[MUSIC] Hi, and welcome to module 19 of Mechanics and Materials I. Today's learning outcome is to show that the principal planes are 90 degrees apart, or what we'd say normal to each other, on mutually perpendicular planes. And so here's where we left off last time. We had our stress transformation equation for inclined planes. We found the angle where max or minimum normal stresses occur. And we said that the angle to those, those are what we call the angle to the principal planes. And that's where we see maximum or minimum normal stresses, which are called principal stresses. And we also made the note that shear stress is zero on these principal planes. And so, here's again, is our expression for the angle for the principal planes, it's a tangent function. If I look at a tangent function, the tangent repeats every 180 degrees. So, I go from like minus 90 degrees to plus 90 degrees minus, or 90 degrees to 270 degrees, etc. And so for the values of, a given set of values for sigma sub x, sigma sub y, and tao sub xy, the values of 2 theta sub p are going to differ by 180 degrees. Or theta sub p, the angles to the principal planes, are going to be 90 degrees apart. And so therefore, again, they're on mutually perpendicular planes or normal to each other. And so here's our stress block in the xy frame. We can rotate to our principal planes. We have normal stresses only. No shear stresses on those principal planes and we see that these principal planes are 90 degrees apart. And so, recalling back to earlier parts of the class, I said that we're using three-dimensional, we can use a three-dimensional state of stress at a point. We've reduced it down to two dimensions for plane stress. But I mentioned earlier that I was going to show that an infinite number of planes can be passed through a point but we really only need three mutually perpendicular planes to completely describe this state of stress. And we're starting to see that now because I can arbitrarily turn my stress block and I have two planes for my perpendicular, excuse me, for my principal stresses, in the xy plane. And I'd have a third plane in the z direction. And so, it's all starting to come together. And we'll continue on with this next module. [MUSIC]