SM-2311: Engineering Mathematics Semester 1, 2023/24 Assignment (50 points) 18 October 2023 Name: Reg. No.: Answer all the questions. Full credit requires showing all working steps and calculations. 1. [15 marks] Consider the interconnected brine tanks as shown in the figure. Tank A contains 10 gallons of brine, and Tank B contains 20 gallons. Both tanks have an inlet and an outlet with a flow rate of 1 gallon per minute. The incoming fluid into Tank A has a concentration of 0.2 pounds of brine per gallon. Let a(t) and b(t) as the functions representing the salt content in pounds for Tank A and Tank B respectively. (a) Show that the system of differential equations that governs the amount of salt in each tank is a′ (t) = 0.2 − 0.1 a(t) b′ (t) = 0.1 a(t) − 0.05 b(t) (b) If initialy there is 5 lbs of salt in tank A and 15 lbs of salt in tank B, use Laplace transforms to determine the amount of salt in each tank as a function of time. 1 2. [15 marks] Given the initial-value problem: 4 5 ′ X = X, −2 6 5 where X(0) = 4 (a) Determine the general solutions of the system of differential equations. (b) Find the phase portrait for the system and describe the nature of the fixed point. (c) Find the particular solution that satisfies the given initial condition. 3. [20 marks] In the system illustrated below, two carts move horizontally with minimal resistance due to air drag or wheel rolling. The connection between the carts is represented by springs, and these springs adhere to Hooke’s law. (a) By considering Hooke’s law: F = −kx and Newton’s second law: F = ma, derive the matrix equation of the form x′′ = Ax that describes the behaviour the system. (b) Determine the coefficient matrix A in the case that k1 = 20 N/m, k2 = 30 N/m, m1 = 10 kg and m2 = 7.5 kg. Assume the solution is of the form x = vert , find the angular frequencies ω1 , ω2 and their corresponding eigenvectors of the two normal modes. (c) At time t = 0, the carts start from rest with displacements x1 = 1 m, and x2 = 0 m. Evaluate the particular solution for the system under these initial conditions. Question Points Score 1 15 2 15 3 20 Total: 50 Page 2