Dimensional Analysis

In science we encounter problem solving where conversion of units are needed.  In conversion of units the easiest way is to use dimensional analysis or factor label method, a method of converting one unit to another unit provided the same quantities are involved.   It is a simple technique that needs a little of memorization of the conversion factors to use in calculation.  Dimensional analysis is based on the relationship between different units that express the same physical quantity either metric to metric conversion, metric to English and English to English conversion.

Let us have an example, dollar is different from that of peso but both are amount of money.  One US dollar is equivalent now to 49.72 in Philippine peso.  What if you have 200 US dollar, what is its value in peso?

First thing to do is to have a conversion factor between dollar and peso,  

1 dollar = 49.72 pesos

If you want to convert dollar to peso the conversion factor to use is

  49.72 pesos 
1 dollar

and  if you want to convert from peso to dollar, the conversion factor to use is

     1 dollar     

49.72 pesos

Since the problem above is asking the value of 200 dollars in peso, the first conversion factor will be used.  As shown in the calculation below:

How about if you are asked to convert 1.5 L to mL?  This problem can be expressed as

? mL = 1.5 L

The conversion factor to use is 

     1000 mL    

1L

since the problem is asking the value in mL, mL must be the numerator and L is the denominator, this is for us to cancel out L unit in the calculation as shown below:

                                                                                       =  1.5 x 10³ mL

To express the answer in proper number of significant figures the answer is converted to scientific notation, expressed in 2 significant figures.  1000 mL/ 1 L  will not matter since those are exact numbers.  

In general, to apply dimensional analysis we follow:

given quantity  x  conversion factor =  desired quantity

The calculation of units is shown below:

In dimensional analysis, we make it sure that the all undesired units are canceled out remaining only the desired unit.

If the desired unit is not obtained in the calculation, meaning something wrong with the calculation and it needs to be reviewed to locate the error.


PRACTICE EXERCISE:

Convert the following:
1.  25.5 mg to g
2.  4.0 x 10^-10 m to nm
3.  0.575 mm to µm
4.  1.48 x 10² kg to g
5.  7.25 x 10^-4 s to ms