Answer Happy

1. Systems of ODES [10 marks] A cascading system of weirs and orifices is used to slow floods in a 3-stage canal system. The flow exiting each stage [LP/T] of the canal is proportional to the depth of water (L) in the stage (the head, h) and an orifice coefficient, kį. The volume [L] of water in each stage is the product of its cross- sectional area, Aï, and the water depth. The incoming flood wave is described by the time function F sin(wt). Stage 1 Stage 3 incoming flood F sin(wt) A1 k2 A3 k4 exiting floodwater ki k3 A2 Stage 2 (a) Write down the system of ODEs that describes how water depth in the three stages evolves over time. [6 marks] The equation of motion for vertical position, z, of a damped harmonic oscillator of mass m is mz" + cz' + kz + mg = 0 where c is a damping coefficient, k is a spring constant, and g is the gravitational acceleration. (b) Rewrite the second order ODE as a pair of first order ODEs, expressed in x' = Ax + b form. [4 marks]