INSTRUCTOR: In each of the following situations, we determine which coordinate system is most appropriate. Describe how we would orient the coordinate axes. There could be more than one right answer or how the axes should be oriented. But we select an orientation that makes sense in the context of the problem. Note-- there's not enough information to set up or solve these problems. We simply select the coordinate system

First, we have part a, find the center of gravity of a bowling ball. Now we have an image here of some bowling balls just for reference. They don't actually give us any context, really. But because a bowling ball is spherical, spherical coordinates might actually make sense. So I'm going to go ahead and label this as spherical.

Now how we might put the axes-- and again, there are multiple answers-- could be multiple answers for these that make sense. But one way to orient this would be to-- I'm looking at it and bowling balls have holes. I might pick the thumb hole or whatever and say, I'm going to stick one of the axes there. And the other two axes really wouldn't make any real difference. So choose how you want to put those. But that might be a good idea.

Also, bowling balls have a weight block inside of them. And so you might choose an axis of symmetry of that weight block for one or more of your axes.

Part b, determine the velocity of a submarine subjected to an ocean current. There's not exactly any rotation with a submarine. They turn left, right, north, south, east, west. They decrease in depth, increase in depth. So there aren't really any symmetries, so it might actually just make sense to use Cartesian coordinates for that.

Now how we'd orient the axes-- it might make sense to make the x-axis being east to west, and the y-axis being north, south, and z being the Depth that would make a lot of sense to me. So that would probably be the choice we'd make there.

Part c, calculate the pressure in a conical water tank. And again, we have a picture of a traffic cone to see what a cone looks like. And when we're thinking about something that's conical, such as this water tank, it occurs to me that a cone is really, well, it's a cylinder, but it's been compressed in a linear fashion. We have a slant to the sides of our cylinder. So it would seem to me that cylindrical coordinates would make sense for that.

Now, as for our orientation-- as for our orientation, it might make sense to make the z-axis go through the axis of the cone-- the center of the cone, as it were. And then, again, the other two, I don't think it's going to really make a difference. We have an x and a y-axis, and they can just be oriented to be perpendicular to that axis of the cone.

If there's a bottom of the cone, since it's a conical tank-- so at the bottom of that could be the center. That would be our origin. And we could have the two emanating from there at 90-degree angles to each other.

All right, find the volume of oil flowing through a pipeline. Here is a picture of a pipeline. Pipelines are, as we just said there, they are-- it actually doesn't say that, but they're horizontal in nature along the ground. So one, it's cylindrical. So cylindrical coordinate system probably makes the most sense.

When we think about the axes, though, it might make sense to put an axis-- I would think the x-axis or possibly the y-- we could put that being directly in the center of that cone. And if we're thinking about the direction that our pipeline's facing-- I don't think that, yeah, the directionalities aren't really going to make a difference. So the other two axes are almost irrelevant. But we would want to put them, if there's a start to the pipeline or something, we could the origin there and put the other axes from there.

So I'll go ahead and call that cylindrical. And, again, because the object we're looking at actually has a cylindrical shape to it.

Part e, determine the amount of leather required to make a football. I'm going to scroll down to the fifth image here that's the football. And they are this, well, shape. There's some rotational symmetry about them that's important. So let's see. We have some sort of rotation. I can tell you one of the axes is going to be going directly through the axis of the ball. The origin could be the center of the ball or even one of the ends.

And because we only have rotational symmetry about one axis, we'd probably want to go with cylindrical, as well, for this one.

OK, so hopefully this helps. As you think through what sort of axes or what sort of coordinate system what I want to use for a situation, hopefully this will get you pushed in the right direction.