MIT OpenCourseWare http://ocw.mit.edu 8.01 Physics I: Classical Mechanics, Fall 1999 Please use the following citation format: Walter Lewin, 8.01 Physics I: Classical Mechanics, Fall 1999. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsMIT OpenCourseWare http://ocw.mit.edu 8.01 Physics I: Classical Mechanics, Fall 1999 Transcript – Lecture 9 The exam on Wednesday will cover our first five lectures and the first two homework assignments. And so I list here the topics the way we discussed them. Of course, it is not possible to discuss all of them today but I will make a selection. I recall that we discussed scaling and we used the interesting example of Galileo Galilei-- an animal, and the animal has legs. And we defined the overall size of the animal as yea big-- we called that "s." And then we said, well, there is here the femur and the femur has length l and thickness d. It was completely reasonable to say well... that l will have to be proportional to S. If an animal is ten times larger than another its legs will be typically ten times longer. Since the mass of the animal must be proportional to its size to the power three it will also be proportional to the length of the femur to the power three, and then came in this key argument-- namely, you don't want the bones to be crushed. Which is called "yielding" in physics. If I take a piece of concrete, a block of concrete, and I put too much pressure on it, it starts to crumble. And that's what Galileo Galilei may have had in mind. And in order to protect animals who get bigger and bigger and bigger against this crushing, we argued-- and I will not go through that argument now anymore-- that the mass will have to be proportional to d squared, which is the cross-section of the femur. And so, you see immediately that d squared has to be proportional to l to the third so d must be proportional to the length of the femur to the power one and a half. So this would mean that if you compare an elephant with a mouse the elephant's overall size is about 100 times larger than a mouse. You would expect the femur to be about 100 times larger, which is true.But you would then expect the femur to be about 1,000 times thicker and that turns out to be not true, as we have seen. In fact, the femur of the elephant is only 100 times thicker, so it scales just as the size. And the answer lies in the fact that nature doesn't have to protect against crumbling of the bones. There is a much larger danger, which we call "buckling." And buckling is the phenomenon that the bones do this and if now you put too much pressure on it the bones will break. And if that's the case, you remember that, in fact, all you have to do is you have to scale d proportional to l, which is not intuitive-- that's not so easy to derive-- but that's the case. And so the danger, then, that nature protects animals against is this buckling, and when the buckling becomes too much then, I would imagine, the bones, at some point in time-- well, these are tough bones, aren't they?-- [snaps] will break, and that's what nature tries to prevent. So that was a scaling argument. And let's now talk about dot products. If I look there... I scan it a little bit in a random way over my topics, so let's now talk about dot products. I have a vector A... Ax times x roof, which is the unit vector in the x direction, plus Ay y roof plus Az Z roof. So these are the three unit vectors in the x, y and z direction. And these are the x components, y and the z component of the vector A. I have another vector, B. B of x, x roof, B of y, y roof, B of z, z roof. Now, the dot product... A dot B-- also called the scalar product-- is the same as B dot A and it is defined as Ax Bx plus Ay By plus Az Bz. And it's a number.It is a scalar, it is a simple number. And so this number can be larger than zero-- it can be positive-- it can be equal to zero, it can also be smaller than zero. They're just dumb numbers. There is another way that you can define... You can call this method number one, if you prefer that. There is another way that you can find the dot product. It would give you exactly the same result. If you have a vector A and you have the vector B and the angle between them is theta, then you can project B on A-- or A on B, for that matter, it makes no difference-- and that projection... the length of this projection is then, of course, B cosine theta. And so A dot B... and that is exactly the same. You may want to go through a proof of that. It is the length of A times the length of B times cosine of theta. And that will give you precisely the same result. What is interesting about this formulation, which this lacks, that you can immediately see that if the two are at 90-degree angles or 270 degrees, for that matter, then the dot product is zero. So that's an insight that you get through this one which you lack through that other method. Let us take a down-to-earth example of a dot product. Suppose A equals 3x and B equals 2x plus 2y, and I am asking you, what is the dot product? Well, you could use method number one, which, in this case is by far the fastest, believe me. Ax is 3 and Bx is 2, so that gives me a 6. There is no Ay, there is no Az, so that's the answer. It's just 6-- that's the dot product. You could have done it that way.It's a little bit more complicated but I certainly want to show you that it works. If this is the x direction and this is the y direction-- we don't have to look into the z direction because there is no z component-- then this would be vector A and this point would be at 3. B... this would be 2, and this would be 2 and so this would be the vector B. And it's immediately clear now that this angle... 45 degrees. That follows from the 2 and the 2. So if we now apply method number two, A dot B. First the length of A, that's 3, times the length of B, that is 2, times the square root of 2-- this is 2, this is 2, this is 2... square root 2-- times the …
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