Answer to Question #225176 in Mechanics | Relativity for Nafiz

Question

For a damped oscillator m =250gm, k = 85N/m and b = 70gm/s. (a) What is the period of the
motion? (b) How long does it take for the amplitude of the damped oscillations to drop to half its
initial value?(c) How many oscillations does it complete in life time?

Answer 1

(a)

Damping coefficient $\gamma$ is given by

$\gamma=\dfrac{b}{2m}\\ \gamma=\dfrac{70}{2\times250}=0.14s^{-1}\\$

Undamped frequency $\omega_o$ is given by

$\omega_o=\sqrt{\dfrac{k}{m}}\\ \omega_o=\sqrt{\dfrac{85}{0.25}}=18.4s^{-1}\\$

Damped frequency $\omega_1$ is given by

$\omega_1=\sqrt{\omega_o^2-\gamma^2}\\ \omega_1=\sqrt{18.4^2-0.14^2}\\ \omega_1=18.4s^{-1}$

Period T is

$T=\dfrac{2\pi}{\omega_1}\\ T=\dfrac{2\pi}{18.4}=0.34s\\$

(b)

Amplitude at any time is given by$A(t)=A_oe^{-\gamma{t_1}}\\ A(t)=\dfrac{1}{2}A_o\\ \therefore t_1=\dfrac{\ln{2}}{\gamma}\\ t_1=\dfrac{\ln{2}}{0.14}=4.95s\\$

time is 4.95s for the amplitude of the damped oscillations to drop to half its

initial value

(c)

Number of oscillations to complete in life time is given by

$f=\dfrac{\omega_o}{2\pi}\\ f=\dfrac{18.4}{2\pi}=2.93oscillations\\$