Dimensional Analysis
The comparison different physical quantities by identifying their base quantities (such as length, mass, and time,) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms) and tracking these dimensions as calculations are performed.

The conversion of units from one dimensional unit to another is often easier within the metric or SI system, than in others, due to the regular 10-base in all units. Dimensional analysis is a widely used technique for such conversions using the rules of algebra.

Commensurable quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are originally expressed in differing units of measure, e.g. yards and metres, pounds (mass) and kilograms, seconds and years.

Incommensurable quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are originally expressed in, e.g. meters and kilograms, seconds and kilograms, meters and seconds. For example, asking whether a kilogram is larger than an hour is meaningless.
 

• When working dimensional analysis problems, the answer must always be expressed in the appropriate number of significant digits.

The Problem:
Converting an initial quantity given in one unit to the same quantity in a different unit.

The Example:
Convert 164 pounds to the weight in kilograms.
 

Step 1:	Write down the initial quantity—the actual number you have been given. Initial quantity:  164 lbs. (3 SF’s) (The answer must have 3 significant figures)
Step 2:	Find the equality that gets you from the units you have been given to the units you need. Equality:  2.20 lbs = 1 kg
Step 3:	Multiply the initial quantity given by the proper conversion factor. The units needed goes in the numerator of the conversion factor. The units you want to cancel go in the denominator. 164 lbs x     1kg  2.20 lbs
Step 4:	If the units cancel properly, leaving only the units needed in the numerator, the conversion factors are set up correctly.   164 lbs x     1kg  =   164 x 1 kg  2.20 lbs 2.20
Step 5:	Solve the problem, being sure the answer has the correct number of significant figures and is in the proper units. 164 kg  = 74.5 kg (3 SF’s) 2.20

What is the difference in commensurable and incommensurable quantities?
 

Practice Problems Answer the following questions:

1. The daily recommended amount of potassium in the diet is 3500 mg. How many grams of potassium are needed each day? Be sure to follow all five steps in solving this problem!

Using Two or More Conversion Factors

During a volcanic eruption on Mauna Loa, Hawaii, the lava flowed at a rate of 33 meters per minute.  At this rate, how far in kilometers can the lava travel in 45 minutes?

Tip One: When setting up the problem, be looking for how things can cancel the way you want them to.

Tip Two: Put the units you want in the numerator of the last conversion factor

Why do the units you want to keep go in the numerator of the last conversion factor?
 

Practice Problems Answer the following questions:

2. During surgery, a patient receives 5.0 pints of plasma. How many milliliters of plasma were given? . On a test, problems like this are worth 4 points each: 1 point for setting up the problem correctly, 1 point for the correct answer, 1 point for the proper units and 1 point for the correct number of significant figures