Question

In damped oscillations, damping force is directly proportional to speed to oscilator . If amplitude becomes half of its maximum value in 1s , then after 2 s amplitude will be (intial amplitude =$A_0$)

• A. $\frac{1}{4}{A}_0$
• B. $\frac{1}{2}{A}_0$
• C. $\frac{1}{5}{A}_0$
• D. $\frac{1}{7}{A}_0$

Answer 1

The correct option is A $\frac{1}{4}{A}_0$

In damped oscillations, damping force is directly proportional to speed to oscilator . If amplitude becomes half of its maximum value in 1s , then after 2 s amplitude will be

Amplitude is given by:

$A=A_o{e}^{-\alpha t}$

Where, $A$ is amplitude at time t.

t is time

$A_0$ is initial aplitude

$\alpha$ is constant

At t = 1s

$A=\frac{A_0}{2}$

So,

$\frac{A_0}{2}=A_0{e}^{-\alpha }$

$e^{-\alpha }=\frac{1}{2}$

At t = 2s

$A=A_0{e}^{-2\alpha }$

$A=A_0{\left(\frac{1}{2}\right)}^2$

$A=\frac{A_0}{4}$