The displacement of a damped harmonic oscilla | vedclass

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

Question

(D) The displacement of a damped harmonic oscillator is given by $x(t) = A(t) \cos(\omega t + \varphi)$,where $A(t) = A_0 e^{-bt/2m}$.Comparing this w

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(D) The displacement of a damped harmonic oscillator is given by $x(t) = A(t) \cos(\omega t + \varphi)$,where $A(t) = A_0 e^{-bt/2m}$.Comparing this with the given equation $x(t) = e^{-0.1t} \cos(10\pi t + \varphi)$,we identify the amplitude as $A(t) = A_0 e^{-0.1t}$,where $A_0 = 1$.We need to find the time $t$ when the amplitude $A(t)$ becomes half of its initial value $A_0$.Set $A(t) = \frac{A_0}{2}$,which gives $A_0 e^{-0.1t} = \frac{A_0}{2}$.Dividing both sides by $A_0$,we get $e^{-0.1t} = \frac{1}{2}$,or $e^{0.1t} = 2$.Taking the natural logarithm on both sides: $0.1t = \ln(2)$.Using the value $\ln(2) \approx 0.693$,we get $0.1t = 0.693$.Therefore,$t = \frac{0.693}{0.1} = 6.93 \, s$.Rounding to the nearest integer,we get $t \approx 7 \, s$.

The displacement of a damped harmonic oscillator is given by $x(t) = e^{-0.1t} \cos(10\pi t + \varphi)$. The time taken for its amplitude of vibration to drop to half of its initial value is close to .... $s$

Similar Questions

To understand resonance,describe the experiment of oscillations of five pendulums.

When an external force with angular frequency $\omega_d$ acts on a system of natural angular frequency $\omega$,the system oscillates with angular frequency $\omega_d$. The condition for the amplitude of oscillations to be maximum is

What is resonance? Give its example.

Which of the following figures represents damped harmonic motion?

Vedclass Test Series

Exam Paper Generator

Online Exam Module