# Dimensional Analysis Practice: Calculations & Conversions

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• 0:00 Dimensional Analysis
• 0:46 Single Step
• 2:07 Dimensional Analysis and Speed
• 2:42 Dimensional Analysis…
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Lesson Transcript
Instructor: Laura Foist

Laura has a Masters of Science in Food Science and Human Nutrition and has taught college Science.

Dimensional analysis is a handy tool in unit conversion and can help several different types of questions. We will go over several examples of dimensional analysis in this lesson.

## Dimensional Analysis

Science problems in both physics and chemistry often require conversions between units. Dimensional analysis is the process by which we convert between units and whether we should divide or multiply. You may do simple problems like this frequently throughout the day. For example, when watching the clock and waiting for a boring lecture to be over, you may think to yourself, 'We have only 5 minutes left, which is equal to only 300 seconds, which is really not that long!' You just converted from minutes to seconds.

Dimensional analysis converts between units in such a way as to keep track of the units by helping you keep track of and cancel out units to end up with the correct units.

## Single Step

In the example of changing between minutes to seconds, we know that there are 60 seconds per minute, so we can multiply the number of minutes by 60 and we get the number of seconds. Dimensional analysis would indicate this as such:

The nice thing about dimensional analysis is that we can cancel out units like in math, which you can see in the places where the minutes variable is cancelled out. If we were dividing 9/9, we know this equals 1, so it can cancel out in the equation. The same works with the units. If we are dividing minutes by minutes, then it cancels out into '1'.

Sometimes in conversions it is easy to forget if we need to divide or if we need to multiply two things together. Dimensional analysis fixes this problem because we are also multiplying and dividing units, and if we don't end up with the correct units, then we did it wrong:

In this example, we divided 5/60 instead of multiplying 5*60. But since we weren't able to cancel out the units, we ended up with the wrong units, so we know that we did it wrong and can easily see where our mistake was made!

## Dimensional Analysis and Speed

Let's say we want to know how fast a car is going in miles per hour, but we are given that it is going 25 meters per second. So we need to convert from miles to meters and from hours to seconds. We know that there are 1,609.34 meters in 1 mile, and we know that there are 3,600 seconds in 1 hour, which we'll work out in the equation below:

Notice that since seconds is on the bottom of the equation, we needed to multiply the equation by 3,600 in order to cancel out the seconds on the top and bottom.

## Dimensional Analysis and Questions

Sometimes using dimensional analysis can help you to answer questions if you don't know the formula. For example, let's say that you know that 1 horsepower = 550 lb*ft/s. Now, let's say you have the following question:

What is the horsepower of a conveyor that is moving 3 mph, carrying 50 pounds of weight? If you can't remember the formula to determine horsepower, we can instead use the units in the conversion. We can see that in the conversion we multiply weight (lb) by speed (ft/s). We simply need to convert from mph to f/s and then we can convert to horsepower. There are 5,280 feet in each mile, and 3,600 seconds in each hour, which you can see worked out below:

So, now we know that it is travelling at 4.4 feet per second and carrying 50 pounds, so:

Now, we can use the conversion factor to convert to horsepower, as you can see below:

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