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What is Dimensional Analysis? - Definition & Examples

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  • 0:00 What Is Dimensional Analysis?
  • 1:33 Dimensional Analysis Examples
  • 4:20 Lesson Summary
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Lesson Transcript
Instructor
Nissa Garcia

Nissa has a masters degree in chemistry and has taught high school science and college level chemistry.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

When we measure an object's size or amount, we encounter different units. Sometimes, we need to convert one unit to another, and this is where dimensional analysis comes in. In this lesson, we will learn about dimensional analysis and practice it.

What Is Dimensional Analysis?

Comparing and converting between different units is a very useful and important skill. We do this every day without realizing it. For instance, when we follow a recipe, we may need to do simple conversions, like converting grams to ounces, or quarts to cups.

In science and math, we often convert a number or quantity with a dimensional unit to a different unit, like meters to kilometers. Dimensional analysis, also known as factor-label method or unit-factor method, is a method to convert one different type of unit to another. This way, we can convert to a different unit, but their values are the same. To convert from one unit to another, we make use of a conversion factor, which is a numerical quantity that we can multiply or divide to the number or quantity that we want to convert.

For example, if I want to know how many yards are there in 10 feet, we can recall that 3 feet is equivalent to 1 yard. Then, I can use dimensional analysis and convert feet into yards by using the conversion factor shown below in yellow. If I want to know how many minutes there are in two hours, I can use the conversion factor shown in blue.

Conversion Factors

Dimensional analysis is something that we have done without realizing it. We convert minutes to hours, or days to hours all the time. Also, if we travel to another country that normally measures distance by kilometers instead of miles, then we convert between the two units as well by using the dimensional analysis method.

Dimensional Analysis Examples

Let's go over a few examples of dimensional analysis. In this section, we can practice how to make use of conversion factors correctly to convert between units.

Example 1

Let's say we want to know how many kilograms there are in 55 pounds (lb). This means we need to convert pounds to kilograms (kg). In this case, our conversion factor is 1 kg = 2.2046 lb. This information can be written in two ways, one is shown in yellow in the image below and the other is shown in blue. You will notice that in (1), the kg unit is on top, and in (2), the lb unit is on top. Which one do we pick?

Conversion Factors for Example 1

We want to know how many kg there are, so we will pick (1) because kg is on top and lb is on the bottom. If we multiply this by 55 lb, the unit lb will cancel, and we will be left with kg. When we use conversion factor (1), let's see what happens:

Dimensional Analysis for Example 1

Now, we have successfully converted pounds to kilograms.

Example 2

How many feet (ft) are there in 140 centimeters (cm)?

First, we need to determine our conversion factors. We know that 1 ft = 12 inches (in) and that 1 in = 2.54 cm. In this case, we have two conversion factors. If we want to convert the units all the way to feet, we must convert cm to inches and then to feet:

Conversion Factors to Convert cm to ft

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Additional Activities

Additional Examples Involving Dimensional Analysis

In the following examples, students will get extra practice in converting units using dimensional analysis and will see the benefit of easier comparisons when using dimensional analysis to convert two quantities to the same units. The first two examples are basic examples to give students practice converting between units, and the second two examples have students make comparisons between two pieces of data by first using dimensional analysis to convert the data to the same units.

Examples

1) You are cooking a soup recipe that requires 100 ounces of chicken broth. Unfortunately, the grocery store is only selling chicken broth by the pint. How many pints of chicken broth are required to make the soup? How many pints should you purchase? (Use the conversion factor 1 pint = 16 fluid ounces).

2) Your personal trainer is recommending that you increase your jogging workout to 2 miles per day. Your app which tracks distance for you is on the metric system, and only tracks your distance in meters. How many meters do you need to jog per day to meet your trainer's recommendation? (Use the conversion factors 1 mile = 1.60934 km and 1 km = 1000 meters).

3) Which is heavier: a 0.75 ton horse or a 738 kg camel? (Use the conversion factors 1 ton = 2000 pounds and 1 kg = 2.20462 pounds).

4) You can run 7 miles in one hour, and your friend can run a 5k in 30 minutes. Who has a faster pace? (Use the conversion factors 1 hour = 60 minutes and 1 mile = 1.60934 km).

Solutions

1) To find the number of pints required by the recipe, we can multiply 100 ounces by the conversion factor 1 pint / 16 ounces so that the unit of "ounces" cancels, leaving the answer in pints. We have (100 ounces) * (1 pint / 16 ounces ) = 6.25 pints. Since 6.25 pints of chicken broth is required, you will have to purchase 7 pints from the store.

2) To convert 2 miles per day into meters per day, we will need to multiply the 2 miles by the conversion factors (1.60934 km / 1 mile) * (1000 m / 1 km) so that the units "miles" and "km" cancel out leaving an answer in meters. We have (2 miles) * (1.60934 km / 1 mile) * (1000 m / 1 km) = 3,218.68 meters.

3) In order to compare the weights of the horse and the camel, we'll need to convert their weights to the same units. We'll convert both weights to pounds. For the horse, we need to multiply 0.75 tons by the conversion factor (2000 pounds / 1 ton) to get a weight of (0.75 tons)*(2000 pounds / 1 ton) = 1500 pounds. For the camel, we need to multiply 738 kg by the conversion factor (2.20462 pounds / 1 kg) to get a weight of (738 kg) * (2.20462 pounds / 1 kg) = 1627.00956 pounds. So, the camel is heavier (though the animals are close to the same size!)

4) To compare the speeds, we'll convert both speeds into kilometers per hour (though other options are fine as well). To convert 7 miles in one hour to kilometers per hour, we need to multiply 7 miles in one hour by the conversion factor (1.60934 km / 1 mile) to get (7 miles / 1 hour ) *(1.60934 km / 1 mile) = 11.26538 km/hr. To convert the 5k in 30 minutes to kilometers per hour, we need to multiply by the conversion factor (60 minutes / 1 hour) to get (5 km / 30 minutes) * (60 minutes / 1 hour) = 10 km/hr. So you have a faster pace than your friend.

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