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(D) For a damped harmonic oscillator,the mechanical energy $E$ at time $t$ is given by $E(t) = E_0 e^{-bt/m}$,where $b$ is the damping constant and $m

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(D) For a damped harmonic oscillator,the mechanical energy $E$ at time $t$ is given by $E(t) = E_0 e^{-bt/m}$,where $b$ is the damping constant and $m$ is the mass of the block.We are given that the energy drops to half of its initial value,so $E(t) = E_0 / 2$.Substituting this into the equation: $E_0 / 2 = E_0 e^{-bt/m}$.This simplifies to $1/2 = e^{-bt/m}$,or $2 = e^{bt/m}$.Taking the natural logarithm on both sides: $\ln(2) = bt/m$.Solving for $t$: $t = (m/b) \ln(2)$.Given $m = 0.1\, kg$ and $b = 10^{-2}\, kg\,s^{-1}$,we have $m/b = 0.1 / 10^{-2} = 10\, s$.Thus,$t = 10 	imes \ln(2) \approx 10 	imes 0.693 = 6.93\, s$.Rounding to the nearest provided option,the value is closest to $7\, s$.

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