A block of mass 100 ,g is connected to an ela | vedclass

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

Question

(B) The amplitude of a damped oscillator varies with time as $A(t) = A_0 e^{-\alpha t}$, where $\alpha = \frac{b}{2m}$.Given that the amplitude drops

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) The amplitude of a damped oscillator varies with time as $A(t) = A_0 e^{-\alpha t}$, where $\alpha = \frac{b}{2m}$.Given that the amplitude drops to half its initial value, $A(t) = \frac{A_0}{2}$.Substituting this into the equation: $\frac{A_0}{2} = A_0 e^{-\alpha t} \Rightarrow \frac{1}{2} = e^{-\alpha t}$.Taking the natural logarithm on both sides: $\ln(0.5) = -\alpha t \Rightarrow -\ln 2 = -\alpha t \Rightarrow \ln 2 = \alpha t$.Given $b = 69.3 \,g \,s^{-1} = 0.0693 \,kg \,s^{-1}$ and $m = 100 \,g = 0.1 \,kg$.Calculate $\alpha = \frac{b}{2m} = \frac{69.3 \,g \,s^{-1}}{2 	imes 100 \,g} = \frac{69.3}{200} \,s^{-1} = 0.3465 \,s^{-1}$.Using $\ln 2 = 0.693$, we have $0.693 = \alpha t = 0.3465 	imes t$.Solving for $t$: $t = \frac{0.693}{0.3465} = 2 \,s$.

Similar Questions

For a damped harmonic oscillator governed by the equation $m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0$, find the time $t$ after which the mechanical energy becomes half of its initial maximum value.

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 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

Write the equation of angular frequency and amplitude for damped oscillation.

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module