https://youtu.be/dgpTMe_q20Q
PHYS 1101: Lecture Two, Part Two
I’m always going to start a lecture with two initial lecture quizzes that are going to be based on definitions and, really, quizzes on your reading that you hopefully have done before you watch this lecture. So here are your first two lecture quizzes. Pause here if you’d like. Read these over. You might want to go back to the book to help look these up. But this gives you an idea of the kind of content that I’d like you to focus on as you do all of your reading for this class.
Okay. There are your first two quiz questions. Again, you will find in WebAssign an assignment called “Lecture 2 quiz,” and that’s where you’ll enter the answers to that.
As an aside here, let me tell you, you only have one opportunity. Quiz facts. Here let me just remind you. You only have one submission and I give you 3 points for a correct answer and I’ll give you 1 point as a partial credit for incorrect answers. In general, that’s how all of the lecture quizzes will be graded. There will be some questions sprinkled throughout for which I’ll give everybody 3 or more points just for giving me your opinion.
Okay. Actually let me move this up here a little bit.
The first content that we need to cover in this class is the idea of units. Mathematics really is just straight numbers, but once we apply this language of mathematics to describe real life, now all of those numbers are going to represent real physical quantities, and it’s the units that are attached to those numbers that give them their specific meaning.
I walked you through this idea of this physical translation of an equation between math back and forth and real life. I just wanted to point out that if the mathematical connection is valid between those quantities, then the units will also be valid, that connection will be valid and it’ll make physical sense. So having units and keeping track of them with respect to all of your problems and all of your variables will be very important and helpful to you.
So here’s just a silly example to start you out with that, you know, 5, 23 and 12 here. These would just be specific numbers, but as soon as you attach any physical representation to those numbers, these are what I would call units. Now, these units aren’t very specific to the scientific content that we’re going to go over in class, but nonetheless they’re examples of units.
So, just to give you a quick reminder, too, that when you see a unit, something like “cups of coffee per week,” the “per”, mathematically, when you translate that into math means division. It’s cups divided by week. You’ll see that several times throughout the class.
So for our class, though, the units that are going to be particularly useful to us are going to have to do with ways of describing distance, length, time, and mass. Most of this class, it’s about motion and describing motion, and these are the kinds of physical entities that we’re going to need to focus on, that will make up the scientific content of what we’ll study.
So for length, there’s many units you could use to describe a distance or a length. Here are some examples: meters, inches, miles. For time, again, a whole collection of units: seconds, years, centuries. Mass, as a quantity, we’re less familiar with here in the US compared to countries that already use the metric system, but mass actually is not directly convertible to pounds. We’ll cover that later in the class. The units for mass tend to be kilograms or grams, etc.
What I’ve written here, these SI units, that’s the kilogram for mass, the second for time, and the meter for length. As an aside, one meter is roughly one yard. These units — meter, second and kilogram — are the accepted conventional units. They’re called the “SI units” that are used internationally by the scientific community to describe length, time and mass.
So first rule of thumb will be when you do your problems, when we work through this class, always convert to your SI units first. I’ll remind you of that several times. And how to convert units is something we’re going to cover next.
Another point to make here while we’re on this page is that when you’re converting units or adding numbers or adding quantities together, it only makes sense to convert meters, for example, into miles or vice versa. You can only convert within a group of units. You can’t convert 5 apples to 20 miles. Just physically doesn’t make sense. And likewise, when you go to add two quantities, it only makes sense to add like units.
I’ll touch on that a little bit more later. That would mean, for example, it physically wouldn’t make sense to add 5 apples plus 2 miles. You know, what does that mean? It makes no sense. Put a big circle and an arrow through it. So if you end up with a mathematical equation that has different units here and you would be, your equation says to add these quantities together, you need to go back and check your math. Something’s not quite right about that.
Okay. So we’re going to do a lot of unit conversion. And let’s be sure everyone is up to speed on how to do that. Let’s say we start out with the problem of having to convert a quantity that’s in miles per hour, say, 5 miles per hour, and we need to know how many meters per second that is.
Well, first, as I’ve reminded you that 5 miles per hour translated more specifically into mathematics and units, you would write as 5 miles per hour. So in order to this conversion, you’ll notice that we have — let me just as an aside here write that meters per second means our goal is to convert this into units of meters divided by second.
So to do that, we’ve got two units that have to be converted. You’ll notice that miles has got to be converted to meters, but we also have to convert a time unit here, of hours, to seconds. So we’re going to have to do two unit conversion steps.
To do that, we’re going to need conversion factors for each of those. Here’s what we need, and we need to get this information somewhere. Miles to meters. What’s the equivalent of these two quantities? Well, either in the front cover of your book if you did purchase a hard copy of the book, or you could Google unit conversions, and you’ll find lots of pages that will list these equivalencies for you.
But in essence you need to know how many meters are in a mile or vice versa. Likewise, we need to go from hours to seconds. This one potentially you can calculate on your own. And the conversion factor is there are 3,600 seconds in every hour. Let me write here off to the side here. Get your conversion factors, again, front cover of book or just Google it. Just “unit conversions” will work.
Okay. So we’re going to do this example of this unit conversion, but first I want to point out how this works and why this works. Let’s start out with the 1 mile is equal to 1,609 meters. So this is an equation. This is an equivalency that says these, the left and right hands of this equation are equal. Imagine you do a little algebra here. I’m going to divide both sides by 1,609 meters.
When you do that, I’ve maintained this equality in this equation. Also on the right-hand side now, this ratio simplifies to 1. That tells me then that equivalent to this conversion factor is a statement that the ratio of 1 mile over 1,609 meters is equal to 1. This is the idea that we’re really going to take advantage of to do the unit conversions, because we’re going to take our original numbers, and we’re going to be able to multiply by these ratios of 1 to convert our units and because it’s 1, we’re not changing the value of that quantity.
But let me note, I could just as easily have started with my equation 1 mile equals 1,609 meters and I could have divided both sides by 1 mile, meaning I could have also ended up with also a ratio that’s 1, but in this case, it’s the inverse ratio. It’s 1,609 meters divided by 1 mile is equal to 1.
So the real thing to focus on here is that when you get these conversion factors, you can turn them into a ratio, and whichever fraction you want will depend on the unit conversion direction that you’re doing. So that’s what we’re going to see right now.
Now that we have those unit conversions, I’ve explained to you this ratio idea. We’re ready to do the conversion. So, let’s write it out again. We’re starting out with 5 miles per hour. I recommend writing it like this as a ratio with a clean numerator and denominator. Let’s first convert the miles — I’ll write that up here. Let’s work on getting from miles to meters.
Okay. To do that, I’m going to multiply by one of my expressions here or ratios that’s equivalent to multiplying by one, but I’m going to choose one that’s got miles in the denominator, so I’m going to use that last fraction. 1,609 meters is equivalent to 1 mile. The numerator, denominator equal, so this is like multiplying by 1.
The beauty of this choice is that the mile in the denominator and the numerator cancel each other, so now if I take 5, multiply by 1,609, that number is now in units of meters per — and I still have hour here in the denominator. If you do that multiplication out, you end up with 8,045. At this point, my units are meters per hour.
Okay. Now we can take that quantity and we can do our next conversion. Meters per hour. Now I’m going to multiply by 1. I want hour in the numerator. So it’s going to cancel with the unit here I’m trying to get rid of. And then I know that 1 hour is equivalent to 3,600 seconds. The hours cancel, and now if I take 8,045 meters and then everything in the denominator here now is 3,600 seconds, so 8,045 divided by 3,600 is going to give me my number. And that is 2.24 meters per second. Those are the units that I’m left with, meters and seconds.
Now we’ve done it. We’ve learned that 5 miles per hour is equal to 2.24 meters per second.
Let me point out just as a useful tidbit of information that if you have a number that’s so many miles per hour, which is, of course, the unit we’re used to in the US system, that that miles per hour is roughly equal to twice that in meters per second. I’m sorry. I said that backwards. If you have a number in miles per hour, it’s approximately equal to one-half that number in meters per second.
More often the useful thing we’ll run into is we’ll be given a number in meters per second. That’s a speed, a distance per time. But we don’t work in the metric systems when we’re familiar with miles per hour, so if you want to get a rough idea of how fast that is, you just multiply that by 2, and you’ll have a rough estimate of what it is in miles per hour.
So for example, in a problem if it says something, a car is going 30 meters per second, that’s approximately 60 miles per hour. Just a useful fact there to help give you some physical intuition.
Okay. Let me point out that this unit conversion that we did in these individual steps can be done all at once. In other words, if we start out with 5 miles per hour, we can first multiply by a conversion factor to convert miles to meters. So let’s multiply by one here of the form 1,609 meters is equal to 1 mile. This takes care of converting our miles into meters. We can now go right ahead and multiply by another factor of 1 where we’re going to convert hours into seconds. I’m going to write in the numerator 1 hour is equivalent to 3,600 seconds. And now hours have been converted to seconds. So here I have my units, meters per second.
And now I just carry out this multiplication as this equation tells me to. It’s 5 times 1,609, divided by 3,600. And you’ll punch that in to your calculator, and you’ll see it’s 2.24 meters per second. So you can do all these steps at once.
Okay. There’s one more exercise having to do with units that I want to point out to you. This will be a handy skill to use to check after you’ve done some algebra to be sure that your algebra’s correct and that the physical meaning or translation of your equation is still valid. You can check that by seeing if the units are still valid and make sense with that equation.
Notice in this above exercise, let me just point out that when we were canceling out an hour in the denominator with an hour in the numerator, we were treating these units just like you would any other algebraic variable. And that physically makes sense. Mathematically it’s consistent. That’s how they function or act. So we can take advantage of that, of this simplification of units to see if our equation is valid.
So let me write here just as a title for this last little comment I want to do, I’m going to put here, “Use units to check the validity of an equation.” This is something you couldn’t do in pure mathematics. It’s only now that this equation has physical meaning that it makes sense to do this.
As an example, let me give you an equation that says x is equal to v multiplied by t plus one-half times a times t squared. Let me further tell you that x has units of meter. It’s a measure of a length. v has units of meters per second, and a has units of meters per second squared.
Are these units, as defined, when plugged into this equation, consistent? Is it valid? I would check this by noting for x, I would just put in meters. So is a unit of meter equal to — for velocity, I have meters per second. For time, the unit for time — oh, I should have written that down. It’s perhaps obvious. Maybe not. Seconds. So for time you would plug in seconds. And now plus the one-half, not that critical that we write down. In fact, I’m going to erase it having said that, because this exercise we’re just focused on the units. The numbers out front would be — you know, maybe it was 5 meters per second that you would actually plug in here. Not worried about the numbers. We’re just focusing on the units here to see if the units, at least, mathematically are consistent, if it’s logical.
So focusing on that, the only other unit we have here for this last term, one-half is just a number, but a has units of meters per second squared and time has units of seconds. But I have time or t squared, so that’s s squared.
So what do these units simplify to? Two s‘s here in the numerator. Both of those cancel with the s squared in the denominator. This s cancels with that s in the numerator, so it looks like I have, I’m left with the first terms simplifies to meter, and the second terms simplifies to meter.
First point, adding a certain number of meters to another number of meters, that that logically makes sense. That’s a legitimate thing to do, so that’s good. That’s consistent. Indicating our equation at least makes sense.
But then also notice that it’s saying meters on the right should be equal to units of meters on the left. And that’s also consistent and right. So this equation is valid. So let me just put here for number one, “Note addition of like units.” You always have to do that. Let me move this over here. It doesn’t hurt.
And the second point to note here is units on left and the right side of the equation agree. So the equation physically makes sense. It’s logically consistent.
Okay. You have some homework problems to do where you’ll explore this a little further. And in some of those, you’ll find that it’s not consistent on the right and the left-hand side, meaning that that equation doesn’t make sense physically. If you run into that when you’re solving a problem, then it tells you you have to have made some mathematical mistake or assumption somewhere along the way.