Physics final exam question: temperature differences for heat conducted through different bars

Physics 205A Final Exam, fall semester 2014
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Example 14.10, Practice Problem 14.10, Problem 14.57

A 0.50 m long copper bar has a square cross-section of 0.020 m by 0.020 m. A temperature difference of 20° C is applied to the left and the right edges of the bar. Another copper bar has a 0.25 m length, but has the same volume (and corresponding wider cross-section). Discuss why this shorter 0.25 m long copper bar will require a smaller temperature difference between its ends in order to conduct the same amount of heat per time as the 0.50 m long copper bar. (Ignore the very slight thermal expansion of these bars). Explain your reasoning using the properties of heat, temperature, and heat transfer.

Solution and grading rubric:
  • p:
    Correct. Discusses:
    1. that for the two bars to conduct the same rate of heat per time, the bar with the lower thermal resistance will require a smaller temperature difference;
    2. and second bar has a lower thermal resistance due both its shorter length and greater cross-sectional area.
  • r:
    Nearly correct, but includes minor math errors.
  • t:
    Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. Only discusses how either shorter length or greater cross-sectional area results in the second bar having a lower thermal resistance.
  • v:
    Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Some garbled attempt at applying Fourier's law of conduction.
  • x:
    Implementation of ideas, but credit given for effort rather than merit. Approach other than that of applying Fourier's law of conduction.
  • y:
    Irrelevant discussion/effectively blank.
  • z:
Grading distribution:
Sections 70854, 70855, 73320
Exam code: finalm34T
p: 27 students
r: 9 students
t: 11 students
v: 4 students
x: 8 students
y: 2 students
z: 1 student

A sample "p" response (from student 8984):

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