Sure! Let's go through the detailed steps to solve this problem.
Step-by-step solution:
1. Understand the given equation of motion:
The equation of motion provided is:
[tex]\[
x(t) = 5 \exp(-0.25 t) \cos\left(\frac{\pi}{2} \tau\right)
\][/tex]
2. Identify the terms in the equation:
The equation can be broken down into two main terms:
- The exponential term \(\exp(-0.25 t)\): This represents the damping factor, which indicates that the amplitude of the oscillation decreases over time.
- The cosine term \(\cos\left(\frac{\pi}{2} \tau\right)\): This represents the oscillatory part of the motion.
3. Determine the natural angular frequency (\(\omega_0\)):
The natural angular frequency is the frequency at which the system would oscillate if there were no damping (i.e., if the damping factor were zero).
For undamped oscillatory motion, the cosine term includes \(\omega_0\), the natural angular frequency. In this equation, the argument of the cosine function is \(\frac{\pi}{2} \tau\).
Therefore, the natural angular frequency (\(\omega_0\)) is directly obtained from the argument of the cosine function:
[tex]\[
\omega_0 = \frac{\pi}{2}
\][/tex]
4. Convert the result to a numerical value:
The value of \(\pi\) (pi) is approximately 3.141592653589793.
[tex]\[
\omega_0 = \frac{\pi}{2} \approx \frac{3.141592653589793}{2} = 1.5707963267948966
\][/tex]
Thus, the natural angular frequency (\(\omega_0\)) of the oscillation is approximately \(1.5708 \, \text{rad/s}\).
Final Answer:
The natural angular frequency of the oscillation is [tex]\( \omega_0 = 1.5708 \, \text{rad/s} \)[/tex].
