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Answer:
17 seconds
Explanation:
Given that:
The mass attached to the spring (m) = 0.30 kg
The spring constant (k) = 2.00 N/m
The damping constant (b) = 0.025 kg/s
The initial distance [tex]x_o[/tex] = 5.0 cm
The initial final amplitude [tex]A_f[/tex] = 2.5 cm and not 2.5 m, please note the mistake, if it is 2.5 m, our time taken will be -93.7 sec, and we do not want a negative time value.
To start with the angular frequency damping using the formula:
[tex]\omega_{\gamma}= \dfrac{b}{2m}[/tex]
[tex]\omega_{\gamma}= \dfrac{0.025 \ kg/s}{2(0.3 \ kg)}[/tex]
[tex]\omega_{\gamma}=4.167 \times 10^{-2} \ s^{-1}[/tex]
In the absence of damping, the angular frequency is:
[tex]\omega_o = \sqrt{\dfrac{k}{m}}[/tex]
[tex]\omega_o = \sqrt{\dfrac{2 \ N/m}{0.3 kg}} \\\\\omega_o = 2.581 \ s^{-1}[/tex]
The initial amplitude oscillation can be computed by using the formula:
[tex]A_i = e^{-\omega_{\gamma}t} x_o \sqrt{\dfrac{\omega_o^2}{\omega_o^2-\omega_f^2}}[/tex]
[tex]A_i = e^{-\omega_{\gamma}0} (5.0 \ cm) \sqrt{\dfrac{2.581^2}{2.581^2-(4.167*10^{-2})^2}}[/tex]
[tex]A_i = 5.0006 \ cm \\ \\ A_i = 5.001 \ cm[/tex]
The final amplitude, as well as the initial amplitude, can be illustrated by using the relation:
[tex]A_f = e^{-\omega_{\gamma}t}A_i\\ \\ e^{-\omega_{\gamma}t} = \dfrac{2. 5 \ cm}{5.001 cm}\\ \\ = 0.4999\\ \\ \implies -\omega_{\gamma}t_f = \mathsf{In (0.4999)} \\ \\ t_f = \dfrac{\mathsf{-In (0.4999)}}{4.167*10^{-2} \ s^{-1}} \\ \\[/tex]
[tex]t_f = 16.64 \ sec \\ \\ \mathbf{t_f \simeq 17 sec}[/tex]