20. Heat and Temperature

Linear Thermal Expansion

# Expanding Steel Measuring Tape

Patrick Ford

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Hey guys, so let's get started with this problem here. Hopefully took a shot at, on your own. It's a little tricky to kind of understand the language of this problem. And I think this really benefits from a drawing. So let's get started here. So we have some steel measuring tape and it's calibrated for a measurement accuracy at 20°C. What does that mean? So I like drawing stuff out basically what it just means is that if you have a ruler If you were to be at 20°C, The markings on the ruler would show 50 m. Right? So you measure this thing out, I'm gonna call this L ruler. The measurements on the market would be 50-50.000 m. And if you were to actually take a different measurement instrument, like a laser pointer or something and you were to measure out a line, then that would also be exactly 50.0 m. Basically the measurement that it's showing you is exactly equal to the real life measurement of 50.0 m. That's what it means to be calibrated at a certain temperature, Then what happens is that your measuring tape is going to increase your gonna sort of increase the temperature to 40°C, right? So when a copy paste this? So then a 40°C, what happens? Well, the actual distance of 50 m doesn't change, right. If you were to measure with a laser pointer or something like that would still be the same. But what happens is the steel ruler has increased its length a little bit right? It's made of steel and expands a little bit like this. Now, what happens is the measurements, the markings on the ruler don't change though. There's still show exactly 50 m at this distance here. But the real measurement won't be 50.0 m anymore. This is 0.50 50.0. The actual measurements will actually be this line over here. That's really what we're trying to find here. So, this is what the actual distance is and that's what we're trying to find. All right, So, the distance on the ruler still gonna be 50.0. But the l actual is what we're trying to find. All right, So, that's kind of what's going on this problem. Hopefully, that kinda makes sense. Now, which equation we're gonna use? Well, basically what happens is that this l ruler is kind of like our length initial, right? So this is kind of like r L knots. And this l actual is really kind of like R L final, Right? So we have some initial distance, it expands to some final distance and this is what we're trying to find. So because of that, we're actually gonna use this second equation over here. Right? So this guy, So we're gonna have L F equals L knots and then one plus alpha times delta T. Alright, so basically this is el actual equals L. Ruler And then one plus alpha times delta T. Alright so all we have to do here is just go ahead and plug and chug. So we've got el actual equals uh this is gonna be L ruler which is going to be 50.1 plus. Then we've got the 1.2 times 10 to the minus five. That's the linear expansion coefficient for steel which is given to us. And then we have to figure out the delta T. Write the change in the temperature number delta T. Can be in Celsius or kelvin. It doesn't matter which one you use. So basically the difference between 20 and 40 C, this delta T. Here is just 20. Right? That's so you can plug in. So we've got 20 like this. So when you plug all of this in, what you should get you should get 50. m. So what happens here is that at a higher temperature of 40°C, even though your ruler will say 50. The actual distance that it's measuring Is going to be the length of this line which is 50.012. So your ruler is going to be inaccurate because it's calibrated for a certain temperature. That's kind of what's going on in this problem. Hopefully that makes sense and let me know if you have any questions

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