CALC Suppose that both ends of the rod in Fig. 17.23a are

Chapter 17, Problem 123CP

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CALC ?Suppose that both ends of the rod in Fig. 17.23a are kept at a temperature of 0°C,and that the initial temperature distribution along the rod is given by where ?x ?is measured from the left end of the rod. Let the rod be copper, with length L = 0.100 m and cross-sectional area 1.00 cm2 .(a) Show the initial temperature distribution in a diagram. (b) What is the final temperature distribution after a very long time has elapsed? (c) Sketch curves that you think would represent the temperature distribution at intermediate times. (d) What is the initial temperature gradient at the ends of the rod? (e) What is the initial heat current from the ends of the rod into the bodies making contact with its ends? (f) What is the initial heat current at the center of the rod? Explain. What is the heat current at this point at any later time? (g) What is the value of the ?thermal diffusivity k/pc ?for copper, and in what unit is it expressed? (Here ?k is the thermal conductivity, is the density, and c? i? s the specific heat.) (h) What is the initial time rate of change of temperature at the center of the rod? (i) How much time would be required for the center of the rod to reach its final temperature if the temperature continued to decrease at this rate? (This time is called the relaxation time ?of the rod.) (j) From the graphs in part (c), would you expect the magnitude of the rate of temperature change at the midpoint to remain constant, increase, or decrease as a function of time? (k) What is the initial rate of change of temperature at a point in the rod 2.5 cm from its left end?