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36. Special Relativity

Inertial Reference Frames

36. Special Relativity

Inertial Reference Frames

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concept

Inertial Reference Frames

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Hey, guys, In this video, we're going to talk about one of the most important concepts in special relativity, which is the concept of an inertial reference frame. Okay, let's get to it. Now. You can broadly break up reference frames, which are just corden systems that you construct arbitrarily to measure things right. If we're tryingto measure displacement, let's say if in object starts at some point x y and moves to some new point x one y one. The only way that we can say that it started at some position X y and ended at some new position is if we construct a coordinate system right in this case, a Cartesian X Y regular horizontal vertical coordinate system to measure the initial and final position. So that's coordinate system dependent. And the coordinate system just represents a reference frame, something with which we can refer Teoh right to make a measurement so you can broadly break up reference frames into two categories inertial reference frames and non inertial reference frames. Okay, inertial reference frames are ones that move at a constant velocity. Okay, so the frame itself is either at rest or moving at a constant velocity Ah, perfectly good example is a frame, which is at rest, right? Just you standing somewhere in a lab making measurements. Um, a good example of a moving inertial frame is a car. For instance, if your car is going at 20 miles an hour, when you're sitting in the car, you feel like you're at rest, and you can make measurements inside the car within your reference frame. But to an observer outside the car standing on the side of the road, they see your reference frame moving at a constant velocity. So the measurements that you make in your reference frame aren't going to be the same, necessarily as the measurements the outside observer makes in his reference frame. But there's still inertial frames because your cars moving at a constant velocity and the guy on the sidewalk is at rest. Both of those are inertial frames, non inertial frames. As the name implies, our frames that are not inertial. And if inertial frames without a constant velocity, non inertial frames move at a changing velocity, which means right that they have acceleration. Okay, so accelerated reference frames are non inertial frames. Okay, constant velocity frames are inertial frames. Okay, so, um, inertial frames are typically subdivided into two once again. Broad categories. The lab frame and the moving frame. Okay, or sorry. Rest, frames and moving frames. Rest frames are frames that have a quote unquote zero velocity. And I put the quotes there specifically, and I'll get to those in a second. Moving frames, as the name implies, have a quote unquote non zero right velocity. The lab frame is the most common type off rest frame, and it's just a frame that's at rest with respect to the earth or specifically, the Earth's surface. Okay, that's where you construct a laboratory. You build it on the surface of the earth, you put tables, you put instruments and you're making your little measurements inside that stationary reference frame in the lab. That is by far the most common type or the most common example of arrest frame friends that move at some velocity. Okay, Uh, sorry at the same velocity at some event is specifically how I worded here. And this is typically how special relativity is done. You're interested in events. You're interesting in things that are happening. Okay, so if that event, let's say It's a particle that's unstable. That can decay. Okay, so there's a little Adam that's moving along. It's not very stable, so eventually it's gonna break apart that decay, that breaking apart. That's the event that you're interested in, that Adam can be moving very quickly in the lab frame, like 1% the speed of light, 10% the speed of light. The lab frame is with scientists at rest in the lab, and they're going to see that Adam whizzing by. But if you were to construct a reference frame that moves with the atom, that would be considered what we would call the proper What kind of frame? Okay, the proper frame is the one that moves at the same velocity as an event, right? I put event in quotes. That's exactly what I was just explaining that we care a lot about events when discussing relativity. And so the proper frame is the one that that's going to be going with the event. Um, one of the most common events that you're gonna discuss in relativity is going to be a ticking clock. I know that doesn't really sound like an event, but this is typically how physicists phrase it. A ticking clock can be moving or can be stationary. For instance, you could be standing and on your wrist. You could have a little watch, right? That's ticking. Bye. Or somebody who's in a car moving past. You could be wearing a watch that's ticking by. If you are interested in your watch, then the lab frame or the rest frame is the proper frame. But if you are interested in this guy's watch, then the moving frame, the frame moving with the car is the proper frame. Okay, now something that I need to key in on because I said it specifically. Here are these quotes. Okay, why are they in quotes? Why is it zero and non zero in quotes? Well, it's because that doesn't actually mean anything. What to zero velocity mean? I said the lab frame, which is the most common type of rest frame, is at rest with respect to the Earth. There's no such thing as absolutely at rest or absolutely moving. That's not an actual physical concept. Everything is moving relative to one another or at rest relative to one another. But there's no what sort of universal coordinate system where you could say something is definitely at rest or something is definitely moving. Typically, special activity problems are going to sort of be anchored to the Earth. Okay, because that gives us as people. Right? I mentioned it here. Us as humans, right? Us is people unease. Easier way to understand the problems and what's going on. If you anchor things to the surface of the Earth, you say the surface of the earth is at rest. If you're measuring things relative to the surface of the earth, there, at rest, if you're measuring things moving relative to the surface of the earth than they're in motion, that just conceptually is a lot easier for people to understand. Out in space, though, which there are quite a bit of special activity problems that are not anchored to the earth but occur out in space. Um, you can't use the Earth as a reference point. You can't say that the earth is stationary, and so everything's, uh, stationary relative to the earth is stationary. Everything moving relative to the earth is moving. You can't say that because you don't have that anymore. So you just have to arbitrarily choose one lab frame so popular instance or popular example is, let's say that there is a space ship chasing another spaceship. You can consider both of these to be moving frames if you want, or you can consider one of them to be stationary and then the other is moving relative to the stationary one. Okay, this is all stuff. That's probably really confusing right now, and it's really confusing toe everybody when you first see it. But stick with it and what you'll see as we start covering problems is that it will start to make mawr and more sense it'll start to click. OK, but these reference frames air really hard to get used to. Um, at first. Okay, so near the surface of the earth, right. Um, we would typically consider right a lab frame to be at rest relative to the surface of the earth and a moving frame to be moving relative to the center of the earth. We usually call lab frames as s and the moving frames as s prime. Now, that's entirely up to your professor. Your book, What is what? But that's just the typical convention that I've always come across. And that's the convention that we're gonna use in these videos, okay? And also, you is typically used for the velocity of a frame, whereas V will be used for velocities of things within the frame. Okay, So if you see this, you right here is the velocity of that s prime frame. The moving frame relative to the surface of the earth and conversely relative to the lab frame over there. Frame s right, because S is at rest with respect to the earth. So if s prime is moving at you with respect to the earth, it's also moving at you with respect to the lab frame. Okay, because the lab frame once again is at rest with respect to the earth. So if there is some objects moving out of speed V or Velocity V, I could say in the lab frame, if we were to measure the velocity in the moving frame, it would be a different velocity. The prime. Okay, by the way, don't think about these two reference frames as being a spatially separated. Okay, I just have to show them separated so that you can pictorially understand it so you can visualize it. Um, but imagine if this guy was 1 m into that frame. This guy could be 1 m into this frame as well. Okay, so this is the difference between s the lab frame and s prime, the moving frame and those velocities V and V prime, they're not going to be the same. Okay, now, the last thing to discuss before we wrap up this video on inertial frames is that non inertial frames aren't important. What is important to understand is that they are ignored in special relativity. Okay, Special relativity never deals with non inertial frames. Special relativity. Onley deals with inertial frames. Okay, General Relativity. The second theory of relativity published by Einstein Much later on, actually, 10 years later that deals with non inertial frames. Okay, Now, technically, the earth the surface of the earth is moving in a circle. And since it's technically moving in a circle, it is actually a non inertial reference frame. Right? It's just that the earth rotates very, very slowly, so we don't really notice the rotation of the earth. When you're in a car and the car starts to accelerate, you feel the acceleration of the car But when you're just standing on the surface of the earth, you do not feel that acceleration because it's almost unnoticeable. It's such a small acceleration. Okay, now, the fact, though, that the Earth is a non inertial reference frame does actually have really life ramifications. Specifically, there's the Coriolis force, which is responsible for how hurricanes rotates. And then the centrifugal force, Um, which actually alters slightly the gravitational acceleration at the pole of the Earth and at the equator. So if you have the earth and here's the equator, the gravitational acceleration here is gonna be slightly different than the gravitational acceleration would be at the equator. Because of the centrifugal force, it's almost It's a very, very, very small and hard to measure difference. But the difference is actually there. Okay, But once again, thes aren't, um, important to our discussions there just real life ramifications of non inertial frames. Alright, guys. So that wraps up this introduction to inertial frames. Even if you don't quite understand them at this point, that's okay because we're gonna be using them continuously throughout our discussion of special relativity. And the best way to really understand them is to start seeing problems where we start using those inertial frames. All right, thanks so much for watching guys. And I'll see you in another video.

2

Problem

Problem

Calculate the quantum number for a glycerol droplet with a radius of 1.2 μm while it moves in a 22 μm long box at a speed of 1.2 μm/s. (Assume the density of glycerol is 1261 kg/m³).

A

6.53×10¹⁶

B

5.22×10¹⁷

C

7.23×10⁸

D

2.55×10⁸

3

Problem

Problem

Determine the ionization energy of a hypothetical element that exhibits the energy levels shown in the figure below.

A

1.8 eV

B

2.8 eV

C

3.6 eV

D

8.2 eV

4

Problem

Problem

An 18-nm long box, divided into 3-nm and 15-nm sections by a removable partition, contains an electron at its third energy level (n=2) inside the smaller region. After temporary partition removal and reinsertion, the electron is in the larger section. What's its new quantum state?

A

10

B

20

C

25

D

30

5

Problem

Problem

An electron has the following energies: (i) 0.60 eV, (ii) 1.40 eV, and (iii) 1.80 eV. Determine the penetration distance for this electron in a potential well of depth U₀=4.00 eV.

A

(i) 0.105 nm (ii) 0.121 nm (iii) 0.131 nm

B

(i) 0.121 nm (ii) 0.105 nm (iii) 0.141 nm

C

(i) 0.105 nm (ii) 0.121 nm (iii) 0.161 nm

D

(i) 0.151 nm (ii) 0.121 nm (iii) 0.105 nm

6

Problem

Problem

In a rigid box with a length of L = 1.00 nm and for the quantum state with quantum number n=4, where is the particle most likely to be found along the box's length?

A

1/8 nm, 3/8 nm, 5/8 nm, 7/8 nm

B

1/4 nm, 1/2 nm, 3/4 nm

C

1/8 nm, 2/8 nm, 3/8 nm, 4/8 nm

D

1/4 nm, 3/2 nm, 7/4 nm

7

Problem

Problem

A molecule with O-H bond absorbs infrared light at a wavelength of 2.9 micrometers (μm). Calculate the energy for the first three states of vibration for this O-H bond.

A

E₀=0.21 eV , E₁= 0.64 eV, E₂=1.1 eV

B

E₀=0.21 eV , E₁= 0.71 eV, E₂=1.0 eV

C

E₀=0.21 eV , E₁= 0.58 eV, E₂=1.1 eV

D

E₀=0.21 eV , E₁= 0.64 eV, E₂=1.2 eV

8

Problem

Problem

An electron is confined within a one-dimensional rigid box with a length of 0.159 nm, approximately thrice the Bohr radius. Determine the three lowest electron energy levels.

A

E₁= 14.9 eV, E₂=60 eV, E₃=104 eV

B

E₁= 60 eV, E₂=14.9 eV, E₃=134 eV

C

E₁= 60 eV, E₂=134 eV, E₃=14.9 eV

D

E₁= 14.9 eV, E₂=60 eV, E₃=134 eV

9

Problem

Problem

Identify the wavelengths in the atom's emission spectrum due to quantum shifts between its three lowest energy states. Considering the atom as an electron trapped in a one-dimensional box of 0.159 nm, about three times the Bohr radius. Label each wavelength as λ_n→m to indicate the specific transition.

A

λ_2→1= 28.0 nm, λ_3→1= 10.0 nm, λ_3→2= 20.0 nm

B

λ_2→1= 10.0 nm, λ_3→1= 27.0 nm, λ_3→2= 17.0 nm

C

λ_2→1= 17.0 nm, λ_3→1= 10.0 nm, λ_3→2= 27.0 nm

D

λ_2→1= 28.0 nm, λ_3→1= 10.0 nm, λ_3→2= 17.0 nm

10

Problem

Problem

Consider a particle confined along the y direction and characterized by the following wave function:
Ψ(y)=Ψ₀√(1 -y²/4 mm ²) if |y| ≤ 2.0 mm and Ψ(y)=0 if |y| ≥ 2.0 mm.
Determine the probability of locating the particle within a distance of 0.50 mm from y=0.0 mm.

A

38 %

B

52 %

C

60 %

D

72 %

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