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Tutorial: Introduction to Dimensional Analysis

The assumption is that you've gone over scientific notation, significant figures, and the metric system before you begin this tutorial.

Mention "dimensional analysis", "factor-label method", or "unit analysis" to students and you can sometimes hear the cries of dismay. I think this area is often terribly misunderstood by many students. It's unfortunate that many people only have their eye on the end-result rather than how they got to that point. As a teaching assistant at The Ohio State University, I'll never forget one situation with a student. The university has a room where all the general chemistry TA's hold office hours which allows students to have access to people other than their own TA for help. One day, a student walked up and asked how to get the answer to a lab report question. As I proceeded to work with her to help her understand how to work the question, she stopped me and said- "this is due in 15 minutes and I only want to know the answer." Well, I stopped and told her she should ask someone else because I didn't work that way. And she then walked away in a huff and probably found someone else. A case of not seeing the forest for the trees, I suppose...

Dimensional analysis is one of the most useful tools that students will learn in a science class. Not only does it encourage critical thinking, but it's something that's used every day by everybody. Yes, I mean everybody. You've already used this concept even though you may not have realized it. That quick calculation when you pulled up to the gas station with $5 in your pocket and you estimated how many gallons of gas you could get. Which laundry detergent is the best bargain? Just how many US dollars are in that Canadian dollar you're about to use in Quebec? This is not just a topic for scientific calculations.

An excellent example of just how dimensional analysis impacts our lives came in 1999 with one of NASA's Mars missions. After the loss of the Climate Observer, an investigation board was put together to determine what happened to the mission and a number of things were found to be responsible for inadequate checks and such. But, there was one factor that directly resulted in the mission's failure. I'll quote from the press release so you can read it in NASA's own words:

"The 'root' cause of the loss of the spacecraft was the failed translation of English units in a segment of ground-based, navigation-related mission software, as NASA has previously announced," said Arthus Stephenson, chairman of the Mars Climate Orbiter Mission Failure Investigation Board.

The problem was that one team used English units (feet, pounds, etc.) while another team used metric units which caused the craft to take a different path than intended. In other words, there was a problem in unit conversions- what you're learning right now!

Here's the press release about the preliminary findings.
Here's the press release from the Failure Board (the one from which I quoted).

Improper dimensional analysis led not only to a massive loss of scientific knowledge to the world but millions of dollars of United State's citizens' tax money. See, dimensional analysis really does affect everyone!

Part 1- Basics:

We'll go over the importance of using units and how using them along with dimensional analysis can help you check your calculation

Part 2- Setting Up Conversions:

How can you set up conversions to go from one unit to another? You'll learn here.

Part 3- Basic Conversion Problems:

Now that you've learned how to set up simple problems, you learn how to carry them through to the end and how to check your answer.

Part 4- Complex Conversion Problems:

It's off to more complex conversion problems!

Part 5- Dissecting Conversion Problems I:

To successfully use dimensional analysis, you need to understand how to set up problems when given a certain amount of information. This section will show you how to dissect problems. Part 1 of 3.

Part 6- Dissecting Conversion Problems II:

This continues off of the previous section. Part 2 of 3.

Part 7- Density:

A detailed look at the different types of density problems.

Part 8- Quizzes:

You can get quizzes on floating point/scientific notation, basic conversions, and various density problems.

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