The time taken for the amplitude of vibration | 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 $A(t) = A_0 e^{-\frac{bt}{2m}}$.Given that $A(t) = 0.25 A_0 = \frac{A_0}{4}$ at time $t$,we have:

Vedclass · Accepted Answer

$\frac{t}{4}$

Vedclass · Answer

(C) The amplitude of a damped oscillator is given by $A(t) = A_0 e^{-\frac{bt}{2m}}$.Given that $A(t) = 0.25 A_0 = \frac{A_0}{4}$ at time $t$,we have:$\frac{A_0}{4} = A_0 e^{-\frac{bt}{2m}} \implies \frac{1}{4} = e^{-\frac{bt}{2m}} \implies 4 = e^{\frac{bt}{2m}}$.Taking the natural logarithm on both sides: $\ln(4) = \frac{bt}{2m} \implies 2\ln(2) = \frac{bt}{2m} \implies t = \frac{4m \ln(2)}{b} \quad ... (1)$The mechanical energy of a damped oscillator is given by $E(t) = E_0 e^{-\frac{bt}{m}}$.Given that $E(t') = 0.5 E_0 = \frac{E_0}{2}$ at time $t'$,we have:$\frac{E_0}{2} = E_0 e^{-\frac{bt'}{m}} \implies \frac{1}{2} = e^{-\frac{bt'}{m}} \implies 2 = e^{\frac{bt'}{m}}$.Taking the natural logarithm on both sides: $\ln(2) = \frac{bt'}{m} \implies t' = \frac{m \ln(2)}{b} \quad ... (2)$Dividing equation $(2)$ by $(1)$:$\frac{t'}{t} = \frac{m \ln(2) / b}{4m \ln(2) / b} = \frac{1}{4} \implies t' = \frac{t}{4}$.

The time taken for the amplitude of vibrations of a damped oscillator to drop to $25 \%$ of its initial value is $t$. The time taken for its mechanical energy to drop to $50 \%$ of its initial mechanical energy is

Similar Questions

The amplitude of a wave is represented by $A = \frac{c}{a + b - c}$. Resonance will occur when:

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

In case of damped oscillation,the frequency of oscillation is ............

The amplitude of a damped harmonic oscillator becomes $50 \%$ of its initial value in a time of $12 \ s$. If the amplitude of the oscillator at a time of $36 \ s$ is $x \%$ of its initial amplitude,then the value of $x$ is

The graph between velocity and position for a damped oscillation will be:

Vedclass Test Series

Exam Paper Generator

Online Exam Module