The amplitude of a damped oscillator becomes half in one minute. The amplitude after 4 minute will be (1/Y) times the original, where Y is .

Question

The amplitude of a damped oscillator becomes half in one minute. The amplitude after 4 minute will be (1/Y) times the original, where Y is . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Amplitude of damped oscillator A=A0 exp (-λ t) For t=1 min , (A0/2)=A0 exp (-λ × 1) ∴ exp (λ)=2 For t=4 min , A=A0 exp (-λ × 4) ⇒ A=(A0/[ exp (λ)]4)=(A0/24) ∴ A =(A0/16)

Q. The amplitude of a damped oscillator becomes half in one minute. The amplitude after $4$ minute will be $\frac{1}{Y}$ times the original, where $Y$ is _____.

Amplitude of damped oscillator
$A = A_{0} exp (- λ t)$
For $t = 1 min,$
$\frac{A _{0}}{2} = A_{0} exp (- λ \times 1)$
$∴ exp (λ) = 2$
For $t = 4 min,$
$A = A_{0} exp (- λ \times 4)$
$\Rightarrow A = \frac{A _{0}}{[ e x p ( λ ) ] ^{4}} = \frac{A _{0}}{2 ^{4}}$
$∴ A = \frac{A _{0}}{16}$

Q. The amplitude of a damped oscillator becomes half in one minute. The amplitude after 4 minute will be Y1​ times the original, where Y is _____.

Amplitude of damped oscillator A=A0​exp(−λt) For t=1min, 2A0​​=A0​exp(−λ×1) ∴exp(λ)=2 For t=4min, A=A0​exp(−λ×4) ⇒A=[exp(λ)]4A0​​=24A0​​ ∴A=16A0​​

Q. The amplitude of a damped oscillator becomes half in one minute. The amplitude after $4$ minute will be $\frac{1}{Y}$ times the original, where $Y$ is _____.

Amplitude of damped oscillator
$A = A_{0} exp (- λ t)$
For $t = 1 min,$
$\frac{A _{0}}{2} = A_{0} exp (- λ \times 1)$
$∴ exp (λ) = 2$
For $t = 4 min,$
$A = A_{0} exp (- λ \times 4)$
$\Rightarrow A = \frac{A _{0}}{[ e x p ( λ ) ] ^{4}} = \frac{A _{0}}{2 ^{4}}$
$∴ A = \frac{A _{0}}{16}$