Step 1: Free Vibration Equation
The free vibration equation is expressed as:
\[m X'' + k X = 0\]
Where \( X \) represents displacement, \( m \) is mass, and \( k \) is stiffness. For a harmonic oscillator, the natural frequency \( \omega \) is defined by:
\[\omega^2 = k/m\]
Given the equation \( X + 36\pi^2 X = 0 \), we obtain:
\[\omega^2 = 36\pi^2\]
Step 2: Natural Frequency Calculation The natural frequency \( f \) relates to \( \omega \) according to:
\[f = \frac{\omega}{2\pi}\]
Therefore:
\[f = \frac{6\pi}{2\pi} = 6 \, \text{Hz}\]
Final Answer:
\[ \boxed{6 \, \text{Hz}}\]