Answer:
a) 1.58 kg s^{-1}
b) x_m e^{-1.58t} x_m is initial amplitude
c) 5 kg s^{-1}
Explanation:
given data:
mass =0.5 kg
k = 12.5 N/m
from the data given
a) [tex]w_d = w_o - \frac{0.2}{100}w_o[/tex]
[tex]= w_o - 0.002w_o = 0.99w_o[/tex]
[tex]w_d = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}[/tex]
[tex]0.998w_o = \sqrt{w_o^2 - \frac{b^2}{4m^2}[/tex]
[tex](0.998w_o)^2 = w_o^2 -\frac{b^2}{1}[/tex]
[tex]b^2 = w_o^2 -(0.998w_o)^2[/tex]
[tex]b^2 = w_o^2(1-0.998^2) = 3.996 *10^{-3} w_o^2[/tex]
[tex]b = w_o\sqrt{3.996*10^{-3}}[/tex]
[tex]b = \frac{12.5}{0.5}\sqrt{3.996*10^{-3}} = 1.58 kg s^{-1}[/tex]
b)[tex] x = x_m e^{\frac{-bt}{2m}}[/tex]
[tex]x = x_m e^{-1.58t}{1} = x_m e^{-1.58t}[/tex] where x_m is initial amplitude
c) critical damping amplitude
[tex]c_c =2\sqrt{km} = 2\sqrt{12.5*.5} = 5 kg s^{-1}[/tex]
