Consider an insulated box (a building, perhaps) with internal temperature u(t)

Chapter 2, Problem 14

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Consider an insulated box (a building, perhaps) with internal temperature u(t). According to Newtons law of cooling, u satisfies the differential equation du dt = k(u T (t)), (37) where T (t) is the ambient (external) temperature. Suppose that T (t) varies sinusoidally; for example, assume that T (t) = T0 + T1 cos(t) a. Solve equation (37) and express u(t) in terms of t, k, T0, T1, and . Observe that part of your solution approaches zero as t becomes large; this is called the transient part. The remainder of the solution is called the steady state; denote it by S(t). G b. Suppose that t is measured in hours and that = /12, corresponding to a period of 24 h for T (t). Further, let T0 = 60F, T1 = 15F, and k = 0.2/h. Draw graphs of S(t) and T (t) versus t on the same axes. From your graph estimate the amplitude R of the oscillatory part of S(t). Also estimate the time lag between corresponding maxima of T (t) and S(t). c. Let k, T0, T1, and now be unspecified. Write the oscillatory part of S(t) in the form R cos( (t )). Use trigonometric identities to find expressions for R and . Let T1 and have the values given in part b, and plot graphs of R and versus k.