Tag: free, forced and damped oscillations

Damping is caused by opposing force, which decreases the frequency.

Ans: C

The amplitude, a, at time $t$ is given by $a=a _0\;exp(-\,\alpha t)$

$a _{50}=a _0\;exp(-\alpha\times 50T)=0.80\;a _0$

where $T$ is the period of oscillation

$a _{150}=a _0\;exp(-a\times 150T)$

$=a _0\;(0.8)^3=0.512\,a _0$

Its is a damped oscillation, where amplitude of oscillation at time $t$ is given by $A = a _0e^{-\gamma t}$
where $a _0 = $ initial amplitude of oscillation
$\quad \gamma = $ damping constant
As per question, $\displaystyle\dfrac{a _0}{3} = a _0e^{-\gamma100/v}\quad                    ...(i)$
(where $v$ is the frequency of oscillation)
and $A = a _0e^{-\gamma200/v} \quad                ...(ii)$
From $(i)$; $\quad \displaystyle\dfrac{a _0}{3} = a _0e^{-\gamma\times100/v} \quad            ...(iii)$
Dividing equation $(ii)$ by $(iii)$, we have

$\quad \displaystyle\dfrac{A}{a _0(1/3)} = \displaystyle\dfrac{e^{-\gamma\times200/v}}{e^{-\gamma\times100/v}} = e^{-\gamma\times100/v} = \displaystyle\dfrac{1}{3}$

or $A = a _0\times\displaystyle\dfrac{1}{3}\times\displaystyle\dfrac{1}{3} = \displaystyle\dfrac{1}{9}a _0$

In reality, a spring won't oscillate forever. Frictional force will decrease the oscillation until eventually, the system is at rest.

Underdamped oscillation is practical because there always be resistive force present in reality which will try to make an oscillating body to lose its energy. This loss of energy makes the motion damped motion.

Dampers are found bridges to prevent them from swaying due to wind otherwise this motion can hamper the condition of a bridge.

The force on body oscillating in resistive medium is