A straight elastic rod has reference coordinate X ∈ [−L, L]. The rod rotates with constant angular speed ω about an axis perpendicular to the rod, through the material point X = 0 which is fixed. The reference density is rho0 and is uniform and constant. The equation of motion in the rotating frame of the rod, using the Eulerian description, is rho Dv Dt = rhof + ∂T ∂x + rhoω2x. If the body force density f = 0, the equilibrium equation is dT dx + rhoω2x = 0. (1) Use χ(X, t) = ˆx(X) to describe the time-independent equilibrium deformation. Using both conservation of mass rho(ˆx(X))ˆx ′ (X) = rho0 and that S(X) = T(ˆx(X)) (where S is the stress T but as a function of the reference coordinate) show that S ′ (X) + rho0ω 2xˆ(X) = 0.