DIMENSIONAL ANALYSIS
The language of science deals to a great degree in the description of the physical world in terms of concepts whose basic units are ultimately reducible to dimensions of mass, length, time, and electric charge. Quite often the information available for the calculation of a quantity is given in terms of a particular unit. In order to perform the desired calculation, the given unit needs to be converted into another unit. The transformation between the units can be accomplished by the application of dimensional analysis. An example might be the calculation of the number of seconds in a day - starting with the fact that there are 24 hours in a day.
Notice that the original unit is the denominator of the conversion fraction and that the desired unit is in the numerator portion of the conversion fraction. Furthermore, units in the denominator of one fraction can be used to cancel out the same unit in the numerator of another fraction. If this is done on the left hand side of the above equality, it is seen that the only remaining unit is that of seconds which the desired result is.
PURPOSE OF EXPERIMENT:
The purpose of this experiment is to become familiar with the use of dimensional analysis to perform conversions between different units of measurement. This will be done by taking familiar measurements in the English system and converting them into their metric equivalents.
CONVERSION OF INCHES TO CENTIMETERS
EQUIPMENT: A ruler graduated in inches and centimeters, and a rod* of a definite length.
PROCEDURE:
1. Place the rod on a flat surface, measure its length in inches and record the length.
Length of rod = ___________________________________ inches
2. Place the rod on a flat surface, measure its length in centimeters and record the length.
Length of rod = ___________________________________ centimeters
3. Set the results from step one and step two equal to one another and calculate the number of centimeters in an inch.
___________________________ inches = ______________________ centimeters
1 inch = __________________ centimeters
CONVERSION OF OUNCES TO GRAMS:
EQUIPMENT: Balances capable of weighing in grams and objects of known weight in ounces
PROCEDURE:
1. Record the known weight in ounces of the selected object.
weight of object = ___________________________________ ounces
2. Place the object onto the pan of a balance and determine its weight in grams weight of object = ___________________________________ grams
* Any object of known length is suitable.
3. Set the results from step one and step two equal to one another and calculate the number of grams in one ounce.
___________________________ ounces = _________________________ grams
1 ounce = _________________________ grams
CONVERSION OF QUARTS TO LITERS
EQUIPMENT: An empty plastic milk jug, ½ gallon capacity, a 100 ml graduated cylinder and 1 quart measuring cup.
PROCEDURE:
1. Fill the empty milk jug with water and empty it into the quart measuring cup and determine the number of quarts contained in the jug.
volume of jug = ___________________________________ quarts
2. Fill the empty milk jug with water and empty it into the graduated cylinder and determine the number of liters contained in the jug by counting the number of milliliters and dividing by 1000.
volume of jug = ___________________________________ liters
3. Set the results from step one and step two equal to one another and calculate the number of liters in one quart.
___________________________ quarts = _________________________ liters
1 quart = _________________________ liters
PRACTICE:
If the equality 1 week = 7 days is divided on both sides by the unit week, the result is
In order to calculate the number of days in 40 weeks, we must multiply the quantity 40 weeks by a factor that relates days to weeks:
Using the conversion factors derived above carry out the following conversions. Show all conversion factors and clearly illustrate how they lead to the final answer.
a. 55 miles to centimeters
b. 2000 lbs to grams
c. 40 quarts to liters