Dimensional Analysis: Videos & Practice Problems

Dimensional analysis is a mathematical method used to convert between units by employing conversion factors that relate the units being converted. To perform dimensional analysis, start with the given amount and use conversion factors to cancel out unwanted units, thus isolating the desired end unit. For example, to convert inches to centimeters, use the conversion factor that 1 inch equals 2.54 centimeters. When multiplying, the number of significant figures (sig figs) in the final answer should match the least number of sig figs in the given data. In more complex scenarios involving multiple conversion factors, such as converting minutes to years, each step must cancel out the previous unit until the final unit is reached. The accuracy of the final value is determined by the significant figures of the initial given value, not the conversion factors. Remember, the goal is to systematically cancel out units to arrive at the correct end amount with the appropriate number of significant figures.

Dimensional Analysis

So dimensional analysis is seen as a fail proof process that allows you to convert from one unit to another. You design the problems to begin with your given amount and to finish with the end amount of your unknown. Now just follow the units to ensure the unwanted units are canceled out. In the same way that we did a metric prefix conversions where we lined things up on opposite levels to help them cancel out. We're going to take that and apply it here to the dimensional analysis.

Now the strategy is many conversion problems will utilize your given amount and conversion factors and you're going to use those together to help isolate your end amount. So if we take a look here in this general format, let's say we're given 32 inches, So 32 inches is our given amount and they're asking us to identify how many centimeters this equals O centimeters is our end amount. What we're looking for, the conversion factor is a way of bridging these two ideas together. It's used to convert our given amount into our end amount.

Now to go from inches to centimeters means that we have to use a conversion factor that relates inches to centimeters. When we talked about conversion factors, we said that one inch is equal to 2.54 centimeters. I put inches here on the bottom so that they can cancel out what these inches up here on top so you always want them on opposite levels. Doing that helps me isolate my end amount unit which is centimeters. So then I would just multiply 32 * 2.54. Initially I'll get 81.28 centimeters. But remember when you're multiplying numbers together your answer is the least number of sig figs.

32 has 26 figs in it. Because remember when you don't have a decimal point you move from right to left. You start counting once you get to your first non 0 number and count all the way through. So this has two sig figs. When you have a decimal point you go the opposite way. Our first non 0 number is this two. You start counting here. You go all the way through. So we have three sig figs. So you go with the least number of sig figs which would be two. So this would round to 81 centimeters as our end amount.

But let's say we had to do a type of dimensional analysis with way more steps. What do we do then? Well, let's say here we're given 115 minutes, so that is our given amount and we have to get to years. Years would be our end amount. To connect given to end, we have to utilize conversion factors. I need to cancel out minutes, so I'll put minutes here on the bottom. We know that minutes is connected to hours. We know that one hour has in it 60 minutes and here this will represent our first conversion factor, which I'll abbreviate as conversion factor one.

Sometimes it requires more than one conversion factor to get to our end amount. This is that example. Minutes would cancel out. Now I have hours. We know that hours and days are connected. We know that one day has 24 hours. This would be my second conversion factor or CF two hours cancel out. Now I have days, and then finally we know that days and years can be connected as well. We put days on the bottom so we can cancel out with the days on top and we know that one year has approximately 365 days. This is my conversion factor 3. So days cancel out and now I'm left with years.

So what we're going to have here is 115 on top, multiplying with a bunch of ones which doesn't change anything, but on the bottom we have multiplying 60 * 24 * 365. Now it's incredibly important you know how to plug this into your calculator. My suggestion when you have multiple things on the bottom multiplying is to just multiply them all together and get that sum total. When we multiply 60 * 24 * 365, their total is 525,600 and we still have the 1:15 on top. When we divide 115 by that total, we're going to get 2.19×10-4 we're going to say here, let me put that so 2.19×10-4.

We're going to say here that the number of sig figs within our final answer is based on our given information. Now our only given information was the 1:15. These other numbers were the ones who supplied that. Those numbers were not given to us in any way, so we can't use them to determine the number of sig figs. So again, it's all based on the information that's presented. Look at those numbers presented to you and use those to determine the number of sig figs in your final answer. 115 has no decimal point, so we move from right to left. It has in it three sig figs, so our answer should have three sig figs, which it does. So 2.19×10-4 years would be our answer here.

So everything we've learned up to this point, we're going to use in some way when it comes to dimensional analysis. And remember, our conversion factors are incredibly important because they're a way of connecting our given amount to our end amount. The whole point is to cancel out units and isolate the unit that you're looking for at the end. Now that we've done these example questions, let's move on in our discussion of dimensional analysis.

Dimensional Analysis Example

Here are example states ATA can grade 4 assignments per hour. If each assignment has 12 questions, how many questions can the TA grade in 130 minutes? All right, so if we're going to approach a question like this, let's look at our steps. If present, start with the given amount. That is not a conversion factor. So remember, our given amount is when we have a single unit by itself that isn't tied to another. We're going to say within this question are 130 minutes is our given amount to identify the end amount you want to isolate for your unknown.

So we're going to say here this is our given amount, we have to figure out what our end amount will be. So our end amount is over here. They're asking us how many questions, how many questions? So questions will be an amount. Step 2, write down all the conversion factors. So all our conversion factors. Let's see. We have 4 assignments per hour. Remember, per is the word that connects different units together. So one is 4 assignments every per hour. So that's four assignments for every one hour they tell me. Also, each assignment has 12 questions, so that means one assignment is 12 questions. OK, so 12 questions.

Now this last part. Find the connection between the given amount and the conversion factors in order to isolate the end amount. OK, so let's look at the given amount. The given amount has minutes within it, but neither one of the conversion factors has minutes involved. What we do have is hours. That tells me there's a conversion factor that's even before either one of these two. So conversion factor one actually involves us first changing minutes into hours. So we have minutes here on the bottom, hours here on top. One hour is 60 minutes. Minutes cancel out this way, and the reason we're doing that is because now that we have hours, we can connect it to the hours here within this conversion factor.

So that'll be my second conversion factor. Bring in the one hour for every 4 assignments. Now that we have hours lined up, they cancel out. Now we have assignments and we need to get the questions. Here is our last conversion factor. Now it has assignments and questions within it, but we need assignments to cancel out. So assignments need to be here on the bottom right, and then we need questions. Questions go here on top and it is one assignment is 12 questions. Remember, one of the first things we said is that conversion factors, we can flip them. They can be presented in two different ways. Here we have to flip the initial conversion factor so that assignments can cancel one another out. They have to be on opposite levels to be able to do that.

So in conversion factor 3, what I'll have left at the end is questions. So what we do now is we're going to multiply 130 times 4 * 12 / 60. So when we do all of that, we're going to get here 130. Well, actually, no, not 1:30. We're going to get here as our final answer, 104 questions. So we have 104 questions as our end amount. But remember, we need to worry about significant figures. 130 has two sig figs, 4 has one sig fig, 12 has two sig figs, 60 has one sig fig. So we need to go and have only one sig fig as our final answer. So we'd save roughly about 100 questions is what the TA could do within 130 minutes, which is a lot of questions.

All right. So now that we've done this example where we've set up the basic principles behind dimensional analysis, let's continue onward and do some practice questions.

If the distance between Washington, D.C. and New York City is 224.9 miles, determine the distance in centimeters.

• A. 8.110 x 10⁹ cm
• B. 3.619 x 10⁷ cm
• C. 4.207 x 10^-3 cm
• D. 1.913 x 10^-5 cm
• E. 5.000 x 10¹¹ cm

Answer: (B)

A large scale ranching farm possesses 22.5 million acres with 5.0 x 10^-5 % reserved for housing. Determine how many square kilometers this area for dwelling represents. (1 km = 0.6214 miles) (1 mile = 5280 ft) (1 acre = 43,560 ft²)

• A. 9.2 km²
• B. 0.115 km²
• C. 1.8 km²
• D. 0.046 km²

Answer: (D)

Here’s what students ask on this topic:

What is dimensional analysis?

Dimensional analysis is a mathematical technique used to convert one set of measurements to another, ensuring that the equations used for the conversion are dimensionally consistent. This means that you can only equate or compare quantities that have the same fundamental dimensions (such as length, mass, time, electric current, etc.).

In practice, dimensional analysis involves using conversion factors, which are expressions that relate different units of measurement. By multiplying the quantity by the appropriate conversion factor, you can change the units without altering the actual value of the quantity. This is particularly useful in physics, chemistry, and engineering to check the plausibility of physical equations, to convert units from one system to another, and to compute numerical solutions even without knowing the precise values of the quantities involved.

For example, if you have a speed given in miles per hour and you want to express it in meters per second, dimensional analysis would guide you in setting up the conversion factors to correctly make this change in units.

How to do dimensional analysis?

Dimensional analysis is a method used to convert one unit of measurement to another. To perform dimensional analysis, follow these steps:

1. Identify the starting unit and the desired unit: Determine what unit you're starting with and what unit you want to end up with.
2. Find conversion factors: Look for relationships between the units that express how one unit can be converted into another. A conversion factor is a fraction that represents the equivalence between units.
3. Set up the problem: Write down the quantity you're converting as a fraction, and multiply it by the conversion factor(s). Arrange the conversion factor so that the unit you want to eliminate is opposite of the same unit in your original quantity. This way, the units you want to cancel out will be diagonal from each other.
4. Cancel units: Cross out any units that appear in both the numerator and denominator.
5. Perform the math: Multiply all the top numbers (numerators), then multiply all the bottom numbers (denominators), and divide the top by the bottom to get your final answer.
6. Check your work: Ensure that the units you're left with are the ones you wanted, and the answer makes sense in the context of the problem.

Remember, each step in the multiplication should bring you closer to the desired unit. Keep track of your units throughout the process.

What is the purpose of dimensional analysis?

Dimensional analysis is a mathematical technique used to convert one set of measurements to another, ensuring that equations and calculations are dimensionally consistent. This means that all terms in an equation must have the same dimensional units for the equation to be physically meaningful. It is also used to check the plausibility of derived equations, to plan experiments, and to deduce relationships between physical quantities that are not immediately obvious.

The technique involves using conversion factors that describe the relationship between different units of measurement. By multiplying the quantity by these conversion factors, you can systematically cancel out the unwanted units and end up with the desired units. This is particularly useful in science and engineering where calculations often involve multiple units, and ensuring the correct application of units can be the difference between success and potentially costly or dangerous errors.