The displacement of a damped harmonic oscillator is given by x(t)=e- 0 . 1 t cos(10 π t + varphi) , where t is the time in seconds. The time taken for the amplitude of the oscillator to drop to half of its initial value is close to

Question

The displacement of a damped harmonic oscillator is given by x(t)=e- 0 . 1 t cos(10 π t + varphi) , where t is the time in seconds. The time taken for the amplitude of the oscillator to drop to half of its initial value is close to (A) 13s (B) 7 s (C) 4 s (D) 27 s

Tardigrade Admin · Accepted Answer

(A0/2)=A0e- 0 . 1 t⇒ e- 0 . 1 t=2⇒ 0.1t=ln2 t=(ℓ n 2/0 . 1)=10en2=6.93=7s

Q. The displacement of a damped harmonic oscillator is given by $x (t) = e^{- 0.1 t} cos (10 π t + φ)$ , where $t$ is the time in seconds. The time taken for the amplitude of the oscillator to drop to half of its initial value is close to

$13 s$

$7 s$

$4 s$

$27 s$

$\frac{A _{0}}{2} = A_{0} e^{- 0.1 t} \Rightarrow e^{- 0.1 t} = 2 \Rightarrow 0.1 t = l n 2$
$t = \frac{ℓ n 2}{0.1} = 10 e n 2 = 6.93 = 7 s$

Q. The displacement of a damped harmonic oscillator is given by x(t)=e−0.1tcos(10πt+φ) , where t is the time in seconds. The time taken for the amplitude of the oscillator to drop to half of its initial value is close to

13s

7s

4s

27s