Question

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

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

Answer 1

The correct option is B $0.729$

Amplitude of damped oscillator is $A=A_0{e}^{-\frac{bt}{2m}}$

After $5s$,
$0.9A_0=A_0{e}^{-\frac{b\left(5\right)}{2m}}$..........(i)

After $10s$,
$t=5+10=15s$

$A=A_0{e}^{-\frac{b\left(15\right)}{2m}}$

or $A=\alpha A_0=A_0{\left(e^{-\frac{b\left(5\right)}{2m}}\right)}^3$..........(ii)

From eq. (i) & (ii)

$\alpha =(0.9{)}^3=0.729$

Hence, option $\left(C\right)$ is correct.