Step 1: Understanding the Question:
We are given the differential equation for an undamped free vibration system and asked to calculate its natural frequency (\(f\)) in Hertz (Hz).
Step 2: Key Formula or Approach:
The standard equation of motion for a simple harmonic oscillator (free vibration system) is:
\[ \ddot{X} + \omega_n^2 X = 0 \]
where \(\omega_n\) is the natural angular frequency in radians per second (rad/s).
The natural frequency \(f\) in Hertz is related to \(\omega_n\) by:
\[ f = \frac{\omega_n}{2\pi} \]
Step 3: Detailed Explanation:
We compare the given equation with the standard form:
Given equation: \( \ddot{X} + 36\pi^2 X = 0 \)
Standard form: \( \ddot{X} + \omega_n^2 X = 0 \)
By comparing the coefficients of \(X\), we get:
\[ \omega_n^2 = 36\pi^2 \]
Taking the square root of both sides gives the natural angular frequency:
\[ \omega_n = \sqrt{36\pi^2} = 6\pi \, \text{rad/s} \]
Now, we convert the angular frequency \(\omega_n\) to natural frequency \(f\):
\[ f = \frac{\omega_n}{2\pi} = \frac{6\pi}{2\pi} = 3 \, \text{Hz} \]
Note: The provided answer key in the source code seems to indicate 6 Hz, but the calculation correctly yields 3 Hz, which corresponds to option (A). We will proceed with the mathematically correct answer.
Step 4: Final Answer:
The natural frequency of the system is 3 Hz. Therefore, option (A) is the correct answer.