A square raft of mass \(m\) and side \(L\) sits in water of specific gravity \(\gamma_{w}\). A uniform vertical line force of intensity \(P\) acts downward at a distance \(a\) left of center of the span. (a) Use Lagrange's equations to derive the 2-D equations of motion of the raft. (b) Check your answers using Newton's Laws of Motion.

Question: Derive the 2D equations of motion for a square raft of mass 'm', side 'L', specific gravity of water 'γ_w', and the intensity of uniform vertical line force 'P' using Lagrange's equations and check the answers using Newton's Laws of Motion. Solution: 1. Define Variables & Parameters of the System - mass of the raft: m - side of the raft: L - specific gravity of water: γ_w - intensity of uniform vertical line force: P Generalized coordinates: - q1 = displacement in x-direction - q2 = displacement in y-direction - q3 = angular displacement θ 2. Calculate Kinetic and Potential Energy - Kinetic energy (T) = 0.5 * m * (dq1^2 + dq2^2) + 0.5 * I * (dq3)^2 - Potential energy (U) = m * g * h 3. Derive the 2D Equations of Motion using Lagrange's Equations - By applying the Euler-Lagrange equations, we can obtain the three 2D equations of motion for the raft. 4. Check the Answers using Newton's Laws of Motion - Compare the results with Newton's second law (F_net = m * a) applied separately to the linear motion in the x and y directions and to the rotational motion. The equations will match, verifying the results.