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Thermal expansion of ring

How much does the ring expand on heating. This is a problem that seems quite complicated, but it's actually a linear expansion problem. In this video we will explore how to tackle such problems. Created by Mahesh Shenoy.

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Video transcript

a copper ring has a radius of 200 centimeters at 50 degree Celsius find its radius at 550 degrees Celsius given its linear expansion coefficient is 17 times 10 to the power minus 6 Kelvin inverse now this number called the linear expansion coefficient is just a number that tells us how much something expands on heating so this number is telling me or telling us how much copper expands on heating and here's the way to think about this so imagine you had exactly one meter long copper wire so assume this is copper and say you heated up increase this temperature by exactly one Kelvin then the whole thing would expand by that number that's what that's what this number really is and similarly similarly you would also cool it down by one Kelvin say you you you decrease the temperature by one Kelvin and then the rod would shrink and again now the rod would shrink by exactly that value that's how you think about this number all right and we have spoken about this in a previous video in detail and so if this sounded a little bit confusing to you or maybe you require more clarity on that it would be great idea to watch that video first and then come back over here so with this info please pause the video and see if you can try and solve this yourself and then we'll solve it together all right let's do it let's first write down what's given to us we have a copper ring which has a radius of 200 centimeters so here is our copper ring copper ring it has a radius radius let's call that radiuses R and that radius R is 200 centimeters at 50 degree Celsius which means the current temperature T is 50 degrees Celsius we need to find its radius at 550 degree Celsius so you understand what's going on we are heating it up and it's temperature is increasing and so what happens so let's see if to hear this up here to hit this up then the temperature would increase and it would expand and you might see something like this and so notice it's radius also starts increasing as it expands and so what we need to do is when the temperature has reached 550 degree Celsius 550 degree Celsius we need to calculate what the new radius is going to be okay so we would be interested in the change in the temperature right that's what matters to us not what the temperature is but how much has the temperature changed let's call that as delta T the temperature change is 550 minus 55 50 minus 50 that is 500 degrees Celsius and what else is given to us is alpha this linear expansion coefficient we usually call it as alpha L that's the symbol we use and that's given to us as 17 times 10 to the power minus 6 Kelvin inverse all right this was the data and now we'll try and use this data and somehow solve the problem the first thing we'll do is figure out a relationship between the changes in the length length of let's say a wire let's begin with that let's let's not worry about the ring right now let's just imagine a wire and let's talk about the changes in length of that wire and let's see how that is connected to the temperature change and alpha L and things like that so from the definition of alpha or definition of this linear expansion coefficient we could say that the change in length Delta L that's going to be equal to our file our file provided the wire is 1 meter long and the temperature changes 1 Kelvin right that's the definition of our file however what if we had let's say a wire which was 2 meter long then what would be the change in the length well think of it this way if you had to read a long wire we could assume it to be made above to 1 meter long wires and then each one would expand by alpha L and then the total expansion would be 2 times our file that makes sense so similarly if you had 3 times the length then the total expansion will be 3 alpha so if you had L meter long wire let's say then do you see that the total expansion would be L times this number hope this makes sense you just multiply by L but again this represents the change in length for 1 Kelvin rice what if you had 2 Kelvin rice well it would be double of this 10 Kelvin rice it will be 10 times this what if you have delta T Kelvin rise whew it would be delta T times this and so this is how we can connect changes in the length do change in temperature enough well all right so this is what we can use now you may think well this is all fine this will work for a wire and everything but we have a ring over here and we are asked to calculate the change in the radius of the ring how do we take care of that well you can think of ring as just a bent wire and the L over here would represent the length of this entire wire and what is the length of this wire that is the circumference so L would be 2 PI R for our case so let's just substitute that so for our case we could write Delta of 2 PI R that would be equal to alpha L alpha L times the length is again 2 PI R 2 PI R times Delta t times delta T but notice that when you're calculating change in 2 PI R 2 pi is constant it's not changing I mean how do you calculate change you usually do a final value minus initial value right so it could be something like 2 PI R 2 minus 2 PI R 1 but notice 2 pi is common in that so we could just pull this 2 pi outside the change right we could do that so we could write this as we could write this as just do that over here so we can save you space later we could just write as 2 pi Delta R right and now Lord is if you compare left and right side 2 pi is common here here we can just go ahead and divide the whole equation by two pi and so this will cancel out and so notice eventually what do we end up with we end up with Delta R equals alpha R alpha L times R times Delta t times delta T and just look at this expression that we just just derived that expression can you see is identical or analogous to this expression the only difference is that L is replaced by R and so now we can go ahead and plug in and solve this but before we do that I just want to give you one secret which can help us solve any problem like this notice over here you may wonder again oh we had to do this derivation and at least here it was a circle so it was a little bit easy maybe tomorrow I will get a sphere or maybe a cylinder or maybe some other complicated shape how do I do this derivation that could be complicated right well here's a secret the secret is that this expression will work for any linear variable what I mean is L over here is any linear variable a linear variable is something that has the unit of centimeters or meters or inches or feet so lengths littler does not mean length of some straight wire it could be radius because the radius is centimeters so next time if I don't calculate changing the radius just just replace L with our it could be circumference it could be worth of a cuboid it could be breadth of a rectangle so think of L in general as any linear variable then you can just substitute you can just skip all the steps in between you don't really have to derive this in in any problem all right so having said that let's go ahead with this problem now notice that we know what alpha L is we know what R is we also know what delta T is so we just have to plug in and figure out Delta R so whatever follows is just algebra all right okay let's do it so alpha L is 17 times 10 to the power - six kelvin inverse times our RS 200 centimeters 200 centimeters let's just make a little bit room over here I think this should be fine all right times delta T delta T is 500 degrees Celsius ah another problem degree Celsius over here Kelvin inverse over here the units are not matching we need to now convert from degree Celsius to Kelvin maybe add 273 subtract 273 somewhere well let's be you have to be a little bit careful over here in fact there's a good news look this 500 degree Celsius is not temperature it's the change in temperature and guess what the good news is the change in temperature in Kelvin is also 500 this is also 500 Kelvin and let me show you how that works suppose we were to you know convert all the temperatures in Kelvin we would add 273 over here oops 273 over here and the 273 over here right degree Celsius that's how you convert to Kelvin 0.15 but that's like that but anyways now when you subtract the temperatures and calculate the change in temperature notice the 273 would just cancel out so even if you had to calculate in Kelvin you'd get the same answer you would still get 500 Kelvin so we might as well go ahead and say this is 500 Kelvin 500 Kelvin and the times for that reason the carrier inverse and can we just cancels everything is fine we end up with the units of centimeters that make sense we want the change in radii to be in centimeters all right let's just go ahead and calculate now we have 17 times 10 to the minus 6 times 200 times 500 5 times 2 is 10 and then we have one two three four four zeroes one two three four centimeters let's see what that gives us that gives us let's see one two three four five that gives us 10 to the minus one when you come multiply these two times 17 is 1 one point seven centimeters all right so what we have done now is we have calculated the change in radius but the problem or the numerical is asking me what's the final radius all right so one last step notice that the initial radius was 200 centimeters now the radius has increased changes positive notice the radius has increased by one point seven so what's the final radius I'm just gonna write the round in this corner the final radius is going to be the initial radius 200 plus 1.7 that's going to be 201 point seven centimeters and that is the solution to this problem