 ## Physics

Learn the toughest concepts covered in Physics with step-by-step video tutorials and practice problems by world-class tutors

36. Special Relativity

# Consequences of Relativity

1
concept

## Time Dilation 12m
Play a video:
2
example

## Time Dilation for a Muon from the Atmosphere 4m
Play a video:
Hey, guys, let's see this problem. Okay? We have mu ons which are very, very tiny charged particles similar to an electron, but heavier. They're admitted from very, very high up in the atmosphere when high energy particles from the sun collides with the atmosphere. So if here's the earth's surface, there's a bunch of right atmosphere, just a Ramallah cules. High energy particles coming from the sun collide with atoms inside the atmosphere and produce Mulan's. They're given by the Greek letter mute those Mulan's travel at 90% the speed of light and, as measured in a lab right, measured with respect to the mu on, it's going to decay a 2.2 microseconds right in their rest frame. So that is the proper time, because we're talking about Mulan's moving at 0.9 the speed of light. So this is a mu on, and this is measured in the rest frame. So here's some dude at rest with respect to the surface, watching Um, yuan fly by at 90% the speed of light but in the yuan's own frame as prime right where that frame was moving at 90% speed, the light, the Muan is static. Okay. The muan has no speed. This is technically the prime. It has no speed and it's going to decay in an amount of time of 2.2 microseconds. This time is the proper time. Okay, because the event that we're interested in is the decay of them Yuan. Okay, The moving clock measures time Mawr Slowly. So in the lab frame. When this guy sees them, yuan fly by. He's going to see them. You on live for a longer time because he will be measuring the dilated time. Okay, Now, what exactly is that amount of time? Well, Delta T Prime is going to be gamma times Delta t not. Which is going to be one over the square root of one minus. You squared over C squared times. Delta t not. Okay. Now, most of these problems the speed you is going to be given in terms of the speed of lights. If we look at this term right here, you squared. Divided by C squared is the same as you divided by C squared. So if we plug in 0.9 c divided by C, you'll see that those speeds of light cancel. So this is just gonna be one minus 10.9 square. And this is typically how these problems they're going to be given every now and then instead of giving it something like 0.9 c, they'll say, like three times 10 to the 6 m per second. But most of these problems, they're going to be given in terms of the speed of light because it just makes the calculation more easy. It just makes it easier. Okay, so gamma. The Lawrence factor. If you plug this into your calculator, you're going to get about 2.29 And this is times 2.2 microseconds, which remember 2.2 microseconds, is the proper time. So the Lawrence Factor says that the dilated time is 2. times larger than the proper time, and this is going to be about five microseconds. So the time that an observer on Earth right in the lab frame measures for them you want to decay is five microseconds, whereas the mu on in its rest frame decays in 2. microseconds. Okay, so this is a perfect example of what's the proper frame? What's the lab frame. What's the proper time? What's the dilated time? The event that we're interested in is the decaying of them yuan. And when we measure that time at rest with respect to the mu on, that's the proper time. When we're watching them, yuan zip. By the time that we're going to measure it, taking to decay is going to be the dilated time. Because we're measuring in the lab frame, not in the proper frame. Okay, Alright, guys, that perhaps with this problem, thanks so much for watching.
3
Problem

The international space station travels in orbit at a speed of 7.67 km/s. If an astronaut and his brother start a stop watch at the same time, on Earth, and then the astronaut spends 6 months on the space station, what is the difference in time on their stopwatches when the astronaut returns to Earth? Note that 6 months is about 1.577 x 107 s, and c = 3 x 10 8 m/s.

4
concept

## Length Contraction 7m
Play a video:
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.
5
example

## Length Contraction for a Muon from the Atmosphere 7m
Play a video:
Hey, guys, let's do this problem. Okay? We've already seen a problem basically the same as this with neurons. But we were looking at time dilation. Now we wanna look at the length contraction aspect of it. So once again, we have a bunch of atmospheric particles high up in the atmosphere that encounter these high energy particles emitted from the sun. And every now and then, and not every now and then it actually happens millions of times a second. There is a collision that's going to produce these heavy particles that are like electrons called Mulan's. Now in the Mulan's rest frame, they have a, um they last, I should say, 22 microseconds. They decay after 22 microseconds. We looked at how long the yuan's would last in the lab frame, given the fact that they're traveling at 90% the speed of light. Now we wanna look at how far they will travel in the lab frame, but specifically we want to use length contraction. OK, I'll actually show after I solve this, that you can use time dilation to arrive at the exact same answer. Well, roughly because of rounding errors, you would arrive at the exact same answer if you didn't have to deal with rounding, because length, contraction and time dilation are actually two different sides of the same coin. Okay, so let's look at this from them. Yuan's perspective. So from them yuan perspective, right? It's traveling well. Sorry. It's at rest in a frame that's traveling at 0.9 times the speed of light. So it's not moving, but distance is rushing past it right, and by distance, I mean atmosphere. Okay, so there's some amount of atmosphere right here that's rushing past the mu on. So how much of this atmosphere is going to pass the muan before it decays? Okay, that's pretty easy. The frame is going at the speed of lights. We know that it decays into two microseconds. So let's just figure out how long chunk of atmosphere that is that's going to pass the muan before it decays, right? That's just going to be the speed that it's going. So it's 0.9 times the speed of light three times 10 to the eight times the amount of time that passes. Right distance is velocity times time, so this is going to be 2.2 microseconds. And don't forget Micro's 10 to the negative six. And so this is going to be meters. The question is, Is this the proper time, or is this the sorry the proper length? Or is this the contracted length? This is actually the contracted length, because the proper length would be the one that we measure with respect to the earth, right? We're talking about the Earth's atmosphere, so if we are at rest with respect to the Earth, we would measure the proper length of that atmosphere. So this is actually the contracted length because the muon is not at rest with the strength of the atmosphere, it's moving through the atmosphere at 90% speed of light. So how far would we measure them? You on traveling in the lab frame That's actually the proper length, right? The lab frame represents the proper length because the lab frame is the one at rest with respect to that chunk of atmosphere that them you want is moving through. So the proper length Sorry, let me write out the length contraction equation length Contraction says it's the proper length divided by gamma, so the proper length is going to be gamma times the contracted length. This is gonna be one over the square root of one point one minus 10.9 squared times, 594 m. Okay, now the Lawrence Factor Gamma. We got in the previous problem, and it was equal to roughly two point Okay, multiplying these together, you're going to get a distance of 13 m. Okay, so that is how far the neuronal travel in the lab frame before decayed. Okay? And this is found just using length contraction. No concept of time dilation was used here. But like I said, leading up to this solution, we can still use time violation and not worry about length contraction at all to solve this particular problem. Because in the lab frame. So this is s prime, right? The moving frame, which happens to be the proper frame for the time. Right. But the non proper frame for the distance The lab frame is the proper frame for the distance, but the non proper frame for the time. Okay, so the yuan is traveling at 0.9 times the speed of lights. So what time would we measure in the lab frame before it to case this is the dilated time? Because in the rest frame of the yuan, we measured the proper time in s prime. We measure the proper time in this case. So this is gonna be gamma times Delta t not. And we showed that this waas thought about five microseconds in the previous problem. So we can straight up just measure length in this case. How far does it physically travel before it decays? Right, That's once again going to be velocity times time. So it's going to be a 0.9 times the speed of light three times 10 to the eight times the amount of time that passes, which is about five times 10 to the negative six seconds. And that's going to be 13 50 m. Okay, so we have some rounding error between these, but that's no big deal. The idea here is that time is proper in this frame, right time is proper in this frame, but length is non proper, right? In this frame, length is proper. By the way, this is l not right. That's the proper length. But time is non proper. And so the whole idea is that length. To get it to be contracted, you have to take the proper and divided by gamma time to get to be non proper, you have to take the time and multiply it by gamma. And that divided by gamma and multiplied by gamma, is going to cancel out when you compare the two results. Right. So these should be. If I had carried enough significant figures, these should be exactly equal, not off by 10 m. But that should because a rounding error. All right, so in this particular problem, we can easily see that length contraction is just a consequence of time dilation. Alright, but in a lot of problems length contraction, the equation is much easier to use. All right. All right, guys, Thanks so much for watching that wraps it up for this problem.
6
Problem

In the following figure, a right triangle is shown in its rest frame, S'. In the lab frame, S, the triangle moves with a speed v. How fast must the triangle move in the lab frame so that it becomes an isosceles triangle? 7
concept

## Proper Frames and Measurements 7m
Play a video: 