If the amplitudes of a damped harmonic oscill | vedclass

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

Question

(D) The amplitude of a damped harmonic oscillator at any time $t$ is given by the equation $A(t) = A_0 e^{-bt/2m}$,where $A_0$ is the initial amplitud

Vedclass · Accepted Answer

$\frac{A_1 A_2}{A_0}$

Vedclass · Answer

(D) The amplitude of a damped harmonic oscillator at any time $t$ is given by the equation $A(t) = A_0 e^{-bt/2m}$,where $A_0$ is the initial amplitude at $t=0$ and $b$ is the damping constant.At time $t_1$,the amplitude is $A_1 = A_0 e^{-bt_1/2m}$. Thus,$e^{-bt_1/2m} = \frac{A_1}{A_0}$.At time $t_2$,the amplitude is $A_2 = A_0 e^{-bt_2/2m}$. Thus,$e^{-bt_2/2m} = \frac{A_2}{A_0}$.We want to find the amplitude $A$ at time $t = t_1 + t_2$,which is $A(t_1+t_2) = A_0 e^{-b(t_1+t_2)/2m}$.This can be written as $A(t_1+t_2) = A_0 (e^{-bt_1/2m}) (e^{-bt_2/2m})$.Substituting the expressions for the exponential terms,we get $A(t_1+t_2) = A_0 \left(\frac{A_1}{A_0}\right) \left(\frac{A_2}{A_0}\right)$.Simplifying this,we obtain $A(t_1+t_2) = \frac{A_1 A_2}{A_0}$.

If the amplitudes of a damped harmonic oscillator at times $t=0, t_1$ and $t_2$ are $A_0, A_1$ and $A_2$ respectively,then the amplitude of the oscillator at a time of $(t_1+t_2)$ is

Similar Questions

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 $20 \,s$?

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

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

What is the amplitude of a damped oscillator if time becomes $t = 2m$?

Which of the diagrams shown in the figure represents the variation of the total mechanical energy of a pendulum oscillating in water as a function of time?

Vedclass Test Series

Exam Paper Generator

Online Exam Module