# Mechanics of Materials: Exam 2,  Problem 1, Torsion with Gear Ratios

https://youtu.be/ps1BdJz4bBU?si=Av5V_NN3-bGDbJzT

## Untitled chapter

OK, Yay. Welcome back. We're studying for Exam 2. This is problem number 2. Now this is problem number 1. This is problem number 1 on .... It's in chapter 5, okay? Level 5. Torgent. Now I have picked a problem here. It's very similar one you've seen. I've changed it a little bit. There's not a whole lot of these to pick from or to invent and so I've changed it up a little bit. We're going to work it again. 

Okay, but I've asked you a whole bunch of different things. Number one, find the reactions at the wall at A and B. Okay, find the reactions. Number two, find the shear stress, right? Here do the torsion in shafts A, B and C, D. A, B, C, D. 

Number three, find the angle of twist of gear B. So a lot of equations at play here, right? We've got, number one, we've got gear ratios. Let's put that with this, that's two. That's terrible. Okay, that's just terrible. Let's do that again. Start over, Hanson. Gear ratios. And in this case, what's our gear ratio in this one, right? 

This one here is four times bigger than that one. That was 50. That one's 200. So you've got a four to one gear ratio here. Okay, so how does that work? We've got that. We've got tau, shear stress, due to torsion, which is T C over J. Remember that guy? We've got to figure J. Remember what J is? J is for round shafts. Pi on two are to the fourth. We've got angle of twist, phi. And remember, what's the units for angle of twist? Bam! It's in radians, isn't it, right? T L over J G. So the one torsion equation, we pretty much don't have up here. We've got everything else in the world. But the one thing we don't have up here is power transmission. Okay, and I'll just quickly put that up there for you. Power transmission P equals T times omega, or as my students call it T booty. I don't know why they call it booty. I don't know. But this is just a fancy way, usually. It's a fancy way to have to determine what is T, right? Because it's P divided by omega. And the T goes right there, right? So it's just a fancy way to make you do something else to come up with T, to put in that equation and solve that, right? The one tricky thing that you've got to remember, same tricky, but one HP because America equals 550 foot pounds per second, right? That's the conversion for horse pairs, okay? So you need to remember that. So that's all the technology we'll use, except for this, we won't need that. But there it is. But that's really all the technology to solve a torsion problem on your exam. Let's put it to work and solve this rescue here, okay? 

Now I think I want to start this problem off just like all good problems, right? With a good old-fashioned free body diagram. And I'm going to draw free body of this shaft. And then I'm going to draw one of that shaft, okay? So big gear first, okay, big gear first. 

Yes, I didn't put teeth on my gear. So just get over it, okay? Really? I mean, serious. Okay, let's see here. Here we go. Here's little shaft. All right, what do we got? Let's get a red pin again for our stuff, okay? And what we got here is number one, I've got this, 500 Newton meters. And if I'm looking at this way, that's rotating me. Well, that's counterclockwise, right? So what that's going to do, number one, for the wall is wanting to go the opposite direction. I'm going to call that our D, the reaction at D, the reaction moment at D. Now, as this thing wants to rotate this way, this other gear over here says, I don't want you to rotate, I'm going to resist and push on you back the opposite direction. So right here, there's like this pushing force, it's a tangential force that's trying to spin that gear back the opposite direction, isn't it? And that comes from the teeth over here. Gear push on teeth, teeth push on gear, right? Now over here on this gear, I've got exactly the opposite of that guy, okay? Pushing on that gear, trying to rotate it that way, which means it over here at the wall, I've got our A trying to, it has to oppose that, right? This is trying to rotate that shaft. Again, if I look at it, counterclockwise, this has to rotate clockwise to prevent that. So I've got those things going on, okay? Now, that should allow me to write some, some of the moments for these two guys here, okay? So number one, we'll try this guy here. If you add up all the moments on this thing, it should equal zero, because it's not spinning, because it's wall, wall, it can't rotate, right? So what do you have trying to do me here to spin me? 

Okay, so I've got our D, which goes, R, R, D, that's positive, and then minus 500. Now that's a Newton meters, okay? So be careful this next bit. I've got F, but F is not a moment. I've got to multiply it by distance. Well, the distance is here, right? Four times distance, and that distance is 25, isn't it? Okay? So that's going to be plus 0.25, no, no, no, no, no, 0.025F, isn't it? Right? Because it's a radius, this has a diameter of 50, the radius is 25, right? Because we just want from here over. Well, there you go, okay? Now, well, let's just leave that alone, let's come over here, do the exact same thing for over here. What do we got over here? Well, I got this guy going clockwise, so minus RA, and then plus F times, how far away? Well, the diameter is 200, so the radius would be 100, so that would be 0.1. 

Now, why did I change this to meters? Because that guy right there was in meters, right? So now, my units are all the same. So, RA, well, let's just do F, shall we? F equals calculator. So, F equals, it's moved the RA to the other side, and then there's a 1 in front of the RA, right? Divide 1 by 0.1, it goes in there 10 times, doesn't it? 10 RAs, okay? Let's go over here, now, and substitute F in for over there. 

So, we get RD is equal to 500 minus 0.025, right? I just moved this stuff to the other side, times F, which is 10 RA, okay? And so, well, we can do this, can't we? 

This is going to be RD is equal to 500 minus 10 times that is just 0.25 RA. 

So there, there's a relationship between RA and RD. That doesn't help at all, okay? Well, shoot, maybe this will help, okay? Let's look at this guy right here, the gear ratios, okay? 

Now, we have to make this statement true. We've got two gears, we've got gear C, we've got gear B, okay? Gear C, gear B, B, C, okay? 

Now, we know these, the gear ratios is 4 to 1. Let's just say this, if this guy here rotated one time around, what would this do? Well, it takes four of those, this go around one time, so that goes around one time, that only goes a fourth. So, to make this statement true, right? The rotation of gear C equals the rotation of gear B. How do I make that statement true? And you make that statement true, here's, here it goes. The big number in your gear ratio, 4, okay? Goes on the big gear. So, put A, guess what? 4 right there. Now, does that make sense to you? It might not make sense, right? If this makes one rotation, then how much does B make? If this makes, if this is 1, then, then what does B divide 1 by 4? It's a fourth, isn't it? So, I can just do a quick little substitution in there and kind of sanity check my relationship there. So, this relationship here is what I need to do next to kind of come up with a second equation for this here, okay? 

Now, one thing I need to know here is I need to know G, okay? And I'm going to give you G. G, the sheer modules of rigidity, is for this material, 200 giga passgales. I really don't have to give it to you, but I need it for, well, I need it for you in a minute here, okay? 

Okay, so now, here's our figure relationships, okay? So, phi, what is phi? What is phi? Well, it's TL over JG, isn't it? So, what I'm going to have is this, I'm going to have TL over JG for this shaft is equal to, or for this gear rather, TL over JG for this one up here. But what do we know that B is four times? So, this got on here, it's four, let's say this one like this one B, this is gear B over here, okay? There's a four there, there's a one over here, right? Okay? This is gear C. All right, next, okay? What? Let's just go for this guy right here. What is the torque that gear C is going to see? Okay? Now, what we can do, remember, cut it in half, so let's cut the beam in half and let's cover it up. Now, how much torque is in that shaft right there? It's RD, isn't it? Okay? So, RD counts, gotta make our units work out here again, times L, which is 2.25 meters, let's put everything in millimeters, okay? 2,250 millimeters, divided by J. Ooh, we got two J's here, don't we? Okay? This first J, this guy right here, is, because J is pi over two, R to the fourth is pi over two times 15 to the fourth, that's mill up meters to the fourth, okay? Got it? Okay, now let's go to this guy over here. We got our four there, don't forget about that guy, times T, what's the torque in this shaft? Again, cover it up, what do you got? It's RA, I know, right? RA times L of this guy, which is 1200 millimetres, okay? T L over J. Oh, no, what's J? Same thing, pi over two times, that one's 40, so the radius is 20 to the fourth, millimetres to the fourth, okay? That's going to give us a relationship between RA and RD, isn't it? Let's see what that gives me here, on 22.50, oh man, divided by 15 to the fourth power equals times 2 equals divided by 3.14159 equals 0.028294236, I'm going to put them all in there, okay? Now let me do that side over there, 4 times 1200 equals divided by 20 to the fourth power equals times 2 equals divided by 3.14159 equals 0.019, okay? I'm going to take that and divide it over here, okay? I'm going to take that number and divide it into that number, okay? So 0.028294236 divided by answer is 1.48, so that whole thing boils down to 1.48RD equals RA, okay? So I'm going to plug that business in for RA over there, so what are we going to get? We get RD is equal to 500 minus 0.25 times 1.48RD, okay? So 1.48 times 0.25 equals 0.37, now let's move that to the other side, so I got 0.37 and then I move it over here, there's a one in front of him, right? So that becomes 1.37RD equals 500, RD equals 500 divided by answer equals 1350, no? 500 divided by 1.37 equals how about 364.96? 

Okay, now how can I get RA? 

Well, I can plug that number into there and so for RA, or I mean RA should just be that number times 1.48, right? 

Times 1.48, you should get the exact same thing, either way you go, 540.14, so let's go 540.1, and that is, what is that unit? That is Newton's, right? Newton meters, okay, boom. 

Okay, I think that that there, there's a reaction at A and D, that is the answer, the part A, what's the answer to part B? What's the shear stress in A, B and CD? 

Well, here's our equation right here, let's make us a little bit of room here, let's do it right up here, okay? We got four, let's do A, B. The shear stress is Tc, what's the torque in that section there? The torque in that section is RA, so 540.1 Newton meters, now I don't want that meter in there, let's get rid of that guy, there are 1000 millimeters in a meter, and then we've got C, what is C? This is from the neutral axis to the outside of the part, which is a radius, which is 20 millimeters, okay? The divided by, this is Tau AB, divided by, well J, okay, so that's just Pi, on two times 20 to the fourth millimeters to the fourth, right? And look, millimeters to the fourth, there's two of them, cancel out with two of those, the meters cancel out, it's going to leave me with Newton's over millimeters squared, so my answer's going to be in what, you'll, mega pass counts, that's right, okay? So 540.1 times 1000 equals times 20 equals times 2 equals divided by 3.14159 equals divided by 20 to the fourth equals 42.98, let's just call it 43, right? So Tau AB equals 43 mega pass scales, groovy, okay? What about the next one? I'm going to erase your equations of the day over here so we can finish our problem. Oh, no, we're not doing what we're doing, okay? So Tau BC equals same equations, okay? But now we have what? Our D, the torque has changed, so that's going to be 364.96 times 1000 times C, which is 15 divided by Pi over 2, 15 to the fourth. So Tau BC is equal to, here we go, 364.96 times 1000 times 15 equals times 2 equals divided by 3.14159 equals divided by 15 to the fourth equals. 68.84. Now, sanity check, you might ask yourself, how come the stress is so much higher in this one than that one? Do you know the answer? I think I do. And it's because this is a much smaller cross section, right? So it's going to be subjected to a higher shear, that one's got a lot more material to carry the load. That one's going to be worse, isn't it, okay? All right, and now what do we got left? What do we have left? Oh, find the last one, find the angle of a gear B, okay? So phi for gear B is going to equal. Well, however much this shaft right here is going to rotate, is how much B is going to rotate. So I just need to do phi for this section of shaft right there. And remember, it's TL over JG. So T, the torque in this shaft is RA, which is 540, 0.1, okay? And again, that's Newton meters. So I'm going to multiply that again by 1000 mms per meter. And then L is in what? 1.2 meters. I'm going to call it 1,200 millimeters, okay? Divide it by, because this, what's the unit for phi? Radiance. So all of my units, every single unit in here needs to cancel out, right? 

Okay, over J, which is pi over 2, I'm just 20 to the fourth millimetres to the fourth. That's a lot of millimeters. And then G, right? What is G? Gigapascals. Let's see, I don't want it in that. I want it in megaascals, because mega is, right? 200,000 is Newton's over millimetres square. There's my Newton, there's all the millimeters. Everything cancels out. No units left. That's radians. So last answer of the day, remember, this is in radius, not in degrees. 540, oh, clear? 540.1 times 1000 times 1200, well, not 12,000 equals. And then times 2. And then divided by, what? pi, 3.14159 equals divided by 20 to the fourth, 20 to the fourth. And then divided by 200,000, 123. 0.01289, 0.0129 radians. And there you go. Monster problem, y'all, monster problem. Good problem, though, because it has all of the sections that we need to test you over torsion in one problem. 

Man, that could show up on the exam, couldn't it, okay? All right, I hope this helps. I hope you make 100 on your test. Stay tuned for the next sample problem for the test, okay? Let's go.