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Dimensional Analysis

Page history last edited by pinkhamc@... 1 year, 2 months ago

 

 

DIMENSIONAL ANALYSIS

 

With Grateful Appreciation to Professor John (Jack) O’Neil, Chemistry, Norwich University, 1947-1981, who taught me Dimensional Analysis in 1962.

 

You are given the following problem:

 

There are 110 Anabaena cells in the field of the microscope when it is on high power.  Given that the field of the microscope on high power covers 0.5 mm, how wide is an Anabaena cell in mm?

 

How can you solve this and similar word problems without wracking your brain?  Easy; use DIMENSIONAL ANALYSIS.

 

Step 1.  Identify (highlight, then write out) the known values (amounts and units).

 

              An amount is usually a number given to a unit which represents some characteristic, measurable in this 3-dimensional world we inhabit, hence, the phrase, “Dimensional Analysis.”  No amount (number)?  “The,” “a,” or “an” in front of a unit means the amount, “1.”

 

              In the above problem this step gives the following values presented as an amount (bold) followed by a unit (underlined):

 

                                     110 Anabaena cells                      1 field in microscope on high power                       0.5 mm

 

 

Step 1.  Identify (highlight, then write out) the know values (amounts and units).

 

              Usually this relationship will be in the form of an equality.  For example, the statement

 

                                                                                        1 inch = 2.54 centimeters

 

Is an equality.  Here, the equal sign is interpreted as meaning “equivalent to.”  Words like “per,” “covers,” and “equal” signify equalities.  Therefore in the above problem, this step gives”

 

                                                                                     110 Anabaena cells = 1 field

 

                                                                                            1 field = 0.5 mm

 

Step 2b.  Arrange the equalities as division statements which can be used as conversion factors.

 

              Dividing a number by its equal gives 1.  Furthermore, multiplying a value by 1 does not change that value.  Thus, equalities can be written as division statements and multiplied times one another freely.  Each equality generates two division statements.  These division statements are formed by dividing each side of the equality by the value on the other side.  From the first equality in Step 21, the two division statements are:

 

 

              These two division statements can be used as conversion factors.  Note two things here: (1) the two conversion factors are equal to 1, and (2) they are reciprocal to each other.  In the problem you’re facing in this exercise, this step gives:

 

 

Step 3.  Identify the units of the desired answer and arrange them as a division statement.

 

              Look for words like “How” that introduce a question.  The question, “How wide is an Anabaena cell in mm?” is what you are looking for here:

 

 

              Note, you can put the number “1” where it belongs.  But the other amount will be missing.  Sometimes you will only have one unit in the desired answer.  That’s OK, just put it were the “mm” is in the above example and put “1” where “1 Anabaena cell” is.

 

Step 4.  Start to build the solution.

 

              Place the units of the answer on the far right, several lines below the information in Steps 1 and 2.  Moving from right to left, leave a space for the answer, put an equal sign, place a horizontal line, then a multiplication sign.  Your solution should now look like this:

 

Step 5.  Continue the solution.

 

              Look through step 2b to see if there are any conversion factors that include one or both units found in the answer.  Put these units and their amounts so that they occur on the same side (above or below the horizontal lines) as the units in the answer:

 

              Note, you also could have done this step this way:

 

 

Step 6.  Put a horizontal line to the left of the “X”:

 

 

Step 7.  Put the appropriate conversion factor with the line formed in the prior step.

 

Use another relationship from Step 2b as you did in Step 5.  Only this time you are trying to find a unit that matches one of the units in the expression to the right of the “x.”  Place this unit and its amount so that the matching unit is on the other side of the horizonal (division) line:

 

Step 8.  Verify your setup.

 

Now cancel those units that you can cancel to be sure the remaining units on the left of the equal sign are the same as the units on the right of the equal sign:

 

 

Step 9.  Do the math indicated!

 

You are now (finally) ready to do the math.  Simply get out your calculator and do the operations indicated by the formula you have created:

 

 

Cool!  You now have your answer and you’re sure it’s right:

 

0.0045 mm per Anabaena cell

 

This may seem a little cumbersome at first, but practice this on a few simple word problems, following each step in the instructions slowly and methodically.  The more you do this, the more comfortable and natural it will become and sooner or later you will not even have to think about what you do next; it will just happen!

 

Return to the Front Page of Mentiscopia

 

DIMENSIONAL ANALYSIS

With Grateful Appreciation to Professor John (Jack) O’Neil, Chemistry, Norwich University, 1947-1981, who taught me Dimensional Analysis in 1962.

 

You are given the following problem:

 

There are 110 Anabaena cells in the field of the microscope when it is on high power.  Given that the field of the microscope on high power covers 0.5 mm, how wide is an Anabaena cell in mm?

 

How can you solve this and similar word problems without wracking your brain?  Easy; use DIMENSIONAL ANALYSIS.

 

Step 1.  Identify (highlight, then write out) the know values (amounts and units).

 

              An amount is usually a number given to a unit which represents some characteristic, measurable in this 3-dimendional world we inhabit, hence, the phrase, “Dimensional Analysis.”  No amount (number), “the,” “a,” or “an” in front of a unit means the amount, “1.”

 

              In the above problem this step gives the following values presented as an amount (bold) followed by a unit (underlined):

 

              110 Anabaena cells                      1 field in microscope on                        0.5 mm

                                                                              high power

 

Step 1.  Identify (highlight, then write out) the know values (amounts and units).

 

              Usually this relationship will be in the form of an equality.  For example, the statement

 

1 inch = 2.54 centimeters

 

Is an equality.  Here, the equal sign is interpreted as meaning “equivalent to.”  Words like “per,” “covers,” and “equal” signify equalities.  Therefore in the above problem, this step gives”

 

110 Anabaena cells = 1 field

 

1 field = 0.5 mm

 

Step 2b.  Arrange the equalities as division statements which can be used as conversion factors.

 

              Dividing a number by its equal gives 1.  Furthermore, multiplying a value by 1 does not change that value.  Thus, equalities can be written as division statements and multiplied times one another freely.  Each equality generates two division statements.  These division statements are formed by dividing each side of the equality by the value on the other side.  From the first equality in Step 21, the two division statements are:

 

                                                                       1 inch          2.54 cm

                                                                                       =                      (= 1)

                                                                      2.54 cm         1 inch

 

              These two division statements can be used as conversion factors.  Note two things here: (1) the two conversion factors are equal to 1, and (2) they are reciprocal to each other.  In the problem you’re facing in this exercise, this step gives:

 

                                                        110 Anabaena cells                   1 field

                                                                                             =                

                                                                  1 field                     110 Anabaena cells

or

                                                                              1 field          0.5 mm

                                                                                             =   

                                                                             0.5 mm          1 field

 

Step 3.  Identify the units of the desired answer and arrange them as a division statement.

 

              Look for words like “How” that introduce a question.  The question, “How wide is an Anabaena cell in mm?” is what you are looking for here:

 

                                                                                        ? mm

                                                                          

                                                                                 Anabaena cells

 

              Note, you can put the number “1” where it belongs.  But the other amount will be missing.  Sometimes you will only have one unit in the desired answer.  That’s OK, just put it were the “mm” is in the above example and put “1” where “1 Anabaena cell” is.

 

Step 4.  Start to build the solution.

 

              Place the units of the answer on the far right, several lines below the information in Steps 1 and 2.  Moving from right to left, leave a space for the answer, put an equal sign, place a horizontal line, then a multiplication sign.  Your solution should now look like this:

 

                                                                                                             ? mm

                                                                X                                =                               

                                                                                                     1 Anabaena cell

 

Step 5.  Continue the solution.

 

              Look through step 2b to see if there are any conversion factors that include one or both units found in the answer.  Put these units and their amounts so that they occur on the same side (above or below the horizontal lines) as the units in the answer:

 

                                                                 0.5 mm                          ? mm

                                                      X                                =                            

                                                                  1 field               1 Anabaena cell

 

              Note, you also could have done this step this way:

 

                                                                 1 field                          ? mm

                                                 X                                      =                             

                                                      110 Anabaena cells    1 Anabaena cell

 

Step 6.  Put a horizontal line to the left of the “X”:

 

                                                                                      0.5 mm                      ? mm

                                                                           X                                =                            

                                                                                       1 field               1 Anabaena cell

 

Step 7.  Put the appropriate conversion factor with the line formed in the prior step.

 

Use another relationship from Step 2b as you did in Step 5.  Only this time you are trying to find a unit that matches one of the units in the expression to the right of the “x.”  Place this unit and its amount so that the matching unit is on the other side of the horizonal (division) line:

 

                                                  1 field                          0.5 mm                      ? mm

                                                                            X                                =                             

                                        110 Anabaena cells             1 field               1 Anabaena cell

 

Step 8.  Verify your setup.

 

Now cancel those units that you can cancel to be sure the remaining units on the left of the equal sign are the same as the units on the right of the equal sign:

 

                                                  1 field                          0.5 mm                      ? mm

                                                                            X                                =                            

                                        110 Anabaena cells             1 field               1 Anabaena cell

 

Step 9.  Do the math indicated!

 

You are now (finally) ready to do the math.  Simply get out your calculator and do the operations indicated by the formula you have created:

 

                                                  1 field                          0.5 mm                0.0045 mm

                                                                            X                                =                            

                                        110 Anabaena cells             1 field               1 Anabaena cell

 

Cool!  You now have your answer and you’re sure it’s right:

 

0.0045 mm per Anabaena cell

 

This may seem a little cumbersome at first, but practice this on a few simple word problems, following each step in the instructions slowly and methodically.  The more you do this, the more comfortable and natural it will become and sooner or later you will not even have to think about what you do next; it will just happen!

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