Consider an insulated box (a building, perhaps) with internal temperature u(t). According
to Newton’s law of cooling, u satisfies the differential equation
du/dt= ?k[u ? T(t)], (i)
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 Eq. (i) 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).
(b) Suppose that t is measured in hours and that ? = ?/12, corresponding a period of 24 h
for T(t). Further, let T0 = 60?F, T1 = 15?F, 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
Rcos[?(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.
