36. Special Relativity

Consequences of Relativity

# Length Contraction

Patrick Ford

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Hey, guys. Now we're going to start talking about the second consequence of the second part of special relativity, which is length contraction. All right, let's get to it now, because time is measured differently in different inertial frames. So this is actually not its own consequence. Technically, it is just a consequence of time dilation. Okay, because time is measured differently in different reference frames. Length is also going to be measured differently in different reference frames, and this fact is known as length contraction. Okay, so we have time dilation, which said that if you measure time in the proper frame time and the non proper frame is going to be dilated, right, time is going to be longer. If what length contraction says is if you measure the length and the proper frame lengthen, the non proper frame is going to be contracted, it's going to be shorter. Okay, so just be on the lookout for that that you're contracted lengths. Your non proper length should always be less than the proper lengths. OK, now, in order to understand where length contraction comes from, we need to imagine measuring a rod in two different ways. First, we're gonna imagine measuring it in its proper frame, which means at rest with respect to the rod. Okay, at rest, with respect to the distance that we want to measure now, because the frame that the rod is in is moving. We want to imagine a clock that is stationary in the lab frame moving past the rod. Because if the clock a stationary in the lab frame and the rod is moving past it, that's the same in the labs. Sorry. In the rods frame in the proper frame as the clock right, which I'm holding in my right hand moving past that length. Okay. And basically all we're going to do is we're just gonna click the clock when we pass one end, let it pass the other end and click it off. So it's like a stopwatch when it clears the other end. So we're just measuring how much time is elapsing as the clock passes. And given that time, we will get some measured length. Okay? Based on how quickly the rod is moving now in the lab frame, instead of having a moving clock, the clock is stationary. Remember that the clock was always stationary in the lab frame Onley When we are in the proper frame of the moving rod, does the clock appear to be moving? Right now, the clock is stationary and the rod itself is moving past the clock. So same exact idea. The rod. It's moving at the same speed you that the frame was moving. Um, the proper frame. So this rod is going to pass the clock, and we're gonna click it on. When the rod just approaches the clock, start measuring time, click it off just as a rod leaves. And we're gonna measure a different time, right? Because the time is different between the proper and the non proper frame. Right? We have time dilation, so those two times that we measure have to be different. Now, if you actually work through the equations, you get that the length and the proper frame remember, the proper frame is the proper frame for the rod, which means that the rod is rest at rest. Okay, the non proper distance, right? The non proper length is the one measured in this case in the lab frame. And if you put them together, you're going to get something that looks like this. And if you use the time dilation equation, you're going to end up with the proper length divided by gamma. And remember that because the Lawrence Factor gamma is always going to be larger than one, the contracted length L prime is always going to be less than the proper length. L not Okay. This is the opposite logic for time dilation because for time dilation, you get this equation with gamma in the numerator. Since GAM is always greater than one dilated time always larger than proper time for length contraction. Because gamma is always in the did not sorry, because gammas in the denominator and gamma is always larger than one. You always get a smaller, non proper length, right? A contracted length. Okay. Very simple problem here to get us started in length contraction. A spaceship is measured to be 100 m long while being built on Earth. That means that that is the proper length. While it's being built on Earth. We're assuming that the people who are building it and measuring it are at rest with respect to the spaceship. Why would they be building the spaceship as it flew by them right? That doesn't make any sense. So that 100 m should be the proper length. Now, if the spaceship were flying past somebody on Earth, they would measure the contracted length the non proper length off that spaceship. Because now that spaceship is moving past the observer at some speed. Okay, First, let's just sold for gamma. That's one over the square root of one minus. You squared over C squared. And like most problems you, the speed is given in terms off the speed of light, right? 10% speed of light means that U is 0. times. See? So this is one over the square root of one minus 0. squared and this is going to be 1005 Okay. And then this leads us to the conclusion that the contracted length, which is 100 m over gamma, is actually going to be 99. m. Okay, so half a meter short, shorter than it waas, right. Basically half a percent shorter in length, going 10% the speed of light, which is very, very, very fast. You only get a half a percent of drop in length. Okay, Alright, guys, that wraps up this video on length contraction. It's not that big a deal. It's actually much easier than time dilation because proper lengths are easy to recognize its just measured at rest with respect to the object and then applying length Contraction. Super easy. Alright, guys, Thanks so much for watching. And I'll see you guys probably in the next video.

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