Full transcript of “1.4 | Cantilever | 16 marks | PH8151: Engineering Physics : Unit : 1 : Properties of Matter || VIP”. Click a timestamp to open the video at that moment. Need another video? Use the YouTube transcript generator.
0:00 [Music] Hi everyone, good morning. What is the topic we are going to look at now? Cantilever [Music] Okay, very very important 16 Mark Costin. Either a torsion bar will come, or a cantilever will come. Otherwise, you'll have to write it down, either by asking for it or by making up something else . Okay? Tarzan in the first half , and Cantilever in the second half . Okay. So, we said what would a cantilever look like? Please give me another version of this cost. Somehow we fixed one end in our modulus. They even asked if we were loading the other end. So, what does a cantilever
0:45 look like? One end comes and this is the tail, and that tail comes and it becomes an end and two big ones. What else will they do in the end? They will do mass advertising, meaning they will come in droves and advertise. Okay, let's make our own 50 g each . When we do that, we stick a needle in it. Because what will we find with that needle? We will use a microscope to focus on that top and find the depression that we have with it . Let's see how much it goes down. Does it bend every time you add weight? Is it a correct bend? If it bends like that, how much does it go down? Okay, okay, look, how does the force act here? This is how it works, right? Because here I'm giving a load of what I don't give. When you apply load like that, the force comes and
1:32 acts downward. Okay, this is how the Force acts. How does the reaction act? If so, it acts in the opposite direction at this point. There are a lot of reaction correctors on YouTube, just like there are a lot of funny videos . Everything is in trouble. What kind of bunny videos are you watching? They'll come in and tie a rope and put a water can on it . They will fill the water can with water and make bubbles . Both hands are hanging down. What happens when we are stuck like this? What are we doing to ourselves? What are we pulling towards? As soon as they cut off the water, they'll start beating you up and saying, "What the hell is going on?" You're watching that comedy. So it's the same. So when you apply a force load here, the reaction will be upward at this point
2:18 . But because it's fixed, it's stable there. If you don't make corrections, you'll be stuck with the clock. When adding the correct mass, it will rotate clockwise. So this is the cantilever. Okay? What do you mean by this? Equilibrium condition Under equilibrium condition What do you mean by internal bending moment Internal bending
3:03 moment Equal external bending moment Okay, so that's all, starting with talking to the candle. Okay, what do you do when you do this practically? So, you wo n't do this practically on the lab, but how is the practical experiment? So, a needle would come up and fix it. We would come up and put a clamp on it and lock it on the table. We would lock it even if we didn't have a table. If they had done it, they would have locked it. When we were here, we would have brought a microscope. Okay, let's bring a microscope and take a reading and make a note. Okay, in the same way, every mass is
3:52 added, what happens when the needle comes down? It will keep going down. It goes down and down under the microscope, but when you look at it under the microscope, it goes up and down. You know how to use a microscope . You can try it in the lab and see for yourself. Then you will understand. What am I saying? When you look at the needle under the microscope, the needle goes up and down. We'll make it into our correct publication, bring it to the center point, and then we'll come and read it and make notes. Okay, so when we read it like that, how come you're coming to Mass and adding it ? First of all, you just added an anchor. That means this will be an anchor. We will be doing our weightlifting at Anchorage. There will be an anchor. Then add 50 g, then 100 g, then 150 g, and so on . When there is only an anchor in the first anchor, what is the reading?
4:40 Okay, note the reading. What is the reading when you put the weight on? Then this reading? What is the difference in this reading? That is the depression. Okay, difference depression. Okay, so divide that depression by mass and you get m/y. Okay, m/ small y is the depression. Okay, so find the depression. You find the modulus of the angle. Okay, okay. Okay, this is the introduction basics. Okay, next, let's go to the big derivation part. Okay,
5:27 what do they say in the derivation part? Then there is a fixed place in that place. L is the length of a. They're taking candy livers. Okay, L. A candy liver that's about that length. If you look at that candy liver, this is the first and last end, O, and A, okay, so what do you know from this? O, and what is L, okay, okay, now I'll add weight here. Okay, wait a minute, it'll go down. It's been like that for so long. That means
6:15 okay, so I've come this far by adding weight from here. I'm writing this as if it were my own. We are waiting for you. Okay, so oa comes oa, when you add weight, okay, now I'm taking a point at this point, okay, that point name comes, I'm taking a point like that, okay, from that pink point, from this ola to this P, I'm taking a point like that, okay, from ola to the point where B is, what x is that, okay, from ola to the point where B is, what x is that , then how far is A' from P ? If OA comes L and OP comes X, then how much will P A
7:02 be? L-X Very good. So, this is L-X, okay? Okay, is the distance dx from the point p ok? I'll take a point called Q at a distance dx . Okay, now that you've seen this, what shape will it be? If the organ is in the shape of an arc, it will definitely be in a circular shape. So I'll find a center point for that . Correct, I found the center point by drawing a circle like an arcana. So I'm going to add everything to this center point . I am joining the joint wing. Okay, I joined. Now if you come and see this angle d θ
7:50 okay okay now what do I do now I'll draw a tangent to the center okay now this is B this is B I'll draw a tangent to it what will the tangent be 90 degrees correct so I drew 90 degrees when someone comes to that place and I say like that okay is there a similar cq or not I'll draw a tangent to it. Okay, that's it, okay, do you understand how I drew the tangent ? If you look at this distance, the distance between t and s is dy, and the distance between s and a is dy, okay, you
8:35 think it's big, so dy is right, do you understand how to draw this diagram? Okay, now this is d θ, so this is also d θ. Similarly, if you look at this angle, d θ is okay, what angle is this, θ, and this is also θ, right? Does this radius exist or not? That is, now we draw a circle here, so this will have a radius. That means cp comes with a radius and we're going to put ours in it, so we're going to make our sauce with the value R. Okay, so this is a diagram, Candy Libre's theoretical diagram. Okay, this is how it should be put. We will draw the OA and add mass weight to it until it reaches you. It has come all the way to Soda. Next,
9:22 I'll take a point-to-point approach. If it is at a distance of x, then the remaining distance is l-x distance. Then I take a point at another distance dx, say q. I draw it and take a radius, and it comes out like a line with a radius of rn . If that distance is d θ, then I'll take a tangent to both of them. So that's also d θva. That's it. Right? This is what we drew in our drawing. Now, we've seen here, haven't we, internal bending moment = external bending moment, so that's what we 're going to look at now. Here, if you look at the internal bending moment,
10:18 internal bending moment is equal to y, then the internal bending moment is equal to x. Okay, this is the internal bending moment. Similarly, external bending moment is equal to x. Okay, this is equation number 1. This is equation number 2. Okay, what will we do when we expand it? Under equilibrium condition, internal bending, external bending. Okay, so I will equalize the first equation and the second equation. First equation =
11:03 second equation. What do we get? y/a = y, then I want the radius. Okay, then what will I get? r = what will this r go there? Then the rest will come down. So those are the ones. Simple yig/ w e-x Okay, this is equation number 3. Okay, now let's look at the diagram in this picture. In the diagram, what is the angle pcq? d θ. Correct, so the angle
11:49 pcqv= = d θ Okay, okay, then we need to know what is the value of pq. What we already said is pq, let's take dx as the distance. So, pq= dx is okay? Okay. Okay, instead of saying dx, can we say it another way? We already looked at our circle to find its circumference. What about a semicircle? What about the full circle? What did we find when we looked at the quarter circle ? We saw that 2πr is like that . So using the same technique, if 2π is the angle, what is the radius? Okay, similarly, here, what is the radius, what is the
12:39 angle, what is the angle, d, θ, those are the same. Okay, okay, I mean, what are the 2 things here? Okay, let's use that technique. I've already taken the circumference of a circle in a class. I would have taken it in TABC Mathematics. For those who haven't watched the video about the area and circumference of a circle, go watch it. Okay, we tried that technique. Okay, so now I need d θ from here. So dθ= what comes down? It's dxv/ah. Okay, dθ= dx. Now I'll take this equation number 4, the
13:26 third equation, and substitute it into the 4th equation. If you don't prove it, tell me what you get. Let's see. That means, d θ= dx. What is the value of the divide by y? Divide by e- x. Okay, this is below, below wl-x, can you write it on top like this? Will it turn upside down? So write it down and let's see. Let's say d θ = w e-x e dx divide by y aij
14:18 ok this is equation number 5 ok ok now how we took this angle here same way we take an angle here what is it then q a e ok ok q a e s we take such an angle in this triangle same way what is d θ come here we found our dθ here for this hunger let's find dθ same way this qa what is d θ come dθ= what comes d θ= sin θ what shall we say opposite side div hypotenuse so here opposite side what is dy correct so dy here like dx put here we put dy
15:05 next here what is radius come here then what is this radius only l-x ok ok so l-x ok ok come here dθ is very small ok d θ is very Small SoVR taking dy= one here I will take this dy take dθ I will take dy so dy= l-x and dθ Okay dy= l-x and dθ equation number 6 Okay now dθ answer this subsitute that means I
15:54 will hang the equation and put it in the equation okay subsitute what will get dy= l-x will remain the same. What is dθ? What is w e- x e dx divide/ y iig Correct How can we simplify this further and write dy=da e- esco dx divide/ y iig Okay this is equation number 7 Okay, so we have got the value of dy but what is the small part of dy Correct this is just this minimum
16:41 of part but now I want to know what is the total cantilever depression then what should I do I definitely need to take this value integration Correct if we want to make it bigger or smaller we will differentiate if we want to make it smaller or larger we will integrate. So let's do integration. I rub this place okay now let's do the integration look at the total okay for the total integrating
17:30 equation number 7 okay let's do the integration limit comes to 0 to a okay let's do the integration because what is the total length l that's the length right okay let's do the integration integration a dy=dvai p yij who are these constant values right next integral 0 to l okay l-x e dx okay let's do the integration dy we get y right next
18:21 integral yij in integral 0 to L l-x^2 correct okay now l-x^2 is what shape is it in a- b^2 formula. a- b^2 What is the formula a^2 - 2ab + b^2 correct e dx Okay, so now we are going to take the integral for this equation separately. Let's take y=dl p k yij inkara 0tu l
19:06 ^2 dx - 0tu l 2lx* dx + 0tu l x^2 dx Okay, close the brackets. Now you understand how to take the integral. Let's take y w/ y as the first term. Look at what is the constant value. So I took it outside dx and integrated it. I get x. x oda limit 0du I substitute ok next - 2L come what is
19:54 constant remaining who is x dx x train integrate I get xs2 limit 0du el next p eska incorretion x^ div p3 get limit 0du el ok so substitute again let's see what we get w/ yig e upper limit- lower limit lower limit support 0o ayrum so only upper limit support l^u is correct ok m 2da what will happen cancel remaining who is x lower limit
20:41 support 0o ayrum. So if you just substitute the upper limit, l^2 already has an l. So this is a l^ . Plus upper limit substitution panna l^/3 lower limit substitution panna. So you get it like this . Here + εl^- ε^ is the cancel remainder l^/3 We don't need this now So I'm calling this page now Okay [Music] Look carefully and analyze [Music]
21:29 Okay [Music] Okay Now how can we simplify this and write y Smallv=dl L Cube Divide b 3 y Ij Okay The equation number 8 So this is the depression of cantilever
22:16 at free end Okay What is the free end? Is that the last fixed end or the depression at the free end? Okay Okay This is the end of this Okay Okay Okay But even in this, maybe there are two cases What if that cantilever is round in shape? What if it is like a round rod What if it is like a cylindrical rod Will there be? What would it look like if the same cantilever was rectangular in shape ? Let's see, okay, first we took case number
23:02 one, rectangular cross section, how is it okay if there is a rectangular cross section, is there an iig value or not? If we substitute 12 for that iig value, what will we get if we substitute 12? How will the equation change in 8? y=tall e^v/ 3y substitute substitution pd^/
23:49 12 correct so what will happen if we cancel it? This is 4 so what will happen to this 4? Then we go to y= 4tall el^ divide by y pd so that it will be correct so this equation number 9v is case number one, if there is a rectangular cross section, what if there is a circular cross section? Circular cross section OK, if there is a circular cross section,
24:34 how then what is the iig value we are going to substitute for this? So let's substitute this for 4/4. Let's see what comes up. So equation 8 becomes what comes up. y=t ^ / 3y. Substituting πr^4/4 instead of iig, okay, there is no chance of it canceling out. So I'll just carry this 4 over. So the depression is y= 4 wlq/ 3y πr^4
25:21 Okay, okay, from here, if we want the modulus of the angle, what can we do? If we want the modulus of the angle, we can take the malus of the angle and get y. Because we can find our depression using the experimental method. Hand depression, okay, let's find the needle coming down and going down using a microscope . If we use a traveling microscope to find the S modulus, we can put the S malus there and bring it to y there. If you want y, put y there and S malus so we can find whatever we want, okay, so this is equation number 10 completes with this, okay, with this, the cantilever is finished, according to the calculation, we have finished the first unit, okay, the first unit is complete, okay, the
26:11 rest is practical, that is the practical video, I have already posted, I have already done the modulus of the uniform meeting, the practical method is full, I have done it in the lab, how can we do the same experiment, how can we do torsion in the lab, how can we do it in our lab, how to take notes for it, everything is in the video, I have already posted such a video, I have put it in the folder, go to the playlist and watch it, okay, so this is the unit one finish. Okay, let's go to Unit 2 in the next class . Unit 2 is very, very interesting. Okay. Unit 2 people come and say, "What are waves and fiber optics?" Waves are light. Waves are fiber optics. Optics are light. Fiber optic cable. Fiber optic cable. Fiber optic cable. Okay, so that would
26:57 be very interesting. How is a cable made? That's not all, Papoma. We will look at cable sizes, what cables are used everywhere, how laser light is transferred, how laser light is generated from one place to another, and what types of laser lights there are in the next class. So don't miss his classes, okay, thank you. [Music]
