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The physical system of a composite material which consists of polymer compounds x and y can be represented by the following simultaneous differential equations as shown in Eqns

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The physical system of a composite material which consists of polymer compounds x and y can be represented by the following simultaneous differential equations as shown in Eqns. Q3a and Q3b:

 

dx/dt + y + 2 = e−t  Eqn. Q3a

dy/dt − x + 3 = 5e−t  Eqn.Q3b

 

a and b are indicating the tensile modulus or Young's modulus (unit GPa) of the polymer compounds. 

 

i. by eliminating the y system, construct a second order ordinary differential equation satisfied by x(t). Proceed to solve for the particular solutions by assuming appropriate initial conditions, where one of the initial conditions is given as x(0)=0 and when the t=0, y(t)>0. List down all the assumptions made

 

ii. Apply alternative method to solve the simultaneous differential equations and list down all the assumptions made. Solve for x(2) and y(2). Discuss the advantages and disadvantages of the methods used in part a(i) and a(ii)

 

b. A children swimming pool initially contains 500 L of clean water. An inlet water stream containing chlorine at a concentration of 4 mg/L is pumped into the pool at a rate of 12 L/hr. Assuming well-mixed condition, the water is pumped out at the rate of 10 L/hr. write the equation for the mathematical model of the amount of chlorine in the pool as a function of time. Perform your own research on the acceptable chlorine concentration in a pool of 500 L, use the chlorine concentration that you have selected to identify the time when you need to stop the inlet flow. List down all the assumptions made

 

c. A block with a mass of 2kg is attached to the end of a spring. A force of 36 N is required to keep the spring stretched 1.2 m beyond its original length. The damping constant of the massspring system is measured at 21 N.s/m. If the block is released at a point of 0.3 m below its equilibrium position with an initial velocity of zero. write the equation of motions for the mass-spring system by using two different masses. Perform comparison for the system when subjected to two different masses and illustrate the motion of the respective blocks in a graph. List down all the assumptions made.