Let's break down the problem: ### Problem Statement You are given a system where a mass \( m \) is attached to the end of a rigid bar, which is hinged at one end and supported by a spring (stiffness \( k \)) at a distance \( b \) from the hinge. The mass of the bar is negligible. ### Given Data - \( k = 2500 \) N/m (spring stiffness) - \( m = 10 \) kg (mass) - \( a, b \) are distances as shown in the diagram (with \( a \) being the distance from the hinge to the mass, and \( b \) the distance from the hinge to the spring) ### Objective **Find the natural frequency (\( \omega_n \)) of the system.** --- ## Step-by-Step Solution ### 1. **Moment Equation** When the bar rotates by a small angle \( \theta \): - The mass at distance \( a \) exerts a moment: \( m a^2 \ddot{\theta} \) - The spring exerts a restoring force at distance \( b \): \( k b^2 \theta \) Equation of motion: \[ m a^2 \ddot{\theta} + k b^2 \theta = 0 \] --- ### 2. **Natural Frequency Formula** \[ m a^2 \ddot{\theta} + k b^2 \theta = 0 \] This is of the form: \[ I \ddot{\theta} + k_{eff} \theta = 0 \] Where \( I = m a^2 \) and \( k_{eff} = k b^2 \). Natural frequency: \[ \omega_n = \sqrt{\frac{k_{eff}}{I}} = \sqrt{\frac{k b^2}{m a^2}} \] --- ### 3. **Matching to the Options** Looking at the answer choices in the image, the correct formula is: \[ \omega_n = \sqrt{\frac{k b^2}{m a^2}} \] This matches the **last option** in the list. --- ## **Final Answer** \[ \boxed{\omega_n = \sqrt{\frac{k b^2}{m a^2}}} \] Let me know if you need a numerical answer as well!