Hello miss. Hello Jackson. How are you today? As think it's great, it good. And so we did fluids and viscosity yesterday and I have some banked questions that we can use to solve them. But I thought wego through hooks law today, hook law elasstic and plastic properties, stress, strain and Young modulus and then we should be able to answer all the questions which I have banked and prepared for. So first of all, hooks law. So do you remember hooks law? Yeah the extension between force and. Extension Yeah Yeah extension Yeah. And what's the what's the relationship between them? Extension equal to force times, extend length. So we could say force is proportional to extension. That's one way to say it. Our force is equal to a constant times extension. And that's for a spring. As long as we don't go beyond the limit of proportionality of that spring. So force is proportional to extension. F equals kx, so we can say force is equal to a constant times extension of the spring, and another name for k is the spring constant. So the spring constant, so the stronger the spring, the harder it is to extend it. So k has units of f per unit x. So spring constant is measured in nutions meters to the minus one. So the thicker the spring the Harit is, to the stiffer it is. So stiffness of spring, for example. So hook law, tell me what hook law is. Again, inwords, expressing these equations is proportional to the load. Good exactly as long as the. Elastic limit is not exceeded, so the spring constant k is the force per unit extension of the spring, and then the area under this graph is equal to the energy stored in the stretched spring, so there can be two versions of that, so the energy storage. They streched. Is. The area under this graph. So it's half the base times the perpendicular height. So 0.5 ops, 0.5 half times x times. Okay, or easier to write it as a half fx, a half, the force times the extension, the area under the graph, you might ask, quiet is not at the rectangle. Why all of this is not shaded well. Some heating occurs in the molecules when a spring is being stretched. So hooks law applies to a spring, whether it is being compressed or if it is being stretched extension. So if you're compressing a spring or if you're stretching it, these two equations apply. And the energy stored. So the hook law, this is if you're stretching a spring or if you're compressing it. So the e, the energy is a half fx. But if we substitute f, if we take kx has f and if we substitute it in here, we have a different equation for energy. So this time, energy. Equals 0.5 times kx. Times x so this becomes a half kx squared. A half kx squared is the also the energy stored in a spring. And I substituted kx for this f and I got a half kx squared. So we've two equations for the energy stored in a stretch. Spring p is equal to half fx or a half kx squared. Okay. So. So the energy stored. Stored in a stretch spring or the relationship between force and extension. So. Now we can have springs that are in series or in parallel. So springs in series, the one over the spring constant is one over the spring constant of p plus one over the spring constant of k. So if we have two springs in series. One over k it's a bit like resistors equals one over K1 plus one over K2. It will. So if you have two springs in series like so that's spring one and then spring two. The total spring constant will be equal to the spring cons, one over the spring constant of the first spring plus one over the spring constant of the second spring. So that's when we have two springs in series. And then the other thing we can do is have two springs in parallel with each other. So in parallel, so in this case, so if again, this is spring one, this is spring two, so the total spring constant K T is equal to K1 plus K2. So this obviously means you'll get a harder job to stretch these springs if they're in parallel, but you'll get more extension if they are in series. So parallel series. Okay, do you remember the meaning between elastic and plastic? No. Okay, so you understand springs in series? Yeah and springs and parallel. So what do we mean by elastic properties? So means if the force is removed, the spring goes back to its original shape. Or length. So elastic properties, if you stretch a spring and then take away the force, itgo back to its original shape if it is elastic. Fantastic stic meetings and the force is removed. The spring stays permanently. So. Think of plasticene. If you stretch it, it stays stretched. Or chewing gum. If you pull chewing gum apart, itstay permanently stretched, so it can be said to be plastic. So with hook law. We have a direct proportional relationship between force and extension up to a certain force. If we go too far beyond the limit of the spring, itstay permanently deformed, the spring will no longer obey hook's law. So itstay permanently deformed. So this is the limit of proportionality. Or the elastic limit. The elastic limit. Okay, so say the elastic limit of my spring was ten nusions. What would happen to my spring if I tried to add 14 newtons to the spring? It will be up front force which 40m Yeah. So if I pull the spring with a force greater than its lastic limit of ten nusions, yes, Yeah, the spring will extend permanently. So permanent deformation occurs. The spring will no longer go back to its original shape. So hooks 's law no longer applies if we go beyond the elastic limit of the spring, the limit of proportionality. So in other words, if I add too many loads, two much. Load. So for example, if you're going shopping and you put things in a plastic bag and you keep putting your shopping in the same plastic bag, itbreak because you've gone beyond the strength of the limit of the plastic bag. Okay. So hoo's law applies to wires. The limit of proportionality usually occurs for quite a small extension because wires are particularly stiff. So if we took a wire, Jackson, like a piece of copper wire, and we attached. A heavy load to that wire. The waste, the wire. Will stretch a little bit. So it obeys hooks law initially. So see this region of a wire. We initially have a direct proportional relationship. So hook law ys obeys is obeyed by wires. So force is proportional to the extension. The limit of proportionusually occurs for quite a small extension, because wires are particularly stiff. Actual shape of force extension graph depends on the dimensions of the specimen under the test, so it's usual to plot a stress strain graph instead. This is independent of the dimensions of the specimen under the test, so it depends on the length and the cross sectional area of your wire. Yeah, okay. Yeah. So tell me what you understand by hooks law. Well about hopes long I know that it is proportional to between force and extension and we know that Yeah and we know that the formula is f equal to K X, which means the and the extension force, the total force equal extension times the strength, the spring constant times times X X being the extension. Yeah so the bigger the spring constant, the harder it is to stretch the spring. Okay. So if you have a spring, a very thick spring, it's quite hard to stretch it. Okay, so hooks law for wars can depend on the. Dimensions of the withe length or the cross sectional area. But sometimes we prefer to know. The property of the material instead of the property of a certain length and cross sectional area of wire. So what we learn about is stress, the wires under and the strain. Now if you remember ent stress and strain, we've come across these before. So the stress is the force per unit area measured in pascals newtons per meter squared. And it's given the symbol sigma. And strain. Is the change in length over the original length l zero. And it has no units because it's a ratio of meters over meters. So no units. Maybe that is why the I. Mean stress string and another extension. Maybe there's another saying it same as. Is then lambda for f divided by a stress? Or sigma. Well, when we define. The Young modulus, the Young modulus is the property not of a particular size of spring or wire, is the property of the material that the wire is made of, so the Young modulus is the stress divided by the strain, so the Young modulus is the force over the area. All over the change in length, over the original length. Okay, so. It's easier to talk about. So Yeah. Riculous. Which is. The property of material. We're talking. About it's a bit like resistivity with electricity. We know resistance. Do you need a charger? No, I just want to open. Okay, let's see. It's a little okay. Okay. So property of some material. I so it's a bit like again, if we think of resistance. It's equal to road times length over area. So row e is the resistivity of the material. The Young modulus is like the how much stretching or strength a material has. So that Young modulus, given the symbol e, is equal to the stress over the strain. So sigma over epsilon. So this is force over area. All over change in length over the original length. So that can be equated to. F times. L nose. Over. Area times change in length. Okay, so the cross sectional area. Of the material times the changing in length up and the force times the initial length. So the Young modulus is a good way of gauging the stress and the stress on a material. So the stress strain, what we can do is plot stress against strain for a material, and we can see. How it behaves. This bit of the graph, our material, let's say, copper wire is obeying hook law. So stress is proportional to strain. We have a proportional relationship. And then so we stretch the spring. It stretches very slightly, not much, and then it suddenly yields. So there's a yield stress point. The wire certainly extends for no apparent reason, and then it stretches quite a lot. If more force is added to it, the maximum stress that the specimen can take here is called the ultimate tensile stress strength uts, the ultimate tensile strength of the material. So after that, it will break. So if the load is removed at c itremain permanently deformed plastically, and then if you add any more force to it, it will break. It breaks. So this stress string graph is very similar to the one that youget for copper wire. So stress again, strain. So if we go beyond this line, the length. Of our wire will never return to its original length, and we'll need to be thrown away because it's too weak after this point. The other thing that we can tell from this graph is the area under a stress string graph. So it's storing elastic potential energy or unit volume. Okay, strenergy elastic energy, but you need volume, okay? So that's the area under the graph, and the gradient of the linear part is the Young modulus. So for this bit, if we got the gradient here, so it's sigma over epsilon, which we know to be the Young modulus of the war. So if that is moof the wire, I think that maybe in this graph, that is the gradient of the linear part. Yes, good. The Young modulus. So Young modulus is defined as the tensile stress divided by the tensile strain. So sigma over epsilon, which is force over area, over delta, l over l, naught. And because epsilon doesn't have units, units for Young modulus or pascals, same as pressure. Or stress. So you can get different shaped graphs for different materials. So Young motorles, the unit of Young motors is yes, okay. So same as how to read that symbol. Sigma Sigma. Yep. Sigma stress. The same as stress. Yes, yes, sigma stress. So it is equal to force over area. Which is the units give newtons per meter squared. Which is pascals. Scso, this looks like our wire, doesn't it? So copper wire stretches a little bit, then it gets weak and then it breaks. So this is the type of graph you get for copper wire. This type of graph, which. Is very simple. It's for a brittle material, like a Crystal, like glass, like slate. What other materials like plastic, plastic use stretches. You exert it, you stretch it, and then it just snaps. So there isn't much else to it. And then for a polymer, so what do we mean by a polymer? Polymer maybe it's like a curve, but not fluctuated and not and not straight line either. Yeah. So rubber and. Rubber is an elastic band is made out of a polymer like rubber or plastic, a plastic material. So let's think of a rubber band. So if you stretch it, it increases in length. And when you are taking the floats off it, it goes back to the original shape. But during some of the unloading and loading, it heats up the molecules. Work has done on these molecules, they've gained some internal energy. They've heated up. So. These sometimes permanently deform the rubber material. So so rubber, copper wire and glass will say glass, it tends to be brittle or. A ruler like this, it tends to snap if you try and stretch ches itjust snap. Okay. What's wrong? Your lesson here? So that's so if we wanted to measure the Young modulus of our wire. We could do it this way. So we could. Stretch or word, Oh, this is a control with this gradious scale on it. And this is our withat. We're going to test by adding a greater load to it. And you can see that the this wire will get longer the more the wayou add to it. So and you also need to be able to measure the diameter of this wire, the more force you add to it. So. Forces must be large to stretch a wire. And itextend. So you can read along this scale, against this scale, how much longer this wire will get or linear part of the load extension graph. So you'll get a uniform extension. So the extension is the gradient times the original length of wiof the cross sextional area, so to find the cross sectional area, we use a microrometer to find the diameter of the wire, so the area, the cross sexual area is pi R squared. Or pi d squared over four, this experiment can be done. So another way of doing this experiment is to. Grab your wire. On a bench. So you can work on a bench like so and. Have a pulley here and you stretch your wire. This way. So most schools would probably do it this way, but either way, you can see how the force causes a change in length of your wire. So you can have a pointer here which will move against a ruler. Yeah, I see the ruler. And again, you can measure the diameter to get the crop cross sexarea. So these are good terms to know and to be able to describe. So Jackson, read out these sentences and make sure you understand these words. Heart resists being scratched to all cuts opposite, equal to soft. And strong needs a large force of stress in order to break the spaceman opposite equal to which Yeah. And then that is tone. A lot of work needs to be done in order to break a spacman of the. Material. So if something is not brittle, it's tough, tough, okay? Okay, next brto breaks without going through the plastic. Yeah so think of glass. It just breaks. So this one's opposite is tall. And the yid stress, it means that the value value of stress, of which the strength increases significantly for minimum minimal increase in stress. So remember, let me just go back up. So yield stress. So you get a sudden increase in shape, a sudden lengthening of it. Yeah. And Yeah. Fatigue lectures occur ring in materials undergoing loading or unloading cycles at value below normal breaking stress. Aeroplanes Jackson, the metal, the material that we make aeroplanes out of can go through fatigue because the planes have to experience high forces, then low forces, then high forces again. So they always have to check the material for tiny cracks. So loading on loading cycles. Okay. Crecreep gradual increase in strain when material has been under stress for a long time. So it gradually gets longer and longer if it's under a lot of stress. So maybe a fence, if it has a lot of load against it, it might eventually get longer. Okay, okay, that's the definition of creep. So duct tail, easy drinto wire. So Yeah metals are ductile. They can be easily drawn into a wire without breaking. So if you get a big lump of copper, if you heat it up, you can converse it. It can be drawn into, make lots of long thin wires. So ductile it can be drawn into a wire. Okay and malleable. Malleable changes shape without cracking when sudden large force applied. Yeah, again, a metal like copper can be made into a frying pan for cooking. So the frying pan, you can hammer it into the shape you want. So it's malleable. It doesn't break like. A ceramic cup, you hammer this, it will break. It's brittle that metals are malleable. They don't break, okay? Okay. Okay. So let's have a go at a couple of straightforward questions. Okay. 9 cm, a force of six newtons, so now is four newtons and the extension of the pair of springs. Is that six b? See. See. So. So extension is f equals kx. Now we're told the springs are identical. End to end. And remember, one over the total constant is equal to one over k plus one over K1 plus one over K2. Yeah I forget that K1 plus K2 so one over six so six is equal to k. Times nine. So. Is six over nine isn't it? Yes for the first Yeah. Six divided by nine times two. So six divided by two. So that is if that is two over three and that is two is f and three is its extension. So f divided by not not correct. So so one over K T. Is equal to one over. 0.6? Plus one over 0.6. Equals two over 0.6. Because spring one and spring two are identical theyhave the same spring constant, so kt. Is. No point six over two. Inverting over these, which is 0.3. Three. So so if that is so, the extension and. So that's four nutions. Yeah that is four newtons. So the extension equal to four times extension. So if the Yeah. Or divided by point three. Hmm. So four divided by 0.3. 13 six. Hmm. The extension. So if six nutions give 9 cm, six nutions equals 9 cm miss, maybe we just four times nopoint three. Four times point three Yeah that would give us so four nutions. Equals. 6 cm, Yeah. If six newtons give an extension of 9 cm, four newtons will give an extension of 6 cm. So two thirds of that and each spring has an extension of six centim. So it's 6 cm plus 6 cm as well. So that's the easier way to do so. We didn't need to do all of this, did we? So. Each spring stretches 6 cm, so 6 cm centimeters, the total extension is twelve. Okay. So a spring extends by 10 cm. When a force of eight nutons is applied, the limit of proportionality is not exceeded. If two of these springs are arranged side by side, and the force of four nutions is applied, the extension of this spring is what? Well this one extension arrangement of the spring so that is too string parallel. So that is K T equal to K1 plus K2. So that is 10 cm and fourth of eight newtons. And the and the kt is no, maybe maybe not. Maybe that is maybe that is. So just for one spring, 10 cm. Ter, finally, Yeah, 10 cm. And the extension, yes, that is the extension times force. Equal to 80. So each spring will share the 5 cm stretching because there's two of them. So I think it's B A 2.5 because there's two springs. They have the extension. Okay, maybe simple than what I think. I think Yeah. The effect of spring constant is two k, the combination extends by x over two, so in each spring, the extension is x over two, so the total extension is shared. So I suppose if 5 cm is our total extension divided by two, we get half the extension, 2.5 cm. Were too worried about the spring constant. And if they're in if they're in series, the total extension is double the extension because each spring behaves as if it's separately when they're in series. So that the total extension is 5 cm, so it's two and a half centimeters each. And then explain why the wiused, when measuring the Young modulus of copper in a school laboratory, is long and thin. Well, it's long and thin. To get a larger stress, don't need such a large force and need too much force. If the the wire is not thin, greater extension linked to thinner wire to the strain extension over original length is easier to measure. So we need a long and a thin wire. So it's long and it's thin, so thin for stress. Force over force over area, not the lay yer and. And long to see change in length over length. A strain easier to measure. Okay, Jackson will leave it there and we'll continue with some questions like this tomorrow, okay? Because that's what we need to practice, being able to apply for the rules that we're learning. Okay, I'll talk to you tomorrow morning. Okay, let miss. Bye bye.