Understanding Hooke's Law and Materials

hello in today's class we shall be talking about books law now the first thing we must state here is books law what is books law hoogslaw states that within the limit of proportionality the extension of a material is proportional to the applied force let's put this in writing it states that within the limit of proportionality the extension of a material is proportional to the applied force now this statement can be showed on a force extension graph which goes in this form here's the graph Here is your extension. and here is the force okay now the graph is a straight line graph all right so here is the origin point zero and here is the elastic limit so this is saying that as the applied force is increased the extension is also increased so i can write this in equation becomes extension is proportional to the applied force. Now I can take off this symbol it becomes extension equal to T which is the constant times what the applied force all right this is the hook's law we can use this to solve more questions now i want to derive another equation from hook's law as it relates to stress and strain. In this case, I will take off the force right here and the extension, and we replace it with the force with the stress, and also we replace the extension with the strain. Now, here we are seeing that as the stress is increased, the strain is increased. Okay, so this is a law that we can use to solve questions that involves young modulus so from this new graph we can state another law which states that within the limit of proportionality of a material the strain produced is directly proportional to the stress producing it so I'm saying that From Hooke's law, we can state that within the limits of proportionality of a material, the strain produced producing it that is another statement that we can use to solve for the strength of material so we can put this in equation forms so it becomes the stress is proportional to the strain all right so we can replace this symbol with equal to sign it becomes a stress is equal to a constant this time we use the capital E times the string so capital E here represents Young's modulus of elasticity is the Young modulus of elasticity so we can make the E the subject so now the Young modulus is equal to the stress over the strain so with this equation we can find the strength of a material As it relates to the materials young modulus and please note that all materials as a typical value for young modulus of elasticity so all material like Aluminum brass cast iron they all have different values for young modelers of elasticity now let's derive more equations from this formula Now I want to expand the Young-Modulus equation as it relates to stress and strain. But before I do that, I will draw right here a force extension graph. Here is the extension. And here is the force. Now here is the origin. And the graph can look like a straight line. Then for you to know, it starts to... change behavior all right now this point right here before the material starts to bend is called the elastic limits all right so from zero to point a is the point where the the force is proportional to the extension so from the zero to a as the force increases the extension is increased all right so we call this point this point here where the material stops or starts to bend it's called the limit it's called the limit of proportionality limit of proportionality okay now from this graph we can find the gradient of this graph and the reason why we have to find the gradient of this graph is that sometimes we'll be given a table of values and we'll be asked to find the young modulus so we can take our gradients from any point on this graph okay so i can take my gradients from this point i can take it down to the x-axis take it down to the y-axis okay now gradient is rise over run and we know that gradient is the change in the y-axis which is the force in this case over the change in the x-axis which is the extension in this case so i can call this let me call this my f2 and i call this uh my f1 now here will be my x1 x1 and here will be my x2 now to find the gradient of this curve you can take your gradients from any point on this straight line okay now to find the gradients of this of this graph the gradients is equal to the change in the y-axis in this case is f2 minus f1 over the change in the x-axis which is x2 minus x1 all right so this is how we find the gradient on this curve and please note that the gradient is also known as the stiffness of the curve it's also known as the word stiffness so the gradient is also known as the stiffness of the curve. Now what is stiffness? Now stiffness is the ability of the material to return to its original shape or form after the applied force is removed. I'll repeat that. stiffness is the ability of the material to return to its original shape or form after the applied force is removed now let's derive an equation for young modulus that we can use to solve questions now we know that the young modulus is equal to stress over the strength and in our previous classes we learned that stress is equal to force force over the area And also we learned that the strain is equal. equal to the extension over the original length. Now we can put this into this equation and we can derive equations from this. Now therefore I can say that my modulus is equal to the stress. The stress is force over area. So force over area. Divide is divide by the strain, the strain is x over the original length. So therefore we can say that force over area so in math we change division to times so when we change division to times we have to flip the right hand side it becomes l over s so this gives us the force times the original length of the material over area of the material times the extension all right so we can still rearrange this equation as it relates to the gradient as you can see the gradient in this case is equal to um f over x as you can see so i can rearrange this to become the modulus is equal to f over x times L over A. That's L over A. Here is a new equation that we have right here. And I know that the force over the extension is also known as the gradient of the force extension graph so I can replace force over s with gradient or with stiffness because I say that the gradient of this force extension graph is also known as the stiffness okay so therefore we can say that young modulus is a quarter so I will replace this with stiffness or with gradients so I can use stiffness becomes stiffness times the length of the material over the area. so here is a formula we can use to solve for the modulus of a material sometimes we'll be giving a table of values for force and extension and we'll be asked to find the modulus of elasticity of the material so the first thing we do is to draw a graph and to find the stiffness of the of the graph then we can now solve that question let's look at our two examples on how we can apply these all concepts okay here is our very first question we are told that a copper rod of diameter 20 millimeter and length 2.0 meter has a tensile force of 5 kilonewton applied to it so let's write out our given datas we are given the diameter diameter is given as 20 millimeters we convert these to meter it becomes 0.02 meters all right we are giving the length of the of the material the length is giving us 2.0 meters all right so we are giving the tensile force the force is giving us five kilo newton five kilo newton we convert this to newton becomes five times a thousand that gives us five thousand newton now question we are asked to find the stress in the world so the stress is not given to us we have to find the stress and the next question is that to find the uh how much the world extends so that is the extension so we have to find the extension of the world force was applied and also we are giving the modulus as 96 so our e capital e is giving us 96 gigapascal remember that gigapascal means 10 to the 9 so i can convert this to pascal it becomes a 96 times 10 to the 9th pascal all right so now we can find our stress so the first question is a find the stress so A, we have to find the stress. The stress is equal to the force over area. Now my force is known, but the area is not given. I can find the area. I know that area is equal to pi d squared. over 4 all right now the reason why i'm using this equation is because the rod is in the form of what a cylinder so we use this formula for the area so therefore my area is equal to pi times my diameter is 0.02 squared 0.02 squared divided by 4 so therefore my area is equal to so when you put this in your calculator we should get 3.1416 3.1416 times 10 to the minus 4 meters minus four meters squared so that is our area so therefore i can find my stress so stress therefore becomes the force is 5000 divided by the area which is 3.1416 times 10 to the minus 4. So therefore, 5000 divided by 3.1416 times 10 to the minus 4 would give us 159154. 57.09 57.09 PASCA 209 pascal all right now i can convert these to uh to uh mega pascal it becomes what so i take my decimal point i'll move it six place backwards so it becomes one two three four five six that's a six that's a mega so we now have what 15.9 mega pascal so that is the stress in the world that is our answer for the stress in the world now the next question is that we are asked to find the extension by how much the world extends so to find the extension of the world the first thing i do is this i can use this equation right here to find the extension all i have to do is to transpose or i can find the strain first which i will do now because i know that modulus is equal to stress over strain so i can find the strain force then from there i can find my my extension so when i transpose this equation i get uh strain is equal to the stress over the modulus so i can log in the values my stress is 15.9 megapascal so i can see 15.9 9 mega Pascal means times 10 to the 6 over over the modulus which is 96 times 10 to the 9 remember this is 15 plus 15 all right so therefore my extension in this case when i put this in my when i put this in my calculator i should get 1.6563 1.65 63 times 10 to the minus 4 remember that string has no units so it's not unit for string because string is dimensionless so now I can find my stress or sorry I can find my extension now I know that string is equal to extension over the original length so now I can find my extension with this equation so I can transpose so we have extension is equal to the strain times the original length all right so therefore so therefore my strain my strain is 1.6563 times 10 to the minus 4 times the original length the original length is giving us 2.0 meters so the original length is 2.0 meters all right so therefore the extension of the world when we multiply 1.6563 times 10 to the minus 4 times 2 will give us 3.3125 3.312 okay one two times ten to the minus four meters times ten to the minus four meters so that is the extension of the rod so this is our final answer okay so here is another question This time we are asked to draw the load extension graph of the information extracted from an experiment conducted to determine the modulus of elasticity of a sample of mined steel. The mean diameter of the wire is given so the mean diameter is given as 1.3 millimeters 1.3 millimeters we can convert that to meter it becomes 0.0013 meters 0.0013 meters okay now give me the length of the material the length is 8.0 meters We are asked to find autodetermine, the modulus of elasticity of the sample. We are finding the E is not given to us. And also we are asked to find the stress at the limit of proportionality. So the stress is not given to us. all right now to find this or the service question the first thing i do is that i have to use the formula that relates to the graph and i know that the e is equal to the gradient or the stiffness stiffness times the original length which is 8.0 over the area now to get the area i can use the diameter so again it's a minus t i will use the formula area to get my area now is the area equal to pi d squared over four so I have to get my area and to get my stiffness I will have to plot a graph to get my stiffness so we have area equal to pi times my diameter is 0.0013 meters 0.0013 meters or squared divided by 4 so therefore my area becomes what so when i put this in my calculator pi times pi times 0.0013 squared divided by 4 that will give us 1.327 as 1.327 times 10 to the minus 6 meter squared minus 6 meter squared so now i've gotten the area for this uh material what i do next is to find this stiffness which I say that to find the stiffness we have to plot a graph so I will have to plot the graph so I come to my graph so here is my my graph okay now I'm to plot this graph let me adjust my my lines because i have a lot of values so i can adjust my lines i can take off this part of it because i have no negative numbers all right good so uh what i do next is to choose a scale so i can choose a scale remember that uh okay so what next i choose a scale now let me use a dark blue marker so let's choose a scale uh let's okay step one here is my this is my uh my load or quality the force here in newton and here is the extension which is in millimeters all right so uh yes zero okay on the on the uh x axis i will use a scale of 2 cm to 2 cm to 2 millimeters so that is that uh here is two this is four this is six eight here is ten twelve fourteen and here is a 16. 18 and now we have 20 there then for my y axis in this case is the load or the force i will use a scale of um this is 1 cm so 20 newton all right so uh 1 cm so 0 is 1 so here is 20. here is 40 okay to make my work neat i will space it after uh 10 centimeters so here is 20 here's 40 this is a 60 this is 80 it should be 100 120 here is 140 yes 160 this is 180. this is 200 this is 220. this is 240 this is 260. this is 280. This is 300. this is 320 so this is 340. i believe the last value on our table for the load is 340 so i can stop right here all right so now i have to plot the graph so when the load was zero my extension was zero so i will mark the origin so i have a point right here i have a point right here when the load was 40 it's 20 it is 40 right here when it was 40 we have a 1.2 extension so this is one this is one point two point four point six point eight two so So 0, yes, yes, 1, so it is 1, right? This is 1, this is 1 right here. So 1.2, so go up to 4, this is 0, this is 20, this is 40, so 40, so 1.2 right here. please mark your points all right so i have two points so when it was one ten my it was uh 3.3 extension so here is two if here is two yes three this is three this is three uh three three point two four six eight four so three point three four between uh 3.2 and 3.4 so it's in between this line so uh i have one ten as load is 100 104 108 and 112 so 110 for between 8 and 12 so come right here is the android the Android right here wonder for wonder it's wonder at 12 so it's for between 8 and 12 so it is 2 4 so this is our yes our point right here come again is 100 104 108 so it falls between 8 and 112 and yes this is 3 3.2 0.4 so 3.34 between that so that is our point right there so i'm counting my load in using four units because here is 20 if here is 20 here is 24 26 20 24 28 32 all right plus 4 36 plus 4 that's 40 plus 4 44 plus 4 48 plus 4 52 plus 4 56 plus 4 60 so that's how we are counting the nodes now the next load is 160 and we have our 4.8 extension so this is 4 this is 140 here is our 160 right here 160 this is 4 this is 4.2468 4.8 right here so 160 140 this is 160 these are 4 4.2468 is our point right here please mark your points all right so when it was 200 we have 6.0 so 6 plus 6 plus 2 is 180 there's 200 200 so 6.0 is a six mark the points all right we have went towards 250 we have 7.5 here is six so yeah it should be uh it should be seven it should be seven seven point two four six so seven points 5 for between 4 and 6 and the load was 250 so here is 220 okay it's 220 take it down to 7 this is 7 right here 220 224 224 228 plus 4 252 so 254 between this and this so take it down 7.2 7.4 so it's forced right there that is our a point right there okay next one is 290 to 10 it's 10 goes up this is 260 260 264 here is 280 okay it's 280 it's 280 284 plus four two two two uh 280 plus four 284 plus four we have uh 288 plus four we have 292 so it's first so between 288 and 22 here's our point right here at the middle so mark our point right there you have to be very careful when doing this you must ensure that you get your exact point so the next one is 340 to 340 as a load the load is 340 as you can see on the graph when it's 340 is 16.2 when it's 340 it is 16.2 16.16 right here this is 340 16.2 right here all right so i have plotted uh this graph or this table of values on my graph okay so we finished plotting the graph what we do next is to join the lines as you can see i have a straight line from 0 to 250 so let me draw the lines from 0 to 250 you can see i have a straight line right there i have a straight line right there from zero to 250 all right a straight line from zero to 250 what next after 250 the the lines begins to begins to bend as you can see right here okay now this point i call this point my a now this here is the elastic limit now from from this point where the the line begins to bend the the force is no longer proportional to the extension but from zero to a the force and extension are proportional so this point is called this point where the line begins to bend is called a limit of proportionality it's called a limit of proportionality all right what next we can now find our stiffness let me use a different marker okay so let's now find the stiffness so as i was saying earlier you can take your gradient or your stiffness from any point on a straight line so let me take my my gradient from uh point 160.160 take it down take it down to this is a ruler when doing this take it down to to 4.8 yes 4.8 and i take it down to my 160. it's my gradient here's my f2 which is 160 and here's my f1 which is uh zero then here is my 4.8 which is my x2 and here is my x1 so from this now we can find the value of our gradient so what next i can never say that the gradient or the stiffness stiffness or the gradient of the force extension graph is equal to F2 minus f1 over x2 minus x1 now my f2 in this case my f2 goes to the y-axis which is the fourth which is 160 is 160 minus my f1 my f1 goes to the y-axis which is zero my x2 goes to the x-axis here is 4.8 remember it is 4.8 millimeters so we convert to meters it gives us 0.0048 so we converted 4.8 here to meters it becomes 0.0048 0.0048 minus x1, x1 is right here which is 0, so therefore we have 160 minus 0 is 160 divided by 0.0048 minus 0 is 0.0048 so therefore our stiffness for this uh for this graph we'll put that in our calculator 160 160 divided by 0.0048 we give us three three point three three three three three point three so we have three three three three three point three newton per meters you know why because it's the force over extension so it's newton per meter so now i've got my stiffness so i can put that in the equation so therefore my modulus e is equal to the stiffness three three point three times the length length is eight point zero divide by the area the area is one point three to seven one point three to seven times ten to the minus six all right so we can put that in our calculator three three three three three point three times eight point zero divide by one point three two seven times ten to the minuses that gives us two points zero zero nine five four three times 10 to the 11 pascal 11 pascal Now we can convert this to gigapascal. How do we convert this to gigapascal? I know that gigapascal is 10 to the 9, so I can move these points backwards. 1, 2. That becomes 2, 0, 1. All right? 2, 0, 1. times 10 times 10 to the minus 2 times 10 to the minus 2 times 10 to the power 11. so that gives us 2 0 1 times 10 remember in maths minus 2 plus 11 gives us 9 i give us 9 pascal so therefore we can say that our modulus e is equal to 2 0 1 giga pascal so that is our modulus but it's not finished right here we are asked to find find the stress at the limit of proportionality so the stress we know is stress is equal to the force over the area now the stress at the limit of proportionality is the point on the graph where the line starts to ah bang so this point right here is the limit of what proportionality and the force at this point is 250 so the force at that point is 250 so therefore my stress is equal to what 250 over the area the area is 1.327 times 10 to the minus 6 so therefore put that in the calculator our stress is 1 it's it's 3 9 4 8 7 5.7 Pascal, 0.7 Pascal. Now we can also convert this to a Mega Pascal so I can see that my stress becomes, I'll move the decimal point 6 place, becomes 1, 2, 3, 4, 5, 6, 6. So that gives us 1, 8, 8. points we have three here we have nine take one from nine to three becomes four becomes point four mega pascal so that is the final answer so here is how we solve questions that relates to uh the force extension graph so this is our graph and here is the values we got using the graph so thanks for watching

In this case, I will take off the force right here and the extension, and we replace it with the force with the stress, and also we replace the extension with the strain. Now, here we are seeing that as the stress is increased, the strain is increased. Okay, so this is a law that we can use to solve questions that involves young modulus so from this new graph we can state another law which states that within the limit of proportionality of a material the strain produced is directly proportional to the stress producing it so I'm saying that From Hooke's law, we can state that within the limits of proportionality of a material, the strain produced producing it that is another statement that we can use to solve for the strength of material so we can put this in equation forms so it becomes the stress is proportional to the strain all right so we can replace this symbol with equal to sign it becomes a stress is equal to a constant this time we use the capital E times the string so capital E here represents Young's modulus of elasticity is the Young modulus of elasticity so we can make the E the subject so now the Young modulus is equal to the stress over the strain so with this equation we can find the strength of a material As it relates to the materials young modulus and please note that all materials as a typical value for young modulus of elasticity so all material like Aluminum brass cast iron they all have different values for young modelers of elasticity now let's derive more equations from this formula Now I want to expand the Young-Modulus equation as it relates to stress and strain.

But before I do that, I will draw right here a force extension graph. Here is the extension. And here is the force.

Now here is the origin. And the graph can look like a straight line. Then for you to know, it starts to... change behavior all right now this point right here before the material starts to bend is called the elastic limits all right so from zero to point a is the point where the the force is proportional to the extension so from the zero to a as the force increases the extension is increased all right so we call this point this point here where the material stops or starts to bend it's called the limit it's called the limit of proportionality limit of proportionality okay now from this graph we can find the gradient of this graph and the reason why we have to find the gradient of this graph is that sometimes we'll be given a table of values and we'll be asked to find the young modulus so we can take our gradients from any point on this graph okay so i can take my gradients from this point i can take it down to the x-axis take it down to the y-axis okay now gradient is rise over run and we know that gradient is the change in the y-axis which is the force in this case over the change in the x-axis which is the extension in this case so i can call this let me call this my f2 and i call this uh my f1 now here will be my x1 x1 and here will be my x2 now to find the gradient of this curve you can take your gradients from any point on this straight line okay now to find the gradients of this of this graph the gradients is equal to the change in the y-axis in this case is f2 minus f1 over the change in the x-axis which is x2 minus x1 all right so this is how we find the gradient on this curve and please note that the gradient is also known as the stiffness of the curve it's also known as the word stiffness so the gradient is also known as the stiffness of the curve.

Now what is stiffness? Now stiffness is the ability of the material to return to its original shape or form after the applied force is removed. I'll repeat that. stiffness is the ability of the material to return to its original shape or form after the applied force is removed now let's derive an equation for young modulus that we can use to solve questions now we know that the young modulus is equal to stress over the strength and in our previous classes we learned that stress is equal to force force over the area And also we learned that the strain is equal. equal to the extension over the original length.

Now we can put this into this equation and we can derive equations from this. Now therefore I can say that my modulus is equal to the stress. The stress is force over area. So force over area.

Divide is divide by the strain, the strain is x over the original length. So therefore we can say that force over area so in math we change division to times so when we change division to times we have to flip the right hand side it becomes l over s so this gives us the force times the original length of the material over area of the material times the extension all right so we can still rearrange this equation as it relates to the gradient as you can see the gradient in this case is equal to um f over x as you can see so i can rearrange this to become the modulus is equal to f over x times L over A. That's L over A.

Here is a new equation that we have right here. And I know that the force over the extension is also known as the gradient of the force extension graph so I can replace force over s with gradient or with stiffness because I say that the gradient of this force extension graph is also known as the stiffness okay so therefore we can say that young modulus is a quarter so I will replace this with stiffness or with gradients so I can use stiffness becomes stiffness times the length of the material over the area. so here is a formula we can use to solve for the modulus of a material sometimes we'll be giving a table of values for force and extension and we'll be asked to find the modulus of elasticity of the material so the first thing we do is to draw a graph and to find the stiffness of the of the graph then we can now solve that question let's look at our two examples on how we can apply these all concepts okay here is our very first question we are told that a copper rod of diameter 20 millimeter and length 2.0 meter has a tensile force of 5 kilonewton applied to it so let's write out our given datas we are given the diameter diameter is given as 20 millimeters we convert these to meter it becomes 0.02 meters all right we are giving the length of the of the material the length is giving us 2.0 meters all right so we are giving the tensile force the force is giving us five kilo newton five kilo newton we convert this to newton becomes five times a thousand that gives us five thousand newton now question we are asked to find the stress in the world so the stress is not given to us we have to find the stress and the next question is that to find the uh how much the world extends so that is the extension so we have to find the extension of the world force was applied and also we are giving the modulus as 96 so our e capital e is giving us 96 gigapascal remember that gigapascal means 10 to the 9 so i can convert this to pascal it becomes a 96 times 10 to the 9th pascal all right so now we can find our stress so the first question is a find the stress so A, we have to find the stress.

The stress is equal to the force over area. Now my force is known, but the area is not given. I can find the area.

I know that area is equal to pi d squared. over 4 all right now the reason why i'm using this equation is because the rod is in the form of what a cylinder so we use this formula for the area so therefore my area is equal to pi times my diameter is 0.02 squared 0.02 squared divided by 4 so therefore my area is equal to so when you put this in your calculator we should get 3.1416 3.1416 times 10 to the minus 4 meters minus four meters squared so that is our area so therefore i can find my stress so stress therefore becomes the force is 5000 divided by the area which is 3.1416 times 10 to the minus 4. So therefore, 5000 divided by 3.1416 times 10 to the minus 4 would give us 159154. 57.09 57.09 PASCA 209 pascal all right now i can convert these to uh to uh mega pascal it becomes what so i take my decimal point i'll move it six place backwards so it becomes one two three four five six that's a six that's a mega so we now have what 15.9 mega pascal so that is the stress in the world that is our answer for the stress in the world now the next question is that we are asked to find the extension by how much the world extends so to find the extension of the world the first thing i do is this i can use this equation right here to find the extension all i have to do is to transpose or i can find the strain first which i will do now because i know that modulus is equal to stress over strain so i can find the strain force then from there i can find my my extension so when i transpose this equation i get uh strain is equal to the stress over the modulus so i can log in the values my stress is 15.9 megapascal so i can see 15.9 9 mega Pascal means times 10 to the 6 over over the modulus which is 96 times 10 to the 9 remember this is 15 plus 15 all right so therefore my extension in this case when i put this in my when i put this in my calculator i should get 1.6563 1.65 63 times 10 to the minus 4 remember that string has no units so it's not unit for string because string is dimensionless so now I can find my stress or sorry I can find my extension now I know that string is equal to extension over the original length so now I can find my extension with this equation so I can transpose so we have extension is equal to the strain times the original length all right so therefore so therefore my strain my strain is 1.6563 times 10 to the minus 4 times the original length the original length is giving us 2.0 meters so the original length is 2.0 meters all right so therefore the extension of the world when we multiply 1.6563 times 10 to the minus 4 times 2 will give us 3.3125 3.312 okay one two times ten to the minus four meters times ten to the minus four meters so that is the extension of the rod so this is our final answer okay so here is another question This time we are asked to draw the load extension graph of the information extracted from an experiment conducted to determine the modulus of elasticity of a sample of mined steel. The mean diameter of the wire is given so the mean diameter is given as 1.3 millimeters 1.3 millimeters we can convert that to meter it becomes 0.0013 meters 0.0013 meters okay now give me the length of the material the length is 8.0 meters We are asked to find autodetermine, the modulus of elasticity of the sample. We are finding the E is not given to us.

And also we are asked to find the stress at the limit of proportionality. So the stress is not given to us. all right now to find this or the service question the first thing i do is that i have to use the formula that relates to the graph and i know that the e is equal to the gradient or the stiffness stiffness times the original length which is 8.0 over the area now to get the area i can use the diameter so again it's a minus t i will use the formula area to get my area now is the area equal to pi d squared over four so I have to get my area and to get my stiffness I will have to plot a graph to get my stiffness so we have area equal to pi times my diameter is 0.0013 meters 0.0013 meters or squared divided by 4 so therefore my area becomes what so when i put this in my calculator pi times pi times 0.0013 squared divided by 4 that will give us 1.327 as 1.327 times 10 to the minus 6 meter squared minus 6 meter squared so now i've gotten the area for this uh material what i do next is to find this stiffness which I say that to find the stiffness we have to plot a graph so I will have to plot the graph so I come to my graph so here is my my graph okay now I'm to plot this graph let me adjust my my lines because i have a lot of values so i can adjust my lines i can take off this part of it because i have no negative numbers all right good so uh what i do next is to choose a scale so i can choose a scale remember that uh okay so what next i choose a scale now let me use a dark blue marker so let's choose a scale uh let's okay step one here is my this is my uh my load or quality the force here in newton and here is the extension which is in millimeters all right so uh yes zero okay on the on the uh x axis i will use a scale of 2 cm to 2 cm to 2 millimeters so that is that uh here is two this is four this is six eight here is ten twelve fourteen and here is a 16. 18 and now we have 20 there then for my y axis in this case is the load or the force i will use a scale of um this is 1 cm so 20 newton all right so uh 1 cm so 0 is 1 so here is 20. here is 40 okay to make my work neat i will space it after uh 10 centimeters so here is 20 here's 40 this is a 60 this is 80 it should be 100 120 here is 140 yes 160 this is 180. this is 200 this is 220. this is 240 this is 260. this is 280. This is 300. this is 320 so this is 340. i believe the last value on our table for the load is 340 so i can stop right here all right so now i have to plot the graph so when the load was zero my extension was zero so i will mark the origin so i have a point right here i have a point right here when the load was 40 it's 20 it is 40 right here when it was 40 we have a 1.2 extension so this is one this is one point two point four point six point eight two so So 0, yes, yes, 1, so it is 1, right? This is 1, this is 1 right here. So 1.2, so go up to 4, this is 0, this is 20, this is 40, so 40, so 1.2 right here.

please mark your points all right so i have two points so when it was one ten my it was uh 3.3 extension so here is two if here is two yes three this is three this is three uh three three point two four six eight four so three point three four between uh 3.2 and 3.4 so it's in between this line so uh i have one ten as load is 100 104 108 and 112 so 110 for between 8 and 12 so come right here is the android the Android right here wonder for wonder it's wonder at 12 so it's for between 8 and 12 so it is 2 4 so this is our yes our point right here come again is 100 104 108 so it falls between 8 and 112 and yes this is 3 3.2 0.4 so 3.34 between that so that is our point right there so i'm counting my load in using four units because here is 20 if here is 20 here is 24 26 20 24 28 32 all right plus 4 36 plus 4 that's 40 plus 4 44 plus 4 48 plus 4 52 plus 4 56 plus 4 60 so that's how we are counting the nodes now the next load is 160 and we have our 4.8 extension so this is 4 this is 140 here is our 160 right here 160 this is 4 this is 4.2468 4.8 right here so 160 140 this is 160 these are 4 4.2468 is our point right here please mark your points all right so when it was 200 we have 6.0 so 6 plus 6 plus 2 is 180 there's 200 200 so 6.0 is a six mark the points all right we have went towards 250 we have 7.5 here is six so yeah it should be uh it should be seven it should be seven seven point two four six so seven points 5 for between 4 and 6 and the load was 250 so here is 220 okay it's 220 take it down to 7 this is 7 right here 220 224 224 228 plus 4 252 so 254 between this and this so take it down 7.2 7.4 so it's forced right there that is our a point right there okay next one is 290 to 10 it's 10 goes up this is 260 260 264 here is 280 okay it's 280 it's 280 284 plus four two two two uh 280 plus four 284 plus four we have uh 288 plus four we have 292 so it's first so between 288 and 22 here's our point right here at the middle so mark our point right there you have to be very careful when doing this you must ensure that you get your exact point so the next one is 340 to 340 as a load the load is 340 as you can see on the graph when it's 340 is 16.2 when it's 340 it is 16.2 16.16 right here this is 340 16.2 right here all right so i have plotted uh this graph or this table of values on my graph okay so we finished plotting the graph what we do next is to join the lines as you can see i have a straight line from 0 to 250 so let me draw the lines from 0 to 250 you can see i have a straight line right there i have a straight line right there from zero to 250 all right a straight line from zero to 250 what next after 250 the the lines begins to begins to bend as you can see right here okay now this point i call this point my a now this here is the elastic limit now from from this point where the the line begins to bend the the force is no longer proportional to the extension but from zero to a the force and extension are proportional so this point is called this point where the line begins to bend is called a limit of proportionality it's called a limit of proportionality all right what next we can now find our stiffness let me use a different marker okay so let's now find the stiffness so as i was saying earlier you can take your gradient or your stiffness from any point on a straight line so let me take my my gradient from uh point 160.160 take it down take it down to this is a ruler when doing this take it down to to 4.8 yes 4.8 and i take it down to my 160. it's my gradient here's my f2 which is 160 and here's my f1 which is uh zero then here is my 4.8 which is my x2 and here is my x1 so from this now we can find the value of our gradient so what next i can never say that the gradient or the stiffness stiffness or the gradient of the force extension graph is equal to F2 minus f1 over x2 minus x1 now my f2 in this case my f2 goes to the y-axis which is the fourth which is 160 is 160 minus my f1 my f1 goes to the y-axis which is zero my x2 goes to the x-axis here is 4.8 remember it is 4.8 millimeters so we convert to meters it gives us 0.0048 so we converted 4.8 here to meters it becomes 0.0048 0.0048 minus x1, x1 is right here which is 0, so therefore we have 160 minus 0 is 160 divided by 0.0048 minus 0 is 0.0048 so therefore our stiffness for this uh for this graph we'll put that in our calculator 160 160 divided by 0.0048 we give us three three point three three three three three point three so we have three three three three three point three newton per meters you know why because it's the force over extension so it's newton per meter so now i've got my stiffness so i can put that in the equation so therefore my modulus e is equal to the stiffness three three point three times the length length is eight point zero divide by the area the area is one point three to seven one point three to seven times ten to the minus six all right so we can put that in our calculator three three three three three point three times eight point zero divide by one point three two seven times ten to the minuses that gives us two points zero zero nine five four three times 10 to the 11 pascal 11 pascal Now we can convert this to gigapascal. How do we convert this to gigapascal? I know that gigapascal is 10 to the 9, so I can move these points backwards. 1, 2. That becomes 2, 0, 1. All right? 2, 0, 1. times 10 times 10 to the minus 2 times 10 to the minus 2 times 10 to the power 11. so that gives us 2 0 1 times 10 remember in maths minus 2 plus 11 gives us 9 i give us 9 pascal so therefore we can say that our modulus e is equal to 2 0 1 giga pascal so that is our modulus but it's not finished right here we are asked to find find the stress at the limit of proportionality so the stress we know is stress is equal to the force over the area now the stress at the limit of proportionality is the point on the graph where the line starts to ah bang so this point right here is the limit of what proportionality and the force at this point is 250 so the force at that point is 250 so therefore my stress is equal to what 250 over the area the area is 1.327 times 10 to the minus 6 so therefore put that in the calculator our stress is 1 it's it's 3 9 4 8 7 5.7 Pascal, 0.7 Pascal.

Now we can also convert this to a Mega Pascal so I can see that my stress becomes, I'll move the decimal point 6 place, becomes 1, 2, 3, 4, 5, 6, 6. So that gives us 1, 8, 8. points we have three here we have nine take one from nine to three becomes four becomes point four mega pascal so that is the final answer so here is how we solve questions that relates to uh the force extension graph so this is our graph and here is the values we got using the graph so thanks for watching

Transcript for:Understanding Hooke's Law and Materials

Transcript for:
Understanding Hooke's Law and Materials