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Current time:0:00Total duration:11:14

Video transcript

we have a wire of length 10 meters which is clamped between two wall and it's temperature right now is 30 degree Celsius we increase the temperature from 30 to 130 and the question is in doing so what's the thermal stress generated inside the wire we've been given the linear expansion coefficient and the Young's modulus all right this term called a thermal stress is very new to us so the goal of this video is to understand exactly what this thing means and we'll do that by solving this numerical alright let's do it so we have a wire of length 10 meters and it's temperature is increasing and when the temperature of something increases it tends to expand when you heat things they expand the only problem that we have here is that there are these walls they are not going to allow this wire to expand so what's going to happen well this is new for us we have not talked about things like this before so what we'll do first is we'll forget about the wall for now let's forget about that and let's just do what we're familiar with we are familiar with calculating changes in the length when the temperature changes right and we can do that by knowing this number alpha L the linear expansion coefficient the number just tells us that if we say if we had one with a long wire and if we increase the temperature by one degree Celsius then the wire would expand if it was free the wire would expand by 20 times 10 to the power minus 6 meters that's given to us so let's first calculate how much this wire expands if there was no wall well it would be a great idea to pause the video because we have talked about this before so pause the video and see if you can try this on your own and if you feel like you require more clarity it would be great to watch that video first and then come back and try this yourself alright so pause the video try it okay let's do it the first thing we'll do is we'll build a relationship between changes in the length and since the temperature very quickly let's do that the change in the length we already know it's alpha L provided we have 1 meter long wire and we increase temperature by one degree Celsius but if we have say two with a long wire it will be twice 4l meter long wire it would be L times more so this would be the change for one degree Celsius one degree Celsius rise what if you have two degree Celsius rise will be twice of this what if you have delta T degree Celsius rise will be delta T times this all right so here is the expression and now we can just plug in these numbers we know what our final is we know what L is we also know what delta T is right we can calculate that delta T is the change in the temperature we can just plug in and figure out the change in the length if the wire was free to expand let's just do that first so in that case we would get Delta L equal to our file that's 20 times 10 to the power minus 6 degrees Celsius inverse times 10 meters times delta T what is delta T well change in temperature and that change in temperature is 130 minus 30 which is 100 degree Celsius so this is 100 degree Celsius let's just go ahead and do the math we have 20 times let's see we have a thousand and we have 10 power minus 6 when you multiply you get 10 power minus 3 the degree Celsius inverse in the degree Celsius cancels that good because you want the change in length to be meters so we get so many meters that's about 20 millimeters 20 millimeters so what we understand from this is that at 130 degree Celsius the wire would be a little bit longer it would be 20 millimeters longer this would be 20 millimeters all right now let's come back to our our problem in our problem we've been given that there is a wall so what's going to happen well the wall is not going to allow the wire to expand so because of that wall even at 130 degree Celsius the length of the wire would still be 10 meters so by not allowing the wire to expand to its necessary length it's as if the walls are come pressing the wire you get that let's write that down so the effect of the wall can be thought of as at 130 degree Celsius the wire should have had 20 millimeters longer length but it doesn't and the reason it doesn't is that the walls are pushing from the sides and making sure that it doesn't expand in other words the wire is being compressed the wire is compressed compressed by by 20 millimeters and now comes the question what's the effect of that so what happens when you compress a wire by some amount well for that let's think of a spring let's make a little bit space over here all right so here is the original wire which has been compressed by 20 millimeters and we're comparing it to a spring what would happen if you had to compress the spring by so amount what would happen well and we all know that the spring snaps back or at least it tries to snap back and it does so by generating what we call as the restoring force so there is a restoring force that gets generated over here which call it as F and you may have already learned that this restoring force this restoring force is proportional is proportional to the amount of compression that we that we provide over here let's call that compression as Delta X amount of compression and the proportionality can be written as equal to some constant K and we call this as the spring constant and a small detail is that since the force and the compressions are in the opposite direction we would have a minus sign over here this is Hookes law but let's not worry about the direction let's let's just work with the magnitude over here now something very similar to happen over here when this wire gets compressed it tries to restore itself and it does so by generating a restoring force within the wire but we don't exactly talk about the restoring force here but instead we talk about a quantity called as restoring force per unit area and when we say unit area I mean a cross-sectional area so if it is zoom you can think of this wire as some kind of a cylinder and so it has some cross-sectional area and so if you take the restoring force divided by the cross-sectional area that's what's important for us and that quantity is what we call as stress that quantity is stress so stress is very closely related to the restoring force in the spring alright and just like how over here this restoring force is proportional to the amount of compression it turns out that the stress is proportional to the amount of compression per unit length and we call that quantity as strain call that quantity as strain and it turns out that even these are proportional to each other or equal to some constant and that constant is Young's modulus all right so stress is force per area that's what stresses and strain is the change in length or the amount of compression per unit length per unit length and Young's modulus is just a constant for a given material now we have actually discussed about this is another chapter called solids as if you require more clarity on this it'll be better to watch that video first and then come back over here but anyways all we need to do now is calculate this stress this stress is called as the thermal stress and the reason it's called thermal is because I mean think about it the stress was generated because of thermal expansion if there was no thermal expansion the wire wouldn't have expanded or wouldn't have tried to expand and this strain wouldn't have been developed in the first place so all we need to do is calculate this thermal stress well let's see what we know we already we already know what Y is that's the Young's modulus we know Delta L that's 20 millimeters we also know the initial length L 10 millimeters so we just go ahead plug in and calculate well thermal stresses so let's just do that they make more space over here all right so if you plug in we get 80 gigapascals x times strength Delta L Delta L is 20 millimeters / the original language is 10 meters 10 meters so that gives us that gives us let's see this 10 cancels and you have a meter cancelling and so what left is 80 times 2 that's 160 160 gigapascals times 10 to the power minus 3 now Giga is 10 to the power 9 times 10 to the power minus 3 is 10 to the power 6 and 6 converted as mega so we could say 160 mega Pascal's 160 mega Pascal's so that is the amount of a thermal stress thermal stress generated in this in this scenario now the effect of the thermal stress can be seen in this picture so notice over here due to the thermal stress the railway track has buckled now you may be wondering well where is the wall to restrict the expansion over here well what may have happened is that only this part of the railway track may have gotten heated up a lot and so it tried to expand a lot more compared to the other parts and so in such case the other parts act like sort of act like a wall sort of like you know do not allow the expansion so much causing the thermal stress which has made the buckling effect over here and if the material is brittle then this buckling can even break the material so suppose you had a glass rod over here and then suppose you had to rapidly heat up one small portion of the glass rod then again that portion would try to expand a lot lot more compared to the other sides other section and again that could cause the whole thing not to buckle this time but just to break and this can also happen when you try to cool something very quickly if you were to cool a small portion of the glass then that portion would try to shrink contract very quickly and that could again break the whole material this rapid cooling or rapid heating could break stuff like glasses and we call such phenomena as a thermal shock and so we need to be careful about things like this when we're dealing with very hot stuff like say hot liquids and we pouring it in say things like glass or other ceramic containers