A body of mass 0.3 kg hangs by a spring with | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The amplitude of a damped oscillator is given by $A(t) = A_0 e^{-(b/2m)t}$.Given that after $n = 100$ oscillations, the amplitude becomes $A_0/e$,

Vedclass · Accepted Answer

$\frac{\pi^2}{600} \%$

Vedclass · Answer

(C) The amplitude of a damped oscillator is given by $A(t) = A_0 e^{-(b/2m)t}$.Given that after $n = 100$ oscillations, the amplitude becomes $A_0/e$, so $e^{-(b/2m)T \cdot n} = e^{-1}$, where $T$ is the time period.Thus, $\frac{b}{2m} T \cdot n = 1$. Since $T = \frac{2\pi}{\omega}$, we have $\frac{b}{2m} \cdot \frac{2\pi}{\omega} \cdot 100 = 1$, which implies $\frac{b}{2m\omega} = \frac{1}{200\pi}$.The damped frequency is $\omega^{\prime} = \omega \sqrt{1 - \frac{b^2}{4m^2\omega^2}}$.Using the binomial approximation $(1-x)^{1/2} \approx 1 - x/2$ for small $x$, we get $\omega^{\prime} \approx \omega \left(1 - \frac{b^2}{8m^2\omega^2}ight)$.Therefore, $\frac{\omega - \omega^{\prime}}{\omega} = \frac{b^2}{8m^2\omega^2} = \frac{1}{2} \left(\frac{b}{2m\omega}ight)^2$.Substituting the value, $\frac{\omega - \omega^{\prime}}{\omega} = \frac{1}{2} \left(\frac{1}{200\pi}ight)^2 = \frac{1}{2} \cdot \frac{1}{40000\pi^2} = \frac{1}{80000\pi^2}$.Converting to percentage: $\frac{1}{80000\pi^2} 	imes 100 = \frac{1}{800\pi^2} \%$.

Similar Questions

Explain the behaviour of the oscillator when the driving frequency is far from the natural frequency in small damped oscillations.

In a time of $2 \ s$,the amplitude of a damped oscillator becomes $\frac{1}{e}$ times its initial amplitude $A$. In the next two seconds,the amplitude of the oscillator is

The amplitude of a damped oscillator is known to decrease to $0.9$ times its original amplitude in $5 \,s$. Approximately, by how many times its original amplitude will it decrease after another $20 \,s$?

Which of the following statements regarding the damping force of a damped oscillator is $NOT$ correct?

Vedclass Test Series

Exam Paper Generator

Online Exam Module