# Problem Restatement - **System:** Two vertical springs (constants \( k_1 \), \( k_2 \)) hang from a slab, attached to the ends of a massless rigid bar of length \( d \). - A third vertical spring (\( k_3 \)) connects the center of the bar to a mass \( m \) below. - The bar is massless, rigid, and can freely rotate about any axis (but since it's massless, only vertical motion matters). - **Goal:** Find the natural frequency (\( \omega_n \)) of small vertical oscillations of mass \( m \). --- # Step 1: Visualize and Define Coordinates Let: - The ends of the bar are attached to the ceiling via springs \( k_1 \) (left) and \( k_2 \) (right), at positions \( x = -\frac{d}{2} \) and \( x = +\frac{d}{2} \). - The mass \( m \) is attached at the midpoint \( x = \), via spring \( k_3 \) (vertical). Let the vertical displacement of the bar's midpoint (and thus the mass) be \( y \) (**down is positive**). --- # Step 2: Relate Bar Ends' Displacement to Center Displacement Because the bar is rigid and massless, any vertical force at the ends must be balanced by the center and the springs. Suppose the bar remains **horizontal** for small vertical motions (since any torque would not be sustained by a massless bar, it instantly aligns). Thus, **all three points move down by the same amount \( y \)**. --- # Step 3: Effective Stiffness Calculation Each spring provides a restoring force when stretched by \( y \): - Left spring: \( F_1 = -k_1 y \) - Right spring: \( F_2 = -k_2 y \) - Bottom spring: \( F_3 = -k_3 y \) The **total restoring force** on the mass: \[ F_{\text{total}} = - (k_1 + k_2 + k_3) y \] --- # Step 4: Equation of Motion Newton's 2nd law for the mass \( m \): \[ m \ddot{y} = - (k_1 + k_2 + k_3) y \] This is a simple harmonic oscillator with effective spring constant \( k_{\text{eff}} = k_1 + k_2 + k_3 \): \[ \ddot{y} + \frac{k_1 + k_2 + k_3}{m} y = \] --- # Step 5: Natural Frequency The natural angular frequency (\( \omega_n \)) is: \[ \omega_n = \sqrt{\frac{k_1 + k_2 + k_3}{m}} \] And the natural frequency (\( f_n \)) is: \[ f_n = \frac{1}{2\pi} \sqrt{\frac{k_1 + k_2 + k_3}{m}} \] --- # Step 6: Final Answer ## **Natural Frequency of the System** \[ \boxed{ \omega_n = \sqrt{\frac{k_1 + k_2 + k_3}{m}} } \] Or, in terms of frequency: \[ \boxed{ f_n = \frac{1}{2\pi} \sqrt{\frac{k_1 + k_2 + k_3}{m}} } \] --- ## **Explanation** - For small vertical displacements, the massless, rigid bar ensures all its attached points move together vertically. - The three springs act in parallel to support the mass, so their stiffnesses sum. - The natural frequency is that of a mass \( m \) on a single spring with the combined stiffness. --- **Diagram (description):** - Two vertical springs (left: \( k_1 \), right: \( k_2 \)) attached to a horizontal bar of length \( d \), suspended from a ceiling. - The bar's center is connected downwards to mass \( m \) via vertical spring \( k_3 \). - All springs remain vertical for small motions.