Let's analyze the question: **Equation of motion of a viscous-damped vibration:** \[ \ddot{x} + 4\dot{x} + 12x = 0 \] We are asked to find the **damping ratio** (\(\zeta\)). --- ### Step 1: Standard Form The standard form of a damped vibration equation is: \[ \ddot{x} + 2\zeta\omega_n \dot{x} + \omega_n^2 x = 0 \] where: - \(\omega_n\) = natural frequency - \(\zeta\) = damping ratio --- ### Step 2: Compare Coefficients Given equation: \[ \ddot{x} + 4\dot{x} + 12x = 0 \] So, - \(2\zeta\omega_n = 4\) - \(\omega_n^2 = 12\) --- ### Step 3: Find \(\omega_n\) \[ \omega_n = \sqrt{12} = 3.464 \] --- ### Step 4: Solve for \(\zeta\) \[ 2\zeta\omega_n = 4 \implies 2\zeta(3.464) = 4 \] \[ \zeta = \frac{4}{2 \times 3.464} = \frac{4}{6.928} \approx 0.577 \] --- ### Step 5: Closest Option Given options: - 2.120 - 0.344 - 1.0 - 0.408 None of the options exactly match 0.577, but **0.408** is the closest. ### **Final Answer:** \[ \boxed{0.408} \]