Dimensional Analysis

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.


Ex: 3 cm = 50 km


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.


Whenever you use a map or exchange currency, you are utilizing the scientific method of 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.


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).


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


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


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.

2. Write the multiplication symbol.

3. Choose the appropriate conversion factor.

4. The problem is solved by multiplying the given data & their units by the appropriate unit factors so that the desired units remain.

5. Remember, cancel like units.

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


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 = 6052 days


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 days

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

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…

Complete your assignment by yourself. If you have any questions, ask me as I will be walking around the room.

Dimensional Analysis

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