Answer:
The damping coefficient of the system is 0.226.
Explanation:
Given:
Mass [tex]m = 0.223[/tex] kg
Time [tex]t = 9.07[/tex] sec
From the formula of amplitude,
[tex]A = A_{o} e^{-\alpha t }[/tex]
Where [tex]\alpha = \frac{b}{2m}[/tex]
Here amplitude decrease to 1%
So we write,
[tex]0.01 = e^{-\alpha 9.07 }[/tex]
[tex]\ln 0.01 = - \alpha 9.07[/tex]
[tex]-4.605 = - \alpha 9.07[/tex]
[tex]\alpha = 0.507[/tex]
Here [tex]\alpha = \frac{b}{2m}[/tex]
Where [tex]b =[/tex] damping coefficient
[tex]b = \alpha 2m[/tex]
[tex]b = 2 \times 0.507 \times 0.223[/tex]
[tex]b = 0.226[/tex]
Therefore, the damping coefficient of the system is 0.226.
The damping coefficient (b) of the system is equal to 0.2264.
Given the following data:
To determine the damping coefficient (b) of the system:
Mathematically, the amplitude for an underdamped harmonic motion is given by the formula:
[tex]A=A_oe^{-\alpha t}[/tex]
Substituting the given parameters into the formula, we have;
[tex]0.01 = e^{-9.07\alpha }\\\\ln0.01 = -9.07\alpha\\\\-4.6052 = -9.07\alpha\\\\\alpha =\frac{-4.6052}{-9.07} \\\\\alpha = 0.5077[/tex]
Now, we can find the determine the damping coefficient (b) of the system by applying this formula:
[tex]\alpha = \frac{b}{2m}[/tex]
Making b the subject of formula, we have:
[tex]b = \alpha 2m\\\\b= 0.5077 \times 2 \times 0.223[/tex]
Damping coefficient (b) = 0.2264
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