|Mass Dilation #1/6 What is mass dilation?|
Hello, everyone, today we're going to be talking about mass dilation. We've learned already about time dilation and length contraction. But mass dilation has the advantages of the fact that it's one of the events that we can regularly observe, and things like particle accelerators. And so, in these large machines that accelerate tiny particles to almost the speed of light, we can notice mass dilation as they move very quickly.
So how do we get to mass dilation Because momentum is a quantity that depends on velocity? And it depends on us. Right? So, it seems a good way to tie in. Now, we know that momentum is conserved in collisions, right? It turns out that even if we go down to a very, very small scale, and then look at individual atoms, we can still see momentum being conserved, no matter how fast they're moving.
So, we're going to assume for the momentum, that the conservation of momentum will always remain the same, even if we're travelling at relativistic speeds, right. Now, according to the principle of relativity, the laws of physics cannot change depending on how fast you're going. Right? You can't see a physical law, like conservation of momentum being violated just because you're moving fast.
So, what this means is that if we notice anything odd happenings in the conservation of momentum, we're going to have to change how we think about momentum. So, to demonstrate up a little example of how that momentum can change, we're going to be doing a thought experiment, except we're not going to be using a very fast-moving train, we're going to be using a collision between two spacecraft. Just because this experiment is a little hard to think of if we only use trains.
So, we have a short animation, showing us two spacecraft coming close to each other and colliding. Right. So, we say at the spacecraft are of equal mass, they're approaching each other at the same speed. Right, we can see the rebounding a little bit afterwards. Now, when they collide, we can see that they start moving away from our line in the middle. Right. So, it means that after the collision, each one of them has a component of motion in the y-direction. And of course, if we conserve momentum, then the component of motion will have to be the same for both, we can say that it's 10 meters per second. Right?
We can say that, that the collision doesn't affect their direction in the x-direction very much, right. So, we can say the velocity in the x-direction is about the same. Makes sense, we can see that the velocity in that direction is not really changing very much, especially if they only gained that amount of speed each way.
Now, let us suppose that instead of looking at it, from this point of view, where the ships are moving toward each other at equal speed, we look at it from the point of view of one of the ships. Now, this means that we are going to be starting to travel at relativistic speeds instead of being stationary. What this means, though, is that our definitions of meters and seconds are going to change. So, we will see what happens to the speeds going up and down, shall we?
So, let us go into the frame of reference of the bottom spaceship. So now as you can see, we start off as stationary. Now, if we look at what the first observer saw, when the spaceships collided, we know that after the collision, we are moving down at 10 meters per second. Right? This makes sense. We can see that the x velocity of this moving spacecraft is going to stay around us. So, we've got momentum conserved in this direction, starts off high to the right, ends up high to the right.
But how fast will this be moving upward? Even that we know about length contraction and time dilation. We don't need to worry about wind contraction. sure, we're moving upward, but we're moving upward at such a slow speed. Probably, that we don't need to worry too much about the distance changing. What we do you need to worry about is the time changing. Because now that we're inside this frame of reference, our seconds are going to be a different length to the first frame of reference.
So, if the first frame of reference says that this ship is going up at 10 meters per second, how fast is it actually going up? Well, we can use time dilation, right. So, if the observer is moving at 0.6 C, relative to us, then one of their seconds turns into 1.25 seconds. So, this ship isn't moving up at 10 meters per second. It's moving up at 10 meters per 1.25 seconds. And of course, 10 divided by 1.25 gives us eight meters per second.
Hello, now we have a problem. We've said the x momentum is conserved, right? Because that pretty much doesn't change. But now we've got two spaceships of equal mass. One's going down at 10 meters per second. One's going up at only eight meters per second. So, it looks like our momentum has changed. So how do we explain this if we still want to keep conservation of momentum? We can't change the velocity. But what else does momentum depend on? That's right, mass.
|Mass Dilation #2/6 Conservation of Momentum|
So, if momentum is not conservative his frame of reference, we need to change how we think of mass. Makes sense? So, this is moving up at eight, this is moving down at 10. This thing's mass is going to have to increase. And exactly how much will it increase by? Well, it'll increase by the same amount, the time increased by, right?
If we have a value in kilogram meters per second for momentum, then if the seconds are increasing, then the kilograms must also increase. And that way, we can keep our conservation of momentum.
So, here's how we end up changing the mass. This equation should start to be looking pretty familiar by now, especially this factor on the bottom, we can see that if we graph this, it looked very similar to our time dilation graph, if not identical.
So, this increase in momentum is called mass dilation. Just like the increase in a time interval is called time dilation. So, we can see that as a particle or any object gets closer and closer to the speed of light, it becomes heavier and heavier and heavier. Once we get to about 85% of the speed of light, we end up weighing or having a mass twice as large as when we were stationary. Right.
And this leads to all sorts of interesting effects, means that if we accelerate a small particle to very, very high speeds when it hits its target, it behaves like it has a huge amount of mass. And a lot of energy is released. So fast-moving, things have more momentum than you would expect. We already know that if an object moves faster, it'll have more momentum, right? p equals mv. So, if we doubled our speed, we doubled our momentum.
Except that because of mass dilation, in reality, if we double our speed, we will slightly more than double our momentum, because the mass will increase as well. And we can observe this in particle accelerators. Let me give you an example.
In the Large Hadron Collider, which is the largest particle accelerator in the world, protons can get up to within a few meters per second of the speed of light. And when their mass is measured, it's measured to be about 7000 times greater than when the protons were stationary. So, it's easy to see that mass dilation is a real effect that really does happen.
Now, these three effects that are time dilation, length contraction, and mass dilation are known as Lorentz transformations. Named after Hendrik Lorentz, of course. So, Lorentz figured out these equations and transformations before Einstein figured out his theory of relativity. He didn't really know what that will be useful for. But obviously, as we can see, they came in handy.
So that's the end of the theory. We've learned about mass dilation. We've seen a thorough experiment as to how it arises. And we know exactly how much the mass will increase based on how fast your brick is moving.
|Mass Dilation #3/6 Relativistic Momentum|
Question 6: A proton and a particle accelerator are travelling at half the speed of light. How does the relativistic momentum, compared to its classical momentum?
So, can you remember what these words mean, relativistic and classical? Classical momentum is the momentum that you learned about back in year 11. When we are talking about p equals mv. Classical momentum says that momentum is proportional to mass and proportional to velocity. And that mass is constant and does not change. relativistic momentum allows for mass dilation.
In relativistic momentum, we have to increase the mass by a factor depending on the velocity. So, let us go through our options. A program does not have relativistic momentum, this is wrong. All objects have momentum, right? Its relativistic momentum will be less equal to or more than its classical momentum.
Now, how does mass change? Does it increase decrease or stay the same? Well, that is right, of course, it increases. So, our answer is that its relativistic momentum will be more than its classical momentum. It'll be larger than Isaac Newton would expect.
Question 7: Which option is correct, the relativistic length of an object is longer or shorter than its proper length, with a length dilation and length contraction, that's right, it's the length contraction. That means the relativistic length is shorter than its proper length.
Remember that the proper length of an object is the length of that object when it is not moving. It's also known as the rest length. Now, the relativistic duration of an event is longer or shorter than the duration in the events rest frame. So, this has to do with duration or time. Now, is it time dilation or time contraction? It is, of course, time dilation.
So, answer here is that the duration of an event is longer than the events rest frame or proper time. Finally, the relativistic mass of an object is larger or smaller than its rest mass. Well, of course, it's mass dilation, right? So, the relativistic mass must be larger than its rest mass.
|Mass Dilation #4/6 Hermit's Rest Frame|
Question eight: A Cobra III spaceship passes a stationary asteroid at 93% of the speed of light. If the ship first mass is 100 tons, find its mass relative to a hermit living on the asteroid. So, what do we use here, of course, we use what we just learned mass dilation?
So, here's our equation for mass dilation, the relativistic mass of an object is increased by this factor. This is a factor, that is always going to be less than one. And if we divide by a number, that is less than one, we end up with a bigger number. So, V over c is going to be 0.93. As we can see, we will square that number. And so, substituting it in we end up with this expression 100 tons of the square root of one minus 0.93 squared.
Now hang on here, you asked, where did the c go? Well, it is quite simple, really. We had defined V as a multiple of c, right? So, V is given by 0.93 c. And we can see that over here, we are dividing that whole thing by c, and then squaring it, the c is just cancelled out.
And so, we end up with just this simple number without units that we can square. All right, so we can use a calculator to evaluate that answer, we will end up at 272 tons, which means that the spaceship when it's moving at 93%, of the speed of light, it is almost three times heavier than in its restaurant. Part B. If the rock government measures that the cover is length is about 24 meters, find its proper length.
So, if the proportion that we worked on before was anything to go by, then it will be a little bit less than three times as long in this rest frame. Right. So, given that three times, 24 is 72, we're expecting an answer. That is a bit less than 72. So, let's use our equations.
Now we know that length contraction is given by this equation, the relativistic length equals rest length times this factor, which is less than one. This means that relativistically will be shorter. Right? But we are already given relativistic length, what we want to find is its proper length, that is the length when it's at rest, l zero. Right? So, we need to rearrange this equation a bit.
There we go. Now looks just like the equation for the last question. Once we substitute in V, we will, of course, have exactly the same denominator for the fraction. So, our equation looks something like this. And simplifying, we end up with 65.0 meters, which just as we expected, is a little bit less than three times as long.
Part C, the pilot of the Cobra waves hello to the hermit on the asteroid. This action takes two seconds in the ship's rest frame. How long does it take in the hermit's rest frame? So now according to the hermit, the pilot, we will call him commander Jameson is moving at 93% of the speed of light. And he performs an event that takes two seconds in his own frame. But we will take a longer time when he's moving. Right.
But that's because time inside this moving frame of reference is going to slow down. So, there's our t zero, we're looking for tV. There is our equation relating the two. Once again, we have a similar factor on the bottom, and we know the factor will be a little bit less than three times as long.
So, we're expecting it. We are expecting the action to take a little bit less than six seconds. Substituting in our numbers, we end up with 5.44 seconds, which is pretty close to the estimate that we gave.
|Mass Dilation #5/6 Speed of Neutron after Collision|
Question 9: An electron with the massive ma is travelling at 0.8c, 80% of the speed of light. When it strikes a stationary neutron and stops, right? conservation of momentum says that the neutron has to start moving. Find the speed of the neutron after the collision.
Now, there are two ways we could be asked to solve this question, we could be used– we could be asked to find the relativistic mass and the relativistic momentum of the neutron. In that case, our equation would look something like this: rest mass of the electron times its velocity over, [sysco] L factor. Except here, instead of V, we will have to have a velocity of the electron, equals and then the [sysco] moments with the neutron. Which will look almost the same.
It turns out that this is really annoying and hard to work out. But the question does not stop there, it says to assume that the neutron does not reach very high speeds. This means that this whole factor at the bottom will be very close to one. And we can forget about it. This equation is much simpler to solve.
And as we'll see, because the neutron does not end up moving very fast, will end up with an answer that is very, very similar to the answer that we would have got had we had the denominator here.
So, let's get started. So, what we're trying to find here is Vn, right? So, we want to make this the subject of the equation. We have another problem, though. We do not know how to how to relate mn to me. Because obviously, we want an answer that is a solid number, and not dependent on one of these masses. Luckily, the question gives us the mass of a neutron, 1800 times the mass of an electron.
So, substituting our numbers, we have 0.8 c, for the velocity of the electron, which was substituted in the denominator as well. And we can see that the mass of a neutron, or at least the rest mass of a neutron, will be given by this number here. Right? Now, we can see that the enemies are going to cancel out. Fantastic.
So, simplifying that and making V the subject of the equation, we have 0.8 C, over 1800 times 0.6. So, this will give us an answer in terms of c, went up getting 0.0007 c. Now, can you see what this bottom factor would become? If we substituted in this number for a velocity, it is not going to be very high, you can do it on your calculator now if you'd like, I can tell you right away, you'll get a number that is approximately one. Dividing or multiplying this number by one is not going to change its value.
It turns out that if you go to the number of significant figures that you can say that is accurate to, it'll end up being at least about five. So, you really don't need to worry about the denominator in this case. This is the reason why when we are talking about classical physics, and collisions between cars or cricket balls or whatever, we don't worry about this denominator.
At everyday speeds, this denominator is so close to one, that even if you left it in, you would get pretty much the same answer. In fact, your experimental error is that is problems with measuring the distance or time of the moving object are probably going to be bigger than the change that this factor would make.
|Mass Dilation #6/6 Cube of Uniform Density|
Questions 10: A cube of uniform density has signs of length two centimetres. So, it's a two centimetre by two centimetres by two-centimetre cube, and a mass of 12 grams in this rest frame. So, it's not moving, calculate the cube density. Density is, of course, mass per volume. So mass is 12 grams, what is the volume?
Well, it's going to be two cubes, because we got a cube. So, we end up with 12 grams over two by two by two. And that, of course, is eight centimetres cubed. So, we are with 1.5 grams per cubic centimetres. Now Part B is going to ask something a little different. Calculate the cube density when it is moving at 60% of the speed of light.
So, in this case, we're going to have to worry about length contraction and mass dilation. Because once the cube starts moving, it is no longer a cube. Instead, it is a square prism. Right, these links are going to be moving– are going to be perpendicular to the direction of motion, so they will not change, the only distance that will change is going to be this one. And because of LinkedIn traction, it will be shorter than two centimetres.
Now. Luckily, we know exactly how much it is going to change by, now the volume isn't the only part of the cube, that's going to change, we have mass dilation to worry about too. That means that our 12 grams are going to change. So, let's write down an expression for the density of the cube now, we have mass over volume equals the relativistically dilated mass times the contracted length of the cube. And then the normal length of the cube which has not changed.
This makes sense, right? This part of the equation is the volume, and the first fraction is the mass. Now hang on, we have in these two denominators, two square roots, which contain the same thing. So, what happens if we multiply two square roots by each other, so we have the square root of y times the square root of y, we're going to end up with y. Right? This means that we can get rid of the square root around these two parts of the denominator. That makes sense, right?
So, we've moved them zero over to the right, because m zero times one times one is still m zero. On the bottom, we have still got the three l zeros. And we have multiplied these two factors together and gotten rid of the square root. Now Hang on, this looks familiar. This is the answer that we calculated in the first part of the question. It is just 1.5. Right.
So now all we need to do is substitute a 1.5 and then a velocity into the left side of the equation, or rather the left fraction in the equation. So, there's our 0.6, we're squaring it, of course, and our 1.5 grams per centimetre per cubic centimetre that we already calculated. And this will evaluate to 1.5625 times our original density, which gives us an answer of 2.34 grams per cubic centimetre.
So, we can see that our density increases not by just that fact that we're used to, but by that factor squared. So, we get a little bit more of an increase in density than we would have expected if we were only using [sysco] linked contraction or using mass dilation. Well, that is the end of the questions. And the section we have covered mass dilation, what it is, how it arises, and quantitatively exactly how it changes when we move at high speeds.