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Physics 101: Help and Review17 chapters | 212 lessons

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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.

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.

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!

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.

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:

Now, we know that it has 0.4 horsepower. So, sometimes, we can use the units and dimensional analysis to answer questions if we don't remember the formulas (it is a nice trick to remember when taking tests!).

Dimensional analysis is used in chemistry in a variety of ways. It can be used to convert between weights or temperatures. But the most common way it is used is to calculate molecules from grams (or vice versa) and with balanced chemical equations.

For each element, we are given the molecular weight, which is the weight given in moles/gram, and it is typically included on periodic tables with each element. We also know that there are 6.022 * 1023 molecules in each mole.

If we have the balanced chemical formula, then we can also use this information to determine the theoretical mass of a product.

Let's look at the chemical formula for aerobic respiration (the breakdown of glucose), which you can see below:

Let's say we are starting out with 100 grams of glucose, and we want to know how much water will be produced. First we need to determine how many molecules of glucose are present in 100 grams, given the molecular weight of glucose is 180.16 grams/mole, which you can see worked out below:

So 3.34 *1023 molecules of glucose are in 100 grams of glucose. Now, looking at the chemical formula, we can see that for every molecule of glucose we form 6 molecules of water, which you can see being worked out below:

So, 2.006*1024 molecules of water are formed. Using the molecular weight of water (18 grams/mole), we can work our way back to determine the weight of water formed, as you can see below:

So we form 59.96 grams of water from respiration of 100 grams of glucose!

Let's briefly review what we've learned!

**Dimensional analysis** is a tool used in unit conversion that helps us keep track of units and whether we should divide or multiply. Remember to cross out units if they can cancel out, to see if we end up with the correct units. If we don't end up with the correct units, then check your work to see what happened.

We can use dimensional analysis to help answer some questions when we don't know the formula by using the units found in conversion.

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Physics 101: Help and Review17 chapters | 212 lessons

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