[MUSIC] This is module 18 of Mechanics of Materials Part two. Learning outcome for today is to find the angle of twist for elastic torsion again, but for a non-prismatic straight cylindrical shaft, and so here is the problem that we've been working on. Last module we found the maximum sheer stress in each section. Now, we're going to look at the angle of twist. And so we've found the torque in each of the sections by doing the free body diagrams. Now we want to find the angle of twist. How would I start? Where do I start? And what you should say is well, we're going to use the angle of twist formula. Let's first of all put a coordinate system on here again. Call this x, y and z, and so my angle of twist formula phi, the angle of twist is equal to TL over GJ and I will go ahead and do the section from C to D. I'm going to find the twist Of D with respect to C, and so phi of from C to D is equal to the torque and C to D is 5 foot keeps and again I'm going to work in inches instead foot, feet. So I'm going to put 12 inches per foot is a conversion. L is four feet, so I've got 4 feet for section C to D and again, converting feet to inches. And I'm going to divide by G. I'm given G for section CD. This is a alloy and G is 9.5 x 10 to the third Ksi. And then, finally, I have to J. J is pi for a circular cylinder. Pi over 2 times the radius, which is 2.5 inches to the 4th. And if I go ahead and multiply that out, I find that all the dimensions cancel out, it's an angle so we'll call it radians, and I get 4.94 time 10 to the minus 3rd radians. And you can see if I am looking at point D with respect to C, that torque is going to cause a twist in this direction, the negative Z direction. So that's going to be -4.94 times 10 to the minus third radians K, K direction. Okay, I want you now to do the same, let's write down just to be clear, this is the twist, angle of twist of D with respect to C. So WRT is with respect to C. And I'd like you to do the same thing now for C with respect to B, and the angle of twist of B with respect to A and come on back and see how you did. So here is the angle of twist of C with respect to B. It's in the positive K direction. Here is the angle of twist for B with respect to A, and it's also in the positive K direction. So if I want to find this total angle of twist of D with respect to A, I'm going to have to add these together. And so I've got Phi total equals phi A to B plus phi B to C plus phi C to D and that's equal to phi A to B was 1.575 times 10 to the minus third, I'm going to keep the units off here, because I'll just put the units in at the end, and then I've got plus FBC which is 1.85 times 10 to the minus third. And then I've got a minus phi CD, 4.94 x 10 to the minus 3. And so the overall answer is -1.515 x 10 to the minus 3rd radians in the K direction. >> Or another way, if instead of the K direction, you could say That arrow in the negative K direction, because its negative, negative K direction same thing. All right, so that's my answerer and we've now been able to find the angle of twist. And we did the maximum shear stress so we're able to deal with elastic torsion of straight cylindrical shafts that are non-prismatic. And we'll see you next time. [SOUND]