplitude decreases by a certain factor.