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Competency Assessment -- Scaling and Dimensions (redirected from Compentency Assessment -- Scaling and Dimensions)

Page history last edited by Joe Redish 11 years, 2 months ago

BERG > NEXUS Project > Development page > Assessing Competencies

 

Units and Dimensions

 

(Topic from the MR5 Report)

 

The competency of understanding units and dimensions also includes the mathematical ideas of functional dependence and scaling.  Here is a list of WSSBATDs (What Students Should Be Able To Do) for this competency.

 

  1. Students should be able to identify the dimensionality (L, T, M, Q) of physical variables that are constructed either by direct measurement or by combination of measurements.
  2. Students should be able to determine whether an equation could possibly be physically legitimate by virtue of its being dimensionally consistent.
  3. Students should be able to distinguish the concepts of dimensionality and units and describe the difference in their own words.
  4. Students should be able to identify units and associate appropriate dimensionalities with them.
  5. Students should know the standard "thousands" prefixes (centi-, kilo-, mega-, milli-, micro-,..) and be able to convert among among units that differ by these factors without having to look anything up.
  6. Students should be able to recognize common non-standard units (feet, calories, degrees Fahrenheit) and be able to convert to standard metric units with the aid of a conversion table.
  7. Students should be able to connect dimensionality with functional dependence.
  8. Students should understand the impact of competing functional dependences for scaling (e.g., surface vs volume) and be able to see the physical and biological implications of them.
  9. Students should be able to connect modeling assumptions for describing a physical or biological system with the functional dependence of equations on various dimensions.

 

Sample test items

 

1. A mass, m, is attached to a rigid steel rod of length L and allowed to swing back and forth.  If it is raised to a height h above its lowest position, when it is hanging straight down it is moving its fastest, a speed vMAX.

The parameters in the problem have the following dimensions:

[m] = M,          [L] = L,                        [h] = L,                        [g] = L/T2

Which of the following expressions could represent its maximum speed from considerations of dimensional analysis alone? Put all possible answers. If none are possible, write N. (Note: you will lose points for including incorrect ones.)

 

 

 

2. The strength of a muscle is proportional to its cross sectional area. As an animal gets bigger, it also gets heavier and its legs have to get bigger to hold it up. Suppose, as it grows, the animal’s torso retains approximately the same shape and proportions, and the length of the legs stays in proportion to its growth in size. How much do you expect the cross sectional area of the legs will need to grow to hold up the animal’s weight when the animal’s torso doubles in all its dimensions?

 

A. The cross sectional area of the legs will have to about double in size.
B. The cross sectional area of the legs will have to about quadruple in size.
C. The cross sectional area of the legs will have to more than quadruple in size.
D. The cross sectional area of the legs will have increase in size but (much) less than double.
E. The cross sectional area of the legs should stay the same size.
F. You need more information to decide. (Say what.)

 

3. When an object is traveling in a circle, it requires a force to keep it turning. That force depends on the following parameters and variables: the object’s mass, m, the distance the object is from the center of the circle, R, and the angular velocity of the object, ω (omega). A force has dimensions, [F] = ML/T2. If our parameters and variables have the following dimensions, [m] = M, [R] = L, and [ω] = 1/T, propose an equation for F that expresses the way it depends on m, R, and ω.

 

4. As part of an examination a few years ago, a student went through the algebraic manipulations on an exam shown in the figure below. You might know what the symbols mean, but it's clear that the final result is wrong, since a velocity squared and a velocity to the fourth power have different dimensions and cannot be legitimately added. Given the information at the left about the dimensions associated with each symbol, decide which of the numbered equations MUST be wrong because it has the wrong dimensions. (This does not guarantee that the earlier equations are correct -- just that they are dimensionally OK.)

 

                         

 

5. You know that 1 cubic centimeter of water has a mass of 1 gram.  What’s the mass of 1 cubic meter of water?

  1. 10 g                             e. 1 kg
  2. 102 g                            f. 10 kg
  3. 104 g                            g. 100 kg
  4. 106 g                            h. 1000 kg

 

6. The rate at which an animal in a cold environment loses heat is proportional to its surface area, but its metabolism generates heat in all of its cells, so the rate of heat generation is proportional to its volume. All other factors being equal (which they often are not!), which implications would follow from these facts?

  1. Because it is smaller, a baby seal is less at risk for hypothermia (body temperature falling to too low a value) than is an adult when it is in an ocean significantly colder than its body temperature.
  2. Because it is smaller, a baby seal is more at risk for hypothermia (body temperature falling to too low a value) than is an adult when it is in an ocean significantly colder than its body temperature.
  3. If you take a baby outdoors on a cold day, how cold it feels to you is a good measure of how warmly you should dress that baby.
  4. If you take a baby outdoors on a cold day, you should dress it more warmly than you would dress yourself because it is smaller.
  5. If you take a baby outdoors on a cold day, you should dress it less warmly than you would dress yourself because it is smaller.

Comments (7)

Ben Dreyfus said

at 9:21 am on Sep 4, 2011

(centi-, kilo-, mega-, milli-, micro-,..): Is this an exhaustive list, or (as the ellipsis suggests) nonexhaustive? If students are expected to memorize some of the prefixes, it seems that we should clearly define which ones those are.

Joe Redish said

at 11:08 am on Sep 6, 2011

This is not a "memorize" task for this class. This is something they are expected to have learned and to become increasingly familiar with. As situations change they should be ready to learn new ones -- for example we all now know what "tera-" is -- something we didn't know before.

Julia Gouvea said

at 2:25 pm on Sep 7, 2011

I am not sure I understand what "have learned and expected to be increasingly familiar with" means. Either it's committed to memory or not. I for one don't always remember these, but it takes 2 seconds to look it up on google. If I had to use them more I would of course be more familiar with them and perhaps have to look them up less often. But I don't see why it's necessary to test whether or not students are at that point. I don't see it as anything else but a memory issue.

Joe Redish said

at 12:42 pm on Sep 8, 2011

Julia -- Not just a memory issue. The issue is associations and access. When they see something given in nanometers do they have to look it up to decide it's smaller than an organic molecule? If they want to convert kilometers to centimeters, do they have to go to Google? Some things you have to have at your fingertips and that isn't just memorization.

Ben Dreyfus said

at 1:40 pm on Sep 8, 2011

How would they do this other than from memory? It's not like they can work these prefixes out from first principles.

Joe Redish said

at 12:44 pm on Sep 8, 2011

Test item 2 turned out to be very confusing to the students at the beginning of the class. I think the difficulty is point 8 in the list at the top.

Julia Gouvea said

at 2:22 pm on Sep 13, 2011

Which part of test item two was difficult? Also are you talking about test item 2 on the quiz you sent out (question 5 above)? Because I don't see how that is connected to point 8.

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