**Step 1: Define the system and select coordinates** Let the displacement of the center of the disc be \( x \). The disc of mass \( m \) and radius \( r \) can translate horizontally and rotate, but since it rolls without slipping, the rotation and translation are related: \( x = r\theta \), where \( \theta \) is the angular displacement. **Step 2: Obtain the Differential Equation of Motion (DEOM)** - Kinetic energy: \( T = \text{Translational} + \text{Rotational} = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}I\dot{\theta}^2 \) - For a disc, \( I = \frac{1}{2}mr^2 \) - Using rolling constraint, \( \dot{x} = r\dot{\theta} \) - So, \( T = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}\left(\frac{1}{2}mr^2\right)\left(\frac{\dot{x}}{r}\right)^2 = \frac{1}{2}m\dot{x}^2 + \frac{1}{4}m\dot{x}^2 = \frac{3}{4}m\dot{x}^2 \) - Potential energy: If both springs have stiffness \( k \), then \( V = kx^2 \) (if both stretch equally; if not, sum both spring energies accordingly). - Equation of motion (using Lagrange's equation): \[ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot{x}}\right) + \frac{\partial V}{\partial x} = 0 \implies \frac{3}{2}m\ddot{x} + 2kx = 0 \] (if both springs act in parallel; adjust if configuration is different). **Step 3: Find the natural frequency \( \omega_n \)** The standard form: \( \ddot{x} + \left(\frac{2k}{\frac{3}{2}m}\right)x = 0 \) So, \[ \omega_n = \sqrt{\frac{2k}{(3/2)m}} = \sqrt{\frac{4k}{3m}} \] If the answer is provided as \( \omega_n = 6k_n \), then check the spring arrangement and constants. For two springs, each of \( k \), acting in parallel, total stiffness is \( 2k \). **Summary:** 1. Let \( x \) be the displacement of the center; relate translation and rotation via rolling. 2. Write energy expressions, apply Lagrange's equation, and get \( \frac{3}{2}m\ddot{x} + 2kx = 0 \). 3. Thus, \( \omega_n = \sqrt{\frac{4k}{3m}} \). If “\( 6k_n \)” is expected, check the given values, but the above is the standard derivation.