1. **Problem Statement:** We have a beam of length $L$ with a fixed pin at the middle, dividing the beam into two equal halves each of length $\frac{L}{2}$. At each free end of the beam, there is a spring attached. We want to derive the equation of motion for the free vibration of this system. 2. **Model Setup:** Let $x(t)$ represent the displacement of the beam at the fixed pin (middle point) from its equilibrium position. 3. **Assumptions and Parameters:** - The beam is symmetric about the fixed pin. - Each spring has spring constant $k$. - The beam has mass $m$ concentrated at the pin (or effective mass for vibration). 4. **Forces on the Beam:** - When the beam displaces by $x(t)$, each spring at the ends stretches or compresses by $x(t)$ because the pin displacement causes equal displacement at both ends due to symmetry. - The restoring force from each spring is $-kx(t)$. - Total restoring force from both springs is $-2kx(t)$. 5. **Equation of Motion:** Using Newton's second law for the mass at the pin: $$ m\ddot{x}(t) + 2kx(t) = 0 $$ where $\ddot{x}(t)$ is the acceleration. 6. **Interpretation:** This is a simple harmonic oscillator equation with angular frequency: $$ \omega = \sqrt{\frac{2k}{m}} $$ 7. **Final Equation of Motion:** $$ m\ddot{x}(t) + 2kx(t) = 0 $$ This describes the free vibration of the beam with springs at both ends and a fixed pin in the middle.