Solved example: Stress and strain

we have a steel wire of cross-sectional area 5 millimeter square and we are hanging a 200 kilogram object from from that and we are asked to calculate the percentage change in length of the steel wire so the wire is going to extend by some amount and we do calculate how much percentage will the length change we are asked to assume that we are within the elastic limits and the Youngs modulus is given to us 200 gigapascals all right let's all this the first thing we'll do is try and understand what the question is we are asked to calculate percentage change in length what do we even mean by that well person to change just means how much is the change out of hundred so if the initial length let's say the relaxed length was L and by hanging the weight let's say the change is Delta L so that we could raise we could be good write that for L length the change is Delta L so if the initial length was 100 what would be the change that's the question right this is what we need to figure out and we can just do this one way by just cross multiplying and we could say well the change 400 would be Delta L times 100 times 100 divided by L so this is what we need to calculate that's the person to change now do we know what Delta L is nope we don't know that do we know what the initial length is we don't know that is as well so how do we do this well if you look at this carefully this is the change in length per unit length and we call this as strain this is our strain and you've spoken a lot about strain and stress and Hookes law and all of that which we'll be using over here in previous videos so if you need some refresher or you know you're not comfortable with this then you'll be great to go back watch that and then come back over here but anyways our goal is really to calculate the strain and once we do that we'll just multiply by 100 so how do we calculate this stream well we can't do it directly because we don't know what Delta L what L is another way we can do that is using Hookes law because Hookes law says that stress is proportional to strain and if you look carefully the stress that we're dealing with is tensile stress because the wire is under tension the two hundred kilogram is pulling downward and in creating tension we call it as tensile stress and whenever we're dealing with tensile stress the elastic modulus that we should use is the Young's modulus all right so let's just write down Hookes law we'll put some right so Hookes law says that the longitudinal stress which is the tensile stress over here is equal to or proportional to the longitudinal strain a strain again generated over here and the proportionality constant is the Youngs modulus we know what Y is we need to calculate this so we really need to calculate the stress now and how do you calculate stress or what is the definition of stress all stress is defined as the restoring force per unit area we know the area so we need to calculate the restoring force that's all we need to do and to do that will this look at a tiny piece of string over here string or a wire okay whatever so you take that tiny piece over here let's just write down all the forces acting on it well there's one force acting downwards because of the weight of this object and the weight is 200 kilograms that's the mass times G that's how you calculate weight mg 200 G and G's if we take it as 10 we'll just assume it to be 10 then the force that is acting on it downward is 2,000 Newtons that's the that's the weight but we know that this piece is not accelerating anywhere right I mean this whole thing is in equilibrium is just staying there that means the total force on this must be zero so someone must be countering this force so there must be an upward 2,000 Newton's of force and who is putting that force well that force is due to the piece of steel that is right about that that's the one that's pulling it up and if you think about it this actually is the restoring force I mean think about it if we get rid of this 200 kilogram this force disappears this force will this tiny piece up and that's how the whole steel wire will snap back to its original length elasticity so this is the restoring force so we have the restoring force we have the area we can calculate the stress and we can plug in and we can figure out what the strain is all right so you're pretty much done with the physics part hole here now you all we have to do is some algebra just plugging in and so feel free to pause the video at this time try to do the algebra and see if we can get the answer all right let's plug in so stress would be 2,000 Newtons that's the restoring force per unit area per unit area and the area is millimeter square 10 to the power minus 3 that's milli meter square square and that's equal to the Youngs modulus that's 200 gigapascals Giga is 10 to the power 9 and Pascal's is just Newton per meter squared that's what we call as Pascal's it was per meter square x times e that's what we need to figure out e all right Newton's cancels there's a meter square over Chien is a meter square or sheared that cancels and so what we left out with now is 2,000 divided by 5 we can quickly go ahead and do that that is 4 times so we're left with 400 in the numerator 400 divided by you have 10 power minus 3 squared notice when I cancel that only meter square got cancelled the 10 power minus 3 is still squared and that's 10 to the power minus 6 and when you put on top you get 10 to the power plus 6 and that's equal to 200 times 10 to the power and 9 times e okay cancel some zeros this 2 goes 2 times a 4 goes 2 times so what we left now is with e equals let's see we have 2 times 10 to the power 6 and divided by 10 to the power 9 and that is equal to let's see what that comes that comes out to be 10 to the power minus 3 and notice there is no units over here that should make sense because strain is unitless right meters from m/m cancels so that's our strain and what does this mean what is the number telling us oh that's telling us that if we had one meter long wire if we had one with a long wire then when you put two eight kilograms then the extension would be 2 times 10 power minus 3 meters or two millimeters I mean think about it let me just write that down let me make some more space and write that down this number is just telling us that 4 for 1 meter length the extension would be would would have been 2 millimeters and that's incredible if you think about it I mean we're taking a very thin steel wire and we're putting 200 kilograms of mass and the extension in the wire is just 2 millimeters that's insane that's so tiny that our human eyes may not be able to see that which is why we probably feel that steel doesn't even extend right I mean that's why you think of it as a perfect rigid body but it isn't and that's why it's so marvelous and we use it all the time and the main reason why we're getting such a small number is because of this insanely high Young's modulus turns out that Steve has one of the highest values of Young's modulus and that's why we like steel all right anyways our final question that we need to do is calculate the percentage change so to do that we just have to take this number and multiply by 100 either that or you can do it directly you we now know that for 1 for unit length the change is 2 times 10 power minus 3 400 how much be the change well you just multiply by 100 right so what is needed now is percentage change percentage change and that is just multiply this by 100 and when you do that 10 power minus 3 and hundred becomes 10 power minus 1 and that uses 0.2 so that's our final answer the percentage change in the length of steel wire is just a minuscule 0.2 person and steel is awesome