In damped oscillations,the damping force is d | vedclass

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(A) The amplitude of a damped oscillator at time $t$ is given by $A(t) = A_0 e^{-bt}$,where $b$ is the damping constant.Given that at $t = 1 \, s$,the

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$\frac{1}{4} A_0$

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(A) The amplitude of a damped oscillator at time $t$ is given by $A(t) = A_0 e^{-bt}$,where $b$ is the damping constant.Given that at $t = 1 \, s$,the amplitude $A = \frac{A_0}{2}$.Substituting these values into the equation: $\frac{A_0}{2} = A_0 e^{-b(1)}$.This simplifies to $e^{-b} = \frac{1}{2}$.Now,we need to find the amplitude at $t = 2 \, s$.$A(2) = A_0 e^{-b(2)} = A_0 (e^{-b})^2$.Substituting $e^{-b} = \frac{1}{2}$ into the expression: $A(2) = A_0 \left(\frac{1}{2}\right)^2 = \frac{A_0}{4}$.

In damped oscillations,the damping force is directly proportional to the speed of the oscillator. If the amplitude becomes half of its initial value $A_0$ in $1 \, s$,then after $2 \, s$,the amplitude will be:

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