Tag: physics

Any oscillation in which the amplitude of the oscillating quantity decreases with time is termed as damped oscilaation

In damped oscillations, the relation of amplitude of oscillations with time is given by $y={ y } _{ o }{ e }^{ -bt }=\frac { { y } _{ o } }{ { e }^{ -bt } } $, where
${ y } _{ o }=$ initial amplitude of oscillation
$t=$time
$b=$damping constant
since $b>0\quad & \quad t>0;$
${ e }^{ bt }>1\ { \Rightarrow e }^{ -bt }<1\ { \Rightarrow { y } _{ o }e }^{ -bt }<{ y } _{ o }\ \Rightarrow y<{ y } _{ o }$
which means that as the time progress, its amplitude will decrease.

Generally, work done by external force goes to total energy of the system. But in forced oscillations, it is dissipated by damping forces.

$\dfrac{A _0}{2} = A _0 e^{-0.1t} \Rightarrow e^{-0.1t} = 2 \Rightarrow 0.1t = \ell n 2$
$t = \dfrac{\ell n 2}{0.1} = 10 \, \ell n2 \approx 6.93 \approx 7s$

$f = 5$
so $T = \dfrac{1}{5}$

$10T = \dfrac{10}{5} = 2$

$\dfrac{A _0}{1000} = A _0 \left(\dfrac{1}{2}\right)^{t/2}$

$(2)^{t/2} = 1000$

$\left(\dfrac{t}{2}\right) log 2 = 3$

$t = \dfrac{6}{log 2} \approx 20 s$

It is our common experience that when a body is made to vibrate in a medium , the amplitude of the vibrating body continuously decreases with time and ultimately the body stops vibrating. This is called the damped vibrations.