Damped oscillation - class-XI

Whenever there is a damping force, it will slow down the motion of a pendulum, and ultimately it will make the pendulum stop. This phenomenon is called damping.

The pendulum on its motion has friction particle. As a result of this frictional force slows down.

In case of dumped vibrations amplitude of oscillation gradually decreases with time.

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.

Forced vibrations: External force is acting on the body. 
Free vibration: Constant amplitude and no external force.
Damped vibration: Amplitude is not constant, it keeps on decreasing due to environmental factors of the system like air resistance.  
Therefore, correct option is C. 

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

For a damped oscillator, amplitude A(t) = A0 * e^(-bt). Given A(6) = A0/27, we have e^(-6b) = 1/27, so e^(-2b) = (1/27)^(1/3) = 1/3. Thus, A(2) = A0 * e^(-2b) = A0/3.

A vibrometer is an instrument used to measure vibration, specifically displacement, velocity, or acceleration.

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.

Mechanical energy E(t) = E0 * e^(-2bt), where b = c/(2m). Given m=0.4 kg, c=0.08 kg/s, b = 0.08 / (2 * 0.4) = 0.1 s^-1. We want E(t) = E0/2, so e^(-2 * 0.1 * t) = 0.5. -0.2t = ln(0.5) = -0.693. t = 0.693 / 0.2 = 3.465 s.

Amplitude A(t) = A0 * e^(-bt). Given A(t) = A0/2, e^(-bt) = 1/2. For 3t, A(3t) = A0 * (e^(-bt))^3 = A0 * (1/2)^3 = A0/8. Thus, n = 8.

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.

A particle oscillating under a force $\bar{F} = - k \bar{x} - b \bar{v}$ is damped oscillator. The first term $-k \bar{x}$ represents the restoring force and second term $-b \bar{v}$ represents the damping force.