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:
  Blank.

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):

Physics final exam question: heat conducted through different-orientation bricks

Physics 205A Final Exam, fall semester 2013
Cuesta College, San Luis Obispo, CA

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

A brick can be placed in either of two orientations. The bottom of each brick is immersed in 20° C water, while the top of each brick is heated to 80° C. (Ignore the very slight thermal expansion of these bricks.) Discuss why these two bricks will not conduct the same amount of heat per time. Explain your answer using the properties of heat, temperature, and heat transfer.

Solution and grading rubric:

• p:
  Correct. Heat is conducted from the top to the bottom of each brick, at a rate (1) proportional to the cross-sectional area, and (2) inversely proportional to the length (in this case, height), such that the wider, shorter brick will conduct heat at a rate faster than the narrower, taller brick.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. Typically argues only (1) area, or only (2) length as a factor in why the wider, shorter brick will conduct heat faster than the narrower, taller brick.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. At least recognizes different factors (1)-(2) and attempts to discuss heat conduction along the length of the bars.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Discussion based on phenomena other than heat conduction along the length of the bricks.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 70854, 70855, 73320
Exam code: finaln0M3
p: 24 students
r: 31 students
t: 1 student
v: 3 students
x: 0 students
y: 1 student
z: 1 student

A sample "p" response (from student 0494):
Another sample "p" response (from student 2395):
A sample "r" response (from student 0825), discussing only how the cross-sectional area A affects the flow of heat conducted through the bricks:
A sample "r" response (from student 1025), discussing only how the length (height) d affects the flow of heat conducted through the bricks:

Online reading assignment: heat transfers

Physics 205A, fall semester 2013
Cuesta College, San Luis Obispo, CA

Students have a weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing a presentation on heat transfers.

Selected/edited responses are given below.

Describe something you found interesting from the assigned textbook reading or presentation preview, and explain why this was personally interesting for you.

"I think that Space Shuttle tiles are the coolest thing in the world."

"The Cooper Cooler™ kind of makes sense. The aluminum can has a pretty high thermal conductivity, so that will increase the amount of heat from the can to the ice. The soda is in a closed system, so the heat transfer goes directly to the ice. The change in temperature is high, so the rate of heat flow will be high. The thin diameter of the can will also increase the rate of heat flow from the can to the ice. I would like to try this."

"Seeing the 'mythbusting' examples makes me excited for the next lab."

"I thought the lava lamp example was cool, it helped make the concept of convection easier to understand."

"Internal (thermal) energy depends on the product of mass, specific heat capacity and temperature."

"I found it interesting that e is different for black-bodied and silver-bodied objects."

"It had never occurred to me that because black objects were good absorbers that they therefore were also good radiators, so it was interesting to learn that and connect the dots of something that should be so obvious."

"I found it interesting that cooling food is considered to be forced convection. When we attempt to cool it by blowing on the surface, thermal energy is released.

Describe something you found confusing from the assigned textbook reading or presentation preview, and explain why this was personally confusing for you.

"The picto-quiz examples of heat transfer was hard to evaluate which heat transfer was taking place. I hope we will go over them in class."

"I don't really understand why the temperature component of Stefan's law is to the power of 4."

"It all made sense."

"Conduction was confusing to me. I think I will need a clearer explanation in class of how changes in temperature, κ and area affect heat flow per time."

"I found Stefan's law of radiation and Wien's law confusing. I need help applying these laws."

"While it's not necessarily confusing, the material in this section definitely feels quite more advanced than what we've covered so far. The equations are just kind of thrown at us in the book, so they're a little hard to grasp immediately."

"Why does an object that is shiny radiate heat less efficiently than objects that aren't as shiny?"

In both Fourier's law of heat conduction and Stefan's law of (net) thermal radiation, briefly describe the parameter "P" ("script P") and its SI units.

"I could not figure it out."

"The rate of heat flow given in watts or J/s."

Indicate how changing these parameters of wall covered with a slab of insulating foam placed between a warm room interior and cool environment will affect the amount of energy it conducts per time. Assume all other parameters remain constant.
(Only correct responses shown.)

Replacing it with a foam slab twice as thick: decreases [70%]
Replacing it with a different material slab with a lower thermal conductivity value: decreases [56%]
Lowering the interior room temperature: decreases [54%]

In Stefan's law of (net) thermal radiation, briefly describe the parameter e and its SI units.

"e is a value from 0 to 1 and is a measure of how good a reflector or radiator/absorber it is. I do not think it has a SI unit."

Indicate how changing these parameters of reflective, shiny balloon that is much warmer than its environment will affect the amount of energy it radiates per time. Assume all other parameters remain constant.
(Only correct responses shown.)

Painting it matte black: increases [43%]
Inflating its size: increases [60%]
Lowering its temperature: decreases [65%]

Indicate how changing these parameters of reflective, shiny balloon that is much cooler than its environment will affect the amount of energy it absorbs per time. Assume all other parameters remain constant.
(Only correct responses shown.)

Painting it matte black: increases [72%]
Inflating its size: increases [52%]
Lowering its temperature: increases [26%]

Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.

"Just to clarify: conduction has to do with heat transfer between objects in some sort of contact, convection has to do with heat transfer with air currents, and radiation has to do with heat transfer by using light?" (Yes, yes, and yes.)

"So what energy transfer heats up food fastest? What about the slowest?" (Sounds like we should set up a kitchen tasting event to verify this.)

"How much energy does an object transfer when the temperature is not uniform everywhere? Is it until the same energy is everywhere?" (Not necessarily--only until there is the same temperature everywhere--i.e., thermal equilibrium is reached.)

"This entire presentation seems to be all over the place. I'm not quite sure what to do with these formulas."

Physics quiz question: wide versus narrow bar

Physics 205A Quiz 7, fall semester 2012
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 14.49

A steel bar has twice the cross-sectional area as another steel bar of the same length. The bottom of each bar is immersed in 0° C water, while the top of each bar is heated to 75° C. (Ignore the very slight thermal expansion of these bars). The wider bar has ________ the thermal resistance of the narrower bar.
(A) one-fourth.
(B) one-half.
(C) twice.
(D) four times.
(E) (There is a tie.)

Correct answer: (B)

The thermal resistance R is given by:

R = d/(Κ·A),

where d corresponds to the (identical) length of these bars, Κ is the (identical) thermal conductivity of these bars, and A is the cross-sectional area of the bars. With A_wide = 2·A_narrow, then for these two bars:

R_wide = d/(Κ·A_wide),

R_narrow = d/(Κ·A_narrow),

The wider bar will have one-half the thermal resistance of the narrower bar:

R_wide = d/(Κ·A_wide) = d/(Κ·(2·A_narrow)) = (1/2)·R_narrow.

Sections 70854, 70855
Exam code: quiz07Di5k
(A) : 0 students
(B) : 24 students
(C) : 18 students
(D) : 3 students
(E) : 5 students

Success level: 48%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.50

Online reading assignment: heat transfers

Physics 205A, fall semester 2012
Cuesta College, San Luis Obispo, CA

Students have a weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentations on heat transfers.

Selected/edited responses are given below.

Describe something you found interesting from the assigned textbook reading or presentation preview, and explain why this was personally interesting for you.

"How you can touch a Space Shuttle tile even when it has a ton of internal energy! I don't quite understand how so it is very interesting to me!"

"How a blackbody is really good at absorbing heat from light and is also a good radiator of heat, and a silverbody being the opposite."

"How Coffee Joulies^TM cools down coffee yet keeps it warm longer."

"The windowpane problems really hit home with how much dual-pane windows conserve heat in the home (and mainly from the air gap)."

Describe something you found confusing from the assigned textbook reading or presentation preview, and explain why this was personally confusing for you.

"I found much of this confusing because I have never dealt with it before."

"Fourier's law of conduction. Stefan's law of radiation. Wien's law."

Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.

"Still trying to grasp the reason why heat flows from hot to cold to reach thermal equilibrium." (This is merely a restatement of the zeroth law of thermodynamics--and this is something that is always observed to happen on its own in nature.)

"How is a kilocalorie the same as a calorie?" (A kilocalorie is a thousand calories. But a kilocalorie is equal to a Calorie (capitalized), also known as a 'food calorie.' No one wants to eat a 100,000 calorie candy bar (i.e., 100 kilocalories), but one would be more than willing to eat the same candy bar labeled as '100 Calories.'")

"What are Coffee Joulies^TM?" (You can investigate how well a crude model of this device works in next week's laboratory.)

"Real-life example of a blackbody?" (Your car radiator, which is painted black in order to most effectively radiate heat, instead with of a light-colored or reflective surface.)

"I thought that if you had an 'A' in the class, you didn't have to take the Final Exam. Since we can earn 600/700 points (85.7%), and before the final 600 points are possible, shouldn't we be excused from the Final Exam if we have earned 514/575 points? That's more than 85.7% so far." (No. Your grade is determined by the total points, not by the average. This is so your grade cannot go down (as an average might), but can only go up (by accumulating more points). That's just how it is.)

"For the Final Exam will we need Scantrons?" (No, the Final Exam will be similar to the midterms (writing your answers on the exam itself), and will consist of four short-answer questions and three worked-out problems.)

Physics quiz question: heat flow rate through window and shade

Physics 205A Quiz 7, fall semester 2011
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 14.56

"Choose the Best Honeycomb Shades"
Hunter Douglas
windowstotheworldinc.com/2014/05/best-honeycomb-shades/.

A Harvey Building Products vinyl casement window [*] has a thermal resistance of 0.33 K/watt. The temperature difference between inside and outside is 30° C. The SymphonyShades™ Virtuoso® window shade[**] is also claimed to have a thermal resistance of approximately 0.33 K/watt. If this window shade completely covers the window, the rate of heat conduction per time will be __________ of the original (uncovered) value.
(A) one-third.
(B) one-half.
(C) two-thirds.
(D) twice.
(E) (No change in rate of heat conduction.)

[*] R-factor = 0.33, ENERGY STAR rated, 19 square feet, harveybp.com/upload/products/literature/Harvey_AccessoryWindows_Brochure.pdf
[**] R-factor = 0.34, single-cellular light-filtering fabric, 19 square feet, symphonyshades.com/single_cell_shades.html.

Correct answer: (B)

The rate at which heat is conducted per time to the environment through the window given by:

Power_window = ((heat flow)/time)_window = ∆T/R_window,

where R_window is the thermal resistance of the window:

R_window = d_window/(κ_window·A_window),

which is already given as 0.33 K/watt, such that the heat flow per time conducted through the uncovered window is:

Power_window = (30° C)/(0.33 K/watt) = 90.909090 watts.

Thermal resistances are additive for materials that are stacked in series, so the heat flow per time through the window and shade will be:

Power_{window & shade} = ((heat flow)/time)_{window & shade} = ∆T/R_{window & shade},

Power_{window & shade} = ∆T/(R_window + R_shade),

Power_{window & shade} = (30° C)/(0.33 K/watt + 0.33 K/watt) = 45.45454545 watts.

Thus increasing the overall thermal resistance by a factor of two will reduce the heat flow per time to one-half of its original (uncovered) value.

Sections 70854, 70855
Exam code: quiz07h3A7
(A) : 9 students
(B) : 27 students
(C) : 8 students
(D) : 7 students
(E) : 1 student

Success level: 52%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.67

Physics final exam question: larger versus smaller surface area coolers

Physics 205A Final Exam, Fall Semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Multiple-Choice Question 14.5

[10 points.] A large styrofoam ice chest has warm temperatures outside, compared to its cold temperature contents. A small styrofoam ice chest has the same wall thickness, and the same warm temperatures outside and cold temperature inside. Which ice chest will have the greatest rate of heat entering its contents per time? Explain your answer using the properties of heat
conduction.

Solution and grading rubric:

• p = 10/10:
  Correct. Both coolers have the same temperature difference and thickness, but the larger cooler has more surface area, and thus has the smaller thermal resistance, and the greater rate of heat entering its contents per time. (Assuming that the larger cooler contains more cool substance that the smaller cooler, the larger cooler would take longer to warm up, due to the mass factor in Q = m*c*delta(T), but the relative amount of contents in the larger cooler does not affect the rate of heat entering its contents.)
  
• r = 8/10:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes.
  
• t = 6/10:
  Nearly correct, but argument has conceptual errors, or is incomplete.
  
• v = 4/10:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Argues that the larger cooler will have a lesser rate of heat entering it per time, based on the mass of its conents.
  
• x = 2/10:
  Implementation/application of ideas, but credit given for effort rather than merit.
  
• y = 1/10:
  Irrelevant discussion/effectively blank.
  
• z = 0/10:
  Blank.
  

Grading distribution:
Section 72177
p: 11 students
r: 1 student
t: 0 students
v: 0 students
x: 0 students
y: 1 student
z: 0 students

Sections 70854, 70855
p: 31 students
r: 6 students
t: 0 students
v: 11 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 1337):

Another sample "p" response (from student 2962):

Yet another sample "p" response (from student 5051):

A sample "v" response (from student 1448):