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Dimensional (Unit) AnalysisConversions
What is Dimensional Analysis?
Have you ever used a map? Since the map is a small-scale
representation of a large area, there is a scale that you can use to convert from small-scale units to large-scale units—for example, going from inches to miles or from cm to km.
What is Dimensional Analysis?
Ex: 3 cm = 50 km
What is Dimensional Analysis?
Have you ever been to a foreign country?
One of the most important things to do when visiting another country is to exchange currency.
For example, one United States dollar equals 1535.10 Lebanese Pounds.
What is Dimensional Analysis?
Whenever you use a map or exchange currency, you are utilizing the scientific method of dimensional analysis.
What is Dimensional Analysis?
Dimensional analysis is a problem-solving method that uses the idea that any number or expression can be multiplied by one without changing its value.
It is used to go from one unit to another.
How Does Dimensional Analysis Work?
A conversion factor, or a fraction that is equal to one, is used, along with what you’re given, to determine what the new unit will be.
How Does Dimensional Analysis Work?
In our previous discussions, you could say that 3 cm equals 50 km on the map or that $1 equals 1535.10 Lebanese Pounds (LBP).
How Does Dimensional Analysis Work?
If we write these expressions mathematically, they would look like
3 cm = 50 km$1 = 1535.10 LBP
Examples of Conversions
60 s = 1 min 60 min = 1 h 24 h = 1 day 365 days = 1 yr
(more accurate 365.25 days = 1 yr)
Examples of Conversions
You can write any conversion as a fraction.
Be careful how you write that fraction. For example, you can write
60 s = 1 min
as 60s or 1 min
1 min 60 s
Examples of Conversions
Again, just be careful how you write the fraction.
The fraction must be written so that like units cancel.
Steps 1. Start with the given value. (what you are converting from)
2. Write the multiplication symbol.
3. Choose the appropriate conversion factor with the units you want to solve for in numerator.
4. The problem is solved by multiplying the given data & their units by the appropriate unit (conversion) factors so that the desired units remain. Multiply top numbers, divide by bottom.
5. Remember, cancel “like” units in numerator and denominator.
6. Conversion factors don’t count for sig figs
Let’s try some examples together…
1. Suppose there are 12 slices of pizza in one pizza. How many slices are in 7 pizzas?
Given: 7 pizzasWant: # of slices
Conversion: 12 slices = one pizza
7 pizzas1
Solution
Check your work…
X 12 slices1 pizza =
84 slices(s.f 90)
Let’s try some examples together…
2. How old are you in days?
Given: 17 yearsWant: # of days
Conversion: 365 days = one year
Solution
Check your work…
17 years1
X 365 days1 year =
6205 days(s.f. 6200)
Let’s try some examples together…
3. There are 2.54 cm in one inch. How many inches are in 17.3 cm?
Given: 17.3 cmWant: # of inches
Conversion: 2.54 cm = one inch
Solution
Check your work…
17.3 cm1
X1 inch
2.54 cm = 6.81 inches
Be careful!!! The fraction bar means divide.
Now, you try…
1. Determine the number of eggs in 23 dozen eggs.
2. If one package of gum has 10 pieces, how many pieces are in 0.023 packages of gum?
Multiple-Step Problems
Most problems are not simple one-step solutions. Sometimes, you will have to perform multiple conversions.
Example: How old are you in hours?
Given: 17 yearsWant: # of hours
Conversion #1: 365 days = one yearConversion #2: 24 hours = one day
Solution
Check your work…
17 years1
X365 days
1 year X24 hours
1 day =
148,920 hours(s.f. 150,000)
Combination Units
Dimensional Analysis can also be used for combination units.
Like converting km/h into cm/s. Write the fraction in a “clean” manner:
km/h becomes km h
Combination Units
Example: Convert 0.083 km/h into m/s.
Given: 0.083 km/hWant: # m/s
Conversion #1: 1000 m = 1 kmConversion #2: 1 hour = 60 minutes
Conversion #3: 1 minute = 60 seconds
83 m1 hour
Solution
Check your work…
0.083 km1 hour
X 1000 m1 km
X1 hour60 min
=
0.023 msec
83 m1 hour
X1 min60 sec
=
Now, you try…
We will do a couple of problems as a group.
Then you will complete the assignment by yourself. If you have any questions, ask me as I will be walking around the room.
