 To get a bit more practice with dimensional analysis, let's try some unit conversions, beginning with a conversion from cubic inches to liters. My dimensional notation requires that I draw a big horizontal line, and I can write 122 cubic inches, and then I will multiply by one in different forms until I get to liters. If I go into my conversion factor sheet and look in volume, I recognize that both cubic inches and liters are representations of volume, so maybe there's a direct conversion here. I look up liters, I see one liters, cubic meters, liter, the cubic feet, and then if I look up cubic inches, I see cubic inches, it represented as cubic centimeters, none of those are what I want directly, which means that I'll have to take some intermediate steps. I could go from cubic inches into cubic feet, and then from cubic feet into liters, that would work. I could also go from cubic inches into cubic centimeters, and then from cubic centimeters into cubic meters, and then from cubic meters into liters, that would also work. Those paths are valid, but what I would encourage you to get into the habit of doing is starting at your destination and working backwards. In our conversion factor sheet, I can have two different representations for liters. I could represent one liter as cubic feet or cubic meters, and then continue to break that down until I can cancel all of my primary dimensions. So I will say one liter is, let's go with feet first, 0.0353 feet, 0.0353 cubic feet. Now nothing cancels yet, which means that I will have to convert from cubic feet into cubic inches. If I jump back to my conversion factor sheet, I see that I don't have a direct conversion from cubic inches into cubic feet either. Now why don't I? Well, it's because it would be a waste of space. I already have a direct conversion between inches and feet, and if I were to cube that conversion, I end up with a conversion from cubic inches to cubic feet. So my approach is going to be writing one feet is equal to 12 inches, and then cubing everything. One cube is one, which is boring, feet cubed, 12 cubed, inches cubed, inches cubed, cancels inches cubed, cubic feet, cancels cubic feet, leaving me with liters. So I pull up my calculator. I can perform the actual computation. Come on calculator, you can do it, 122 divided by 0.0353 times 12 raised to the cubed power. I see that I get 2.0005 liters. Hooray, we have an answer, about two. Let's check it with our oracle. Hey Siri, convert 122 cubic inches to liters. 122 cubic inches is equivalent to about two liters. Hey, about two liters. Look at us. We're amazing. Just for funsies here, let's consider the alternative path. What if we had represented a leader in cubic meters as instead in that alternate reality, I would have written one liter is equal to 10 to the negative third meters, other cubic meters. So 10 to the negative third cubic meters. And then I would have to break cubic meters down until it cancels cubic inches. So again, I could break cubic meters into cubic centimeters and then cubic centimeters into inches, or I could convert cubic meters into cubic feet and then from cubic feet into inches. Just for fun, let's handle the length conversions twice like we did last time. So I will say one meter is equal to 100 centimeters. And then I will cube everything, cubic meters, cancels cubic meters, one cube is boring, centimeters cubed and inches are left. So I will look up the conversion from inches to centimeters in my conversion factor sheet. See that one inch is 2.54 centimeters. So one inch is 2.54 centimeters. And then I cube everything centimeters cubed, cancels centimeters cubed inches cubed, cancels inches cubed, which again leaves me with liters. So I can take 122 multiplied by 2.54 cubed divided by the quantity 10 to the negative third multiplied by 100 cubed. Just be a cube button on this thing. And that will give me 1.99922, which is still about two liters. If we cared more about significant figures, we would have a conversation now about rounding, but we don't for now. So let's move on. Next up is 778.17 feet. Actually, before we move on, let me just point out we got slightly under two this time. Last time we got slightly over two. All of these unit conversion factors, I guess most of the unit conversion factors on this sheet are rounded to some degree or another. So anytime you're converting units, you're going to have a little bit of error as a result. The best thing to do is to get as far as you can symbolically in a problem by simplifying it before you involve unit conversions so you can limit that result. The other best practice is to try to limit unit conversion steps as much as possible. So you'll note that in this form of this Part A solution, we had three steps. In the previous iteration, we had two. Well, two steps is probably going to have less error overall than three steps, even though this one is theoretically perfect. And I guess this one is two. So there's really just one step with error. But general rule, try to limit the steps where you can, except for character building opportunities like these. The next is 778.17 feet pound force to kilojoules. Well, requisite horizontal line, 778.17 feet times pounds of force. And I want to get to kilojoules. So I'm so confident right now. I'm just going to write kilojoules because that's my destination and let's work backwards. I know that a kilojoule and feet pound force are both representations of energy. I know that because kilojoule is a unit of energy and pound force is a unit of force, feet is the unit of distance, distance times force is going to be a representation of energy, primarily work probably, but energy all the same. So if I go into energy, I can look up a conversion per kilojoules into feet times pounds of force. Hey, that's exactly what we want. I also have kilojoules into BTUs. So I will say one kilojoule is equal to 737.56 feet times pounds of force. 737.56 feet times pounds of force. Feet cancels feet, pounds of force, cancels pounds of force. And I will be left with kilojoules. Calculator says SWSWSW really wants me to pay attention to Southwest. 778.17 divided by 737.56 yields 1.055. Part B done. For part C, we are converting from horsepower to kilowatts. So horsepower and then I'll write kilowatts confidently. I will recognize that horsepower and kilowatts are both representations of power, which is energy transfer rate. So in my conversion factor sheet, what I want is the energy transfer rate section. I broke it. Ah, I'm broke. Under energy transfer rate, I have a conversion from horsepower into BTUs per hour, horsepower into feet times pounds of force per second, and horsepower into kilowatts. So one horsepower is equal to 0.7457 kilowatts. Let's see. So I was a little bit too confident. I really have to write one horsepower in the denominator here, and then 0.7457 kilowatts in the numerator. Horsepower cancels horsepower. Is there a conversion from kilowatts? Oh yeah. I also could have written 1.341 horsepower. You know what? Just for fun, let's try it all the ways. So 100 multiplied by 0.7457, 74.57. And I have a conversion for one kilowatt into 1.341 horsepower. So I totally could have done that if I had seen that one first. One horsepower. That's not where the horsepower is written down. One horsepower. One kilowatt is 1.341 horsepower. Horsepower cancels horsepower. Southwest, Southwest, Southwest. 100 divided by 1.341 should yield 74.57. This particular difference isn't that important, and in this situation, if you were looking at the unit conversion factor sheet and thinking about which one to use, I would honestly recommend using whichever form is most convenient for you to get right consistently. Like I have a general propensity to want to write things in the numerator, because I don't like dividing by quantities that I have to use a parentheses for, so I would generally speaking write as many of the quantities in the numerator as possible. There are multiple schools of thought. If you're trying to think through how three decimal points of precision versus four decimal points of precision is going to affect your rounding error, I would say don't worry about it. Our error in these calculations is going to be much wider than the error introduced by the unit conversion steps, especially minor differences like this. I mean, if this calculation were actually being done, 100 horsepower had to have come from somewhere, or the result would be put into application somewhere, and there's no way that we could be that precise with that application. I mean, if you hook your engine up to a dyno and it says that you're producing 100 horsepower on the nose, there's a lot of play in your ability to record that, so it'd be 100 plus or minus a bit. Anyway, a little bit off topic. Since we did it two ways already, let's try more ways, shall we? Let's say that I wanted to convert from horsepower into kilowatts, the longest way that I can come up with at the moment. Let's instead take horsepower into feet pound-force per second, and then make the unit conversion switch for feet, pound-force, and seconds I guess is going to stay the same. And then once I'm in metric, convert those back into kilowatts. Let's try it. So I'm going to say one kilowatt, excuse me, one horsepower is equal to 550 feet times pounds of force per second. Horsepower cancels horsepower. Now I have a representation in feet times pounds of force per second. And then because I am a dutiful student, I want to write kilowatts as our destination and break that down into its component pieces until the primary dimensions can start canceling. So I know one kilowatt is 100 watts, no, 1,000 watts. And a watt is defined as a joule per second. And a joule is defined as a Newton times a meter. Each step along that process can be found in the conversion factor sheet. So a kilowatt using the kilo prefix is going to be 1,000 watts, a watt is a joule per second, and a joule is a Newton times a meter. We could break it down further by looking at force and saying a force of one Newton is equivalent to one kilogram meter per second squared. But because I have force in the form of pounds of force, I can stop here. So watts cancels watts, seconds cancel seconds, joules cancels joules. So I'm trying to get feet to cancel meters and pounds of force to cancel Newtons. So one conversion at a time, one meter is 3.2 808 feet. 3.2 808 bump. And then we want pounds of force and Newtons. That would be under the force category. Go away calculator. Pounds of force 4.4482 Newtons. So one pound of force, 4.4482 Newtons. Now meters cancels meters, feet cancels feet, pounds of force cancels pounds of force, Newtons cancels Newtons, but left with kilowatts and a whole bunch of error. But this is learning. So we take 100 multiplied by 550 times 4.4482 divided by the quantity 1000 times 3.2 808. And I get 74.5705 74.57 ish kilowatts. Part C done. For part D, I'm performing a calculation and converting the result to Newtons. So because I'm multiplying two things together, I will write them in the numerator 10 kilograms and 9.81 meters per second squared. My goal is to get to Newtons and again as a general rule, it's best to start with Newtons and work backwards. So I can break the Newton apart into its components by looking at the force section on my conversion vector sheet and seeing that one Newton is defined as one kilogram meter per second squared. So I can write one kilogram meter per second squared, kilogram cancels kilogram meters, cancels meters, second squared, cancels second squared, and I'm left with Newtons. Easy peasy. The result was already in Newtons. We just didn't realize it. So 10 times 9.81, I'm just going to leave the calculator out of this, 98.1 Newtons. Part E, we are converting kilograms to pounds of force. So what is the issue with this conversion? I'll give you a second. You're right. Kilograms is a representation of mass, pounds of force is a representation of force. In order to perform this conversion, I need dimensional homogeneity, which is a fancy way of saying I need them to be in the same dimension for unit conversion as a concept to even make sense. So there's an implicit assumption to be made in this conversion. I need to assume that this five kilograms is experiencing an acceleration. And since I have nothing else to work on, I'm going to be assuming it's under standard gravitational acceleration. I'm assuming the acceleration is standard sea level earth gravity. Should earth be capitalized? Is that a thing? That's your mark. I mean, it's proper. Right? I don't know. We'll move on. That'd be 9.81 ish meters per second squared. With that assumption in place, I'm going to recognize that my force exerted by this mass under standard gravitational acceleration is going to be represented as mass times acceleration. Therefore, I have five kilograms multiplied by 9.81 meters per second squared. And then my goal is to get to pounds of force. So I will beautifully write pounds of force up in the numerator and make it work. So in my force section, I can write pounds of force as a number of pounds times feet per second squared. Note that that pound written without the F is a representation of mass, not force, which is a whole separate conversation. Pounds of mass is a representation of mass in the imperial unit system that adds confusion. But it also adds convenience because the amount of force in one pound of force under standard gravity is one, excuse me, the amount of force in one pound of mass under standard gravity would be one pound of force. So it's most convenient for us to use pound mass as our default imperial mass unit instead of slugs, say for example, because slugs are arbitrary, not that, you know, all imperial units are arbitrary at this point. But slugs and blobs, though valid representations of imperial mass aren't particularly useful, pound of mass is. So that's going to be the representation we use for mass, the majority of the time. Anyway, pound of force could be written as 32.174 pounds of mass feet per second squared. Or I could write it as 4.4482 Newtons. So I will write that as 4.4482 Newtons. And then I will explode the Newton into its component parts, which we know from last time, one kilogram meter per second squared. So Newton cancels Newton kilogram cancels kilogram meters cancels meters. Second squared cancels second squared. And I'm left with pounds of force. So five times 9.81 divided by 4.4482 will yield a result in pounds of force. So calculator, if you come back please see you will five times 9.81 divided by 4.4482. We get 11.0. Let's call it three pounds of force. So kilograms times meters per second squared two pounds of force. I have mass times length divided by time squared. I know that I'm going to be representing mass times acceleration as a force. So that first computation 10 kilograms times five meters per second squared is going to yield something in the force dimension. So I have dimensional homogeneity. I can get to it 10 kilograms multiplied by five meters per second squared. And this unit conversion is ostensibly the same as the one above. I don't really need to repeat myself. Let's see if we can add a little bit of extra flavor by trying in a different way. Instead of converting from pounds of force into newtons let's try pounds of force into pound mass times feet per second squared. Yeah I think that'd be more fun. So one pound of force is defined as 32.174 pound mass times feet per second squared. So 32.174 pound mass is a good horizontal line per feet second squared. Note that I'm adding an M here to try to limit confusion. It's pounds of mass. And again a pound of mass is the amount of mass in one pound of force. Like I can write this as one pound of force is equal to one pound of mass times 32.174 feet per second squared. That's the conversion that we just used just now. And the equivalent for slugs would be one slug times one foot per second squared. And four blobs would be one blob times one inch per second squared. If you're asking yourself why are slugs and blobs in this game I don't know the imperial unit system is strange. So this is more convenient for our usage than this because we are operating with standard gravity the majority of the time so a pound of mass will exert about a pound of force so it becomes almost interchangeable to talk about. These two are a little bit more convenient for definitions relative to one foot per second of acceleration every second or one blob is one inch per second of acceleration every second. I don't know potato potato. Anyway we are converting from pounds of mass now into kilograms. So if I scroll on up to my mass section a pound of mass note that it's a pound of mass because it doesn't have an m. That's one pound mass is 0.4536. One pound of mass is 0.4536. Second squared cancel second squared and next up would be the conversion between feet and meters but hold up these are both in the numerator something oh you can't see feet and meters are both in the numerator so right now I've broken my dimension I can't end up with a force therefore I must have made a mistake let's scroll on down to our pound of force pound of force is 32.174 pound times feet per second squared pound of mass times feet not divided by okay what I did there was introduce a totally on purpose error so that we could have the opportunity to note how paying attention to your units is another way of checking your work you can get through a lot of engineering problems by just considering the units and if you consider the units while you're working through a problem you can find errors that you made before you actually introduced them into your result a lot of students would just try to figure out the conversion and then just take 10 times 5 times some number in their calculator and just hope for the best that will not fly in thermo because we have so many units we have to pay attention to them but we are almost certainly going to make a mistake every time yep totally on purpose so meters to feet we just looked at one up one meter is 3.2 808 bump feet so I could write 3.2 808 feet one meter feet cancels feet meters cancels meters and I am left with pounds of force cool so calculator come back southwest southwest I have 10 times 5 times 3.2 808 divided by 32.174 times 0.4536 and I get 11.24 so lastly I have part g now second to lastly I have part g taking one half times 5 kilograms times 10 squared meters squared per second squared and my goal is to get to kilojoules so I will start with kilojoules I will break it into its components a kilojoules a thousand joules and I will break that apart a joule is a newton meter a newton is a kilogram meter per second squared joules cancels joules newtons cancels newtons kilograms meter squared cancel kilograms meters meters and second squared cancels second squared giving me kilojoules so one half times five times 10 squared divided by 1000 note that I wrote one half instead of one divided by two like I did last time we encountered this we get 0.25 and then for the ultimate unit conversion we are doing the same thing except we have an imperial velocity this time instead of a metric velocity for that I will clear out the calculator we'll write one half times 50 kilograms times 50 miles per hour quantity squared my goal is to get to kilojoules so I could take kilojoules and convert it into 1000 joules joules newton meter a newton is a kilogram meter per second squared then I could convert meter squared into miles squared and hour squared into second squared but just for character building let's try it the long way again this is not how you would want to actually do this but this is going to be a useful way for us to get practice at dimensional analysis which is what this problem is really all about anyway so instead of converting from kilojoules into joules and newton meters etc let's look at other things that I could do with a kilojoule ooh BTUs I like it let's say one kilojoule is 0.9478 kilojoules no BTUs 0.9478 BTUs I like it nice and imperial then a BTU could be written as feet times pounds of force so one BTU is 778.17 feet times pounds of force BTUs cancels BTUs and nothing else cancels yet I want to break my pound of force apart into its component pieces so for that I will use the definition relative to pounds of mass like we did two or three parts ago I'll write one pound of force is equal to 32.174 pounds of mass times feet per second squared so one pound of force is equal to 32.174 mass times feet per second squared and then I can convert from miles to feet I know one mile it's 5,280 feet I square everything write that squared a little bit better so that hopefully I have a chance of reading it correctly one squared is boring feet and feet cancel square feet square miles cancel square miles and then for my conversion from hours to seconds I'm going to use the same conversion that we had to use when we introduced dimensional analysis 60 squared times 60 squared I could say one hour 60 minutes and then one minute is 60 seconds and square everything or since I'm going to be doing this a lot a little bit more convenient for me to write one hour is equal to 3600 seconds I square everything one squared is boring now we are cooking with whatever it is you cook with second squared cancel second squared we are very nearly there all we have left is the conversion between pounds of mass and kilograms and for that we go back to our conversion factor sheet what sayeth the oh sheet one kilogram is equal to 2.2046 so kilograms needs to be on the denominator in order to cancel so one kilogram is 2.2046 mass and again I'm adding the m for my own convenience kilograms and then the cancels kilograms and the pound of mass cancels the pound of mass that leaves me with kilojoules so calculator if you would please I'm going to write one half as one over two this time so one times 50 times 50 squared times 5280 squared times 2.2046 divided by two times 0.9478 times 778.17 times 32.174 times 3600 squared and I get 12.4904 probably just round that to 12.49 kilojoules and there we have it we are now definitely experts at dimensional analysis and handling units in our problems