Question

The equation of free vibration of a system is \( X + 36\pi^2 X = 0 \). Its natural frequency is:

• A. 4 Hz
• B. 3 Hz
• C. \( 6\pi \, \text{Hz} \)
• D. 6 Hz

Hint:

The natural frequency of a free vibrating system is related to the coefficient of the second derivative term in its equation of motion.

Answer 1

The Correct Option is D

Step 1: Equation of free vibration
The equation of free vibration is of the form: \[ m X'' + k X = 0 \] Where \( X \) is the displacement, \( m \) is the mass, and \( k \) is the stiffness. For a harmonic oscillator, the natural frequency \( \omega \) is given by: \[ \omega^2 = k/m \] Here, the equation is \( X + 36\pi^2 X = 0 \), which gives: \[ \omega^2 = 36\pi^2 \] Step 2: Solve for the natural frequency The natural frequency \( f \) is related to \( \omega \) by: \[ f = \frac{\omega}{2\pi} \] Thus: \[ f = \frac{6\pi}{2\pi} = 6 \, \text{Hz} \] Final Answer: \[ \boxed{6 \, \text{Hz}} \]