Let's break down the problem step by step: ## Given Data - Mass, \( m = 150 \) kg - Spring stiffness, \( k = 45 \) kN/m = \( 45000 \) N/m - Damping is 30% of the critical value: \( \xi = .3 \) ## Logarithmic Decrement Formula The logarithmic decrement (\(\delta\)) for an underdamped system is: \[ \delta = \frac{2\pi \xi}{\sqrt{1 - \xi^2}} \] ## Calculation 1. **Damping Ratio:** \[ \xi = .3 \] 2. **Calculate \(\sqrt{1 - \xi^2}\):** \[ \sqrt{1 - (.3)^2} = \sqrt{1 - .09} = \sqrt{.91} \approx .9539 \] 3. **Plug values into the formula:** \[ \delta = \frac{2\pi \times .3}{.9539} \] \[ \delta = \frac{1.884}{.9539} \approx 1.975 \] ## Closest Option - **1.97** (rounded) is the closest to our answer. --- ## **Final Answer** **The correct option is:** \[ \boxed{1.97} \]