Tag: damped harmonic motion

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.