WEBVTT
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welcome to the video reviewing how to convert from Cartesian
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coordinates to cylindrical polar coordinates. So in the previous
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video, we did just to polar coordinates. Now
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we're gonna add in this Z thing that I talked
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about before we have this Z axis in the vertical
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direction, which we can think of as multiple of
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these polar planes stacked up on top of each other
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. So remember again are variables are are Seda and
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Z in this case now, before we've been thinking
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just in the two dimensional Cartesian, though obviously we
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can't really go from the two to the three very
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well. So I'm gonna throw in our picture from
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the three dimensional Cartesian, which before I label the
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axes as X, y and Z. Okay,
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so those are three variables Now, I should say
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that this isn't necessarily the most common way to think
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about the orientation of of the three axes. In
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fact, very often. Instead, we would label
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them as X here. Why here and see here
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And in fact, this is helpful comparing it to
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our cylindrical polar picture here in that we have the
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X Y plane in which data is measured is in
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the same more or less in in the same plane
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as the cylindrical one here. So let's think about
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it. Uh, this way with that second set
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here now, thinking about how to convert them from
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our cylindrical pull our Cartesian toe are cylindrical polar.
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It's gonna look a lot like it did with E
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with the Cartesian thio simple polar Because all we're gonna
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do is we're gonna say the Z becomes Z So
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we're going from X. Why Z into are Data
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Z And as you might have guessed, Z is
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equal to Z. So we can use the exact
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same formulas that we used before where x is equal
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to r cosine theta. Why is equal to our
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science data? And then we have our is equal
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to the square root of X squared plus y squared
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and feta is equal to the inverse tangent or arc
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tangent of why over X. Remember, you have
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to be careful about which quadrant here in with that
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conversion. Now coming back then over here, we
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could say, Well, let's let's pick a point
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. Let's ah, pick a point on this one
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. We'll say we'll go on to 34 out.
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Okay, so we have an r of four.
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We'll pick an angle of 30 degrees and then a
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z of to or actually, I guess that sea
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of one in this picture. So z of one
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. So that being said in order to get to
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X, Y and Z, then what we need
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to do is we need to say Okay, well
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, I have four times the cosine of 30 degrees
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and then comma four times the sign of 30 degrees
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. And then what? And that's all you have
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to do in the reverse direction. If I were
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to pick a point here, uh, let's say
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I've got a point that's at, um let's see
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. It looks like it's maybe one in the X
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. And then we'll pick three in the UAE and
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five and the Z in order to go into our
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r theta Z coordinates that I would have square root
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of one squared, plus three squared, then the
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arc tangent of one of 3/1 notice. We're in
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the first quadrant there, so it's not a problem
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to just write it down like that. And then
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Z is five, so it looks pretty much identical
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to our polar coordinate conversion. All we're doing is
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we've added in this five where Z is equal to
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see now. The reason you'd want to use cylindrical
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coordinates instead of the spherical polar coordinates is that one
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. It's a little simpler to use because you have
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this Z equals e concept. And then also because
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there's a lot of things out there that actually have
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a cylindrical symmetry. Things like wires, um,
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or cylinders themselves with you thinking about a can rolling
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down a hill or something, which is a classic
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physics problem. Then you can use the cylindrical symmetry
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to solve for some things like the moment of inertia
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, or to use some of the more advanced calculus
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principles later on, when we get into electromagnetism.