https://youtu.be/kfrUAwLKqG0
PHYS 1101: Lecture Three, Part Two
All right. The first thing I’ll do at the beginning of every lecture is an overview of what we covered in the previous lecture so we can more smoothly add to that or move into the next content that we want to cover. This section will always be called “Where Were We?”
So the first part of Chapter 1 we covered a few main topics. The first idea had to do with appreciating the need and the ability to convert units and to always work with these SI units. We’re going to need to work with units that indicate length, time, and mass in this first class of physics. For length we’re always going to use the meter, time will be the second, and for mass we’ll use the kilogram. So always convert to these SI units at the start of a problem.
The other important thing to keep in mind about units is that in your mathematical equations the units have to be logically consistent for the equation to have physical meaning as well. This means when you take the variables that show up in your equation and you substitute in the units for those variables, that they should simplify in a meaningful way. I gave this example last lecture where the simplification led to adding different amounts of meters together, which does make sense. These units have to agree if you’re adding or subtracting, and then whatever units you have on the right side of the equation, of course, logically have to be consistent with or represent what the left side of the equation is meant to be.
We then spent a lot of time in Chapter 1 going over vectors. This is a new concept for many of you. A vector is just a mathematical entity. It’s a tool that we can use that’s very powerful for working with motion and describing motion, the physics of it. A vector is an entity that has two things associated with it: a number or a value and then also a direction. The number or the value will have some physical meaning depending on the type of vector you’re working with. The units for that value will be consistent with that physical interpretation. The direction, however, will always be, as you expect for direction, it will be north, south, east, west, so many degrees north of east, etc.
We’re going to use this notation of an arrow to represent a vector. The nice thing about the arrow is the orientation makes a very nice representation of the direction for this vector, and we can use the length of it like a legend on a map to give us a somewhat abstract representation of how big or small that vector is, or the value of that vector.
Okay, when we work with vectors there are some important things we have to keep in mind. We’re going to be adding vector quantities, subtracting them. We’ll mathematically have need for changing the value of those by what’s called scalar multiplication. A scalar, as an aside, is just a number.
For addition, I walked you through the tail-to-tip method. So if you’re adding two vectors, A and B, you want to start and draw the first vector. The second vector then you, keeping the same orientation, slide it around in the same length, put the tail of vector B to the tip of vector A. The resultant then, the sum goes from the initial tail out to the final tip.
Subtraction you can consider as the addition of the negative of that vector. So here I’ve shown you again, this is vector A, same is it was here but now I’m adding to that the negative of B. So I’ve taken this vector and just flipped the arrow to the other end, points in the opposite direction. So A plus negative B is equal to that.
Scalar multiplication is represented by simply adding a number in front of that vector, and that algebraically, remember, would just mean multiplication. Physically what that means is we keep the direction the same for this vector. The only thing we change is the length, or the value, the magnitude of the vector.
I want to show you two quick movies here to graphically illustrate this addition of vectors. One is going to show you the tail-to-tip method. The other one is called the parallelogram method for adding. Both of them are useful to become familiar with. I’m going to insert those movies right here.
It’s a quick movie that shows these vectors in this animation to do the sum, the head-to-tail addition. If we want to add B to A, slide B so that the tail is at the same position as the head of A. We have to keep it parallel to its original orientation at all times. The sum of the two then goes from the initial tail to the final tip. That’s the third vector. And this animation here shows you how the sum changes as you rotate vector B around.
So because the two have equal magnitude there is a rotation scenario you notice there where the sum actually went to 0. And you’ll see also that the sum can go to twice this length when the two vectors are parallel.
This movie is going to show you the parallelogram way of adding vectors. In this case you put the two tails together of the two vectors and then you sketch in the two lines that are parallel to the two. The sum then is the vector that goes from the initial tail out to the corner of that parallelogram. If you compare the tail-to-tip and this method, you’ll see that the vector sum behaves the same way as it did before. There’s for these two equal-length vectors, one angle orientation that sum adds to 0 versus this orientation it’s coming around to where the sum will add to twice the length when the two vectors are parallel.
Here they’re showing the animation of the sum of these two vectors where the lengths are not equal. You can see in this case when they are parallel but pointing in opposite directions, the sum will still have some contribution in the direction of the larger vector. Then they’re going to swing around here so that this vector is parallel to this. When you add those two together you’re going to end up with a vector that’s longer.
So the adding of vectors plays an important role, or that notion does, in understanding that we can think of any vector that’s off at some angle as being a vector sum of its two components. We’re going to use this concept a lot in this class. In real life, what’s going to make the most sense to us is thinking of a vector as a resultant. It has a certain magnitude, it’s at a certain direction.
To mathematically manipulate that vector and to work with it to solve a problem, it’s going to be useful for us to break it up into two components. These two components are the two smaller vectors that when added together tail-to-tip, putting the tail to this tip, that it combines, it adds together to give you that initial vector. Said another way, by working with the sum of these two vectors that’s equal to, it’s the same thing as working with the resultant vector. We’re going to do this often in this class as I’ve pointed out.
The axes here, I wanted to emphasize to you that this tail-to-tip addition, if you think about it also in terms of that parallelogram addition, that brief movie that you just saw shows you that once you sketch these vectors, it’s just equivalent to imagine sketching the vector vy as over here to the left side where all the tails are together, where you’re adding vx to, say, vy if I were to have drawn it off to the left. And then that sum gets me my resultant vector v. So it doesn’t matter what side you draw these two components on, if you choose to put their tails together or you choose to look at them as this tail-to-tip orientation that adds together to give you the sum.
The last thing we covered from the previous lecture was the trigonometry that one can use to break down a vector into its two components and to get exact numbers for the lengths of these two components, the magnitude of them. You’re going to use these trigonometric functions, the sine, cosine, and tangent. Most often, it will be the sine and the cosine that you’ll use because you’ll be given the hypotenuse. That’ll be the information usually in the problem, and then you’ll want to figure out what the two components are.
For example, this is the angle you know. To get the adjacent side component and the opposite side component, you need to use the two functions sine and cosine. We’re going to do this a lot in this class. Practice doing this so you become really comfortable with it.
In the math bonus point opportunity that I gave you, I’m going to work through some of those examples and make a quick movie to show you how to find components so you can look at that and practice a bit more. The other idea that comes out of a right triangle geometry that we’re going to use a lot is the Pythagorean theorem. This is the thing you want to use when you need to relate or find the lengths of the sides of this triangle when you’re not given an angle. You can go back and forth using the Pythagorean. So it amounts to doing a little algebra on this Pythagorean theorem equation to solve for the variable that you want given the variables that you know.
There’s the summary from our previous lecture and what you would have read for today or before today’s lecture is what we’re going to follow that up with to complete the chapter. This is the little bit more information on components of a vector and then showing you how to add vectors by means of the components.
The last thing which isn’t in your textbook which I’ve added that is a very important addition to the class is to talk about motion diagrams. If we use these a lot throughout the class, they really will help us internalize the concepts we’re going to see.
So if you’ve read these sections then you’d be prepared for these two reading quiz questions. The first one here, I want to be sure you understand the notation. When you’re talking about the x vector component of a vector that’s written by hand or on a piece of paper, we’d write it as R with an arrow over the top. How would you represent that x vector component if this is the hypotenuse, this is the resultant vector? Meaning here is the resultant vector R, and you want to know the notation you would use to describe the x vector component or similar notation would work for the y vector component. What’s the right choice here? How would I write these?
The next question is one of many I’ll emphasize having to do with the sine of the scalar component. This is what we’re going to use to represent one of the component vectors, and what is that sine going to represent for us? You have two choices there.