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Anderson Video - Length Contraction Example

Professor Anderson
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<font color="#ffffff">Okay.</font> <font color="#ffffff">I don't know if I want to do 16 here though, so send them a message back</font> <font color="#ffffff">saying we might not get to it.</font> <font color="#ffffff">Let's take a look at 17 first.</font> <font color="#ffffff">So 17 says astronaut Hans leaves the Earth in a spaceship traveling at a speed</font> <font color="#ffffff">of 0.28 C relative to the observer, Brian, at rest on Earth.</font> <font color="#ffffff">Hans is holding a pencil at an angle of</font> <font color="#ffffff">30 degrees with the direction of travel as seen by Hans.</font> <font color="#ffffff">What angle does the pencil make with the direction of travel as seen by Brian on Earth?</font> <font color="#ffffff">All right, this sounds like a good difficult problem so let's see if we can visualize what's</font> <font color="#ffffff">going on here.</font> <font color="#ffffff">Okay, here's Brian standing here on Earth</font> <font color="#ffffff">and he's observing Hans flying by in a spaceship, so here is our spaceship,</font> <font color="#ffffff">and Hans is standing in there and he's holding a pencil, and that pencil is</font> <font color="#ffffff">making an angle of 30 degrees relative to the direction of travel.</font> <font color="#ffffff">Okay, and the whole spaceship is moved along at speed V which is 0.28 C.</font> <font color="#ffffff">And the question is what angle does the pencil make according to Brian?</font> <font color="#ffffff">Okay. So it's kind of a weird question, right?</font> <font color="#ffffff">I mean this astronauts flying by and he's holding a pencil up at some angle,</font> <font color="#ffffff">it almost seems like this guy should see the pencil at the exact same angle</font> <font color="#ffffff">but that's too obvious an answer, right?</font> <font color="#ffffff">And in fact we're not going to be suckered into answering A, 30 degrees,</font> <font color="#ffffff">so what is really going on here?</font> <font color="#ffffff">This is a little bit related to what we said yesterday in class</font> <font color="#ffffff">when we talked about length contraction.</font> <font color="#ffffff">Okay, length contraction is the following. In the moving frame if you measure L naught</font> <font color="#ffffff">by the guy in the spaceship, then L as measured on Earth is going to be shorter.</font> <font color="#ffffff">Okay, this number is smaller than one, so this</font> <font color="#ffffff">is going to be L not as measured in the spaceship, this is as measured on Earth.</font> <font color="#ffffff">Okay, in this case that would be Brian, Brian's the one that stayed on Earth.</font> <font color="#ffffff">L naught is the measurement in the spaceship,</font> <font color="#ffffff">so that's measured by Hans.</font> <font color="#ffffff">Okay, but they're not asking about the length of the pencil,</font> <font color="#ffffff">they're asking about the angle that it makes relative to this horizontal axis.</font> <font color="#ffffff">Hmm.</font> <font color="#ffffff">What can we say? What we said in class yesterday was imagine you have a box</font> <font color="#ffffff">and that box has height H naught and length L naught,</font> <font color="#ffffff">and now you take that box and you start moving it at some fast speed V.</font> <font color="#ffffff">Does the height of the box change? No.</font> <font color="#ffffff">The length changes, this length contraction is only in the direction of motion.</font> <font color="#ffffff">Okay, so the box shrinks in that direction.</font> <font color="#ffffff">So by analogy here, this pencil that is angled</font> <font color="#ffffff">looks kind of like a triangle.</font> <font color="#ffffff">Here's our pencil.</font> <font color="#ffffff">Okay, but there is a horizontal component and there is a vertical component,</font> <font color="#ffffff">and that horizontal component, let's call it L naught, and the vertical</font> <font color="#ffffff">component, let's call it H naught, and this thing let's call it theta naught.</font> <font color="#ffffff">But when it starts moving and Bryan takes a measurement of it,</font> <font color="#ffffff">L naught shrinks, H naught stays exactly the same</font> <font color="#ffffff">and so in fact the pencil looks like it is more upright,</font> <font color="#ffffff">and let's call that angle theta.</font> <font color="#ffffff">H naught stays the same but the L shrinks.</font> <font color="#ffffff">Alright.</font> <font color="#ffffff">What is the</font> <font color="#ffffff">tangent of theta naught? The tangent of theta naught is H naught over L naught.</font> <font color="#ffffff">What's the tangent of theta? Tangent of theta is H naught over L,</font> <font color="#ffffff">but H naught over L is the same as H naught over</font> <font color="#ffffff">the square root 1 minus V squared over C squared,</font> <font color="#ffffff">all of that times L naught.</font> <font color="#ffffff">And H naught over L naught, we know what that is.</font> <font color="#ffffff">That's just the tangent of theta naught and so now you can put all this stuff together.</font> <font color="#ffffff">And so we have the following. Tangent of theta equals H naught over L naught</font> <font color="#ffffff">1 over square root, 1 minus V squared over C squared, but that is 1 over the square</font> <font color="#ffffff">root 1 minus V squared over C squared times the tangent of theta naught.</font> <font color="#ffffff">And now you know all of that stuff, you know V, you know theta naught,</font> <font color="#ffffff">you can calculate theta. So let's plug in some numbers and see what we get.</font> <font color="#ffffff">We've got 1 over the square root 1 minus 0.28.</font> <font color="#ffffff">That thing is squared, the C's are of course going to cancel out,</font> <font color="#ffffff">and then we have the tangent of 30 degrees</font> <font color="#ffffff">and we're going to take the arctangent of that whole thing,</font> <font color="#ffffff">and that's going to be our theta.</font> <font color="#ffffff">Okay, a few of those answers don't make any sense at all, 30 degrees,</font> <font color="#ffffff">that doesn't make any sense because we know the angle is going to change.</font> <font color="#ffffff">29 degrees doesn't make any sense because we know that it's going to get steeper.</font> <font color="#ffffff">So the other options are 31 degrees, 33 degrees, and 90 degrees,</font> <font color="#ffffff">but 90 degrees, it would be standing straight up and that means that the bottom side of</font> <font color="#ffffff">that triangle would have to shrink to 0, which would happen when you're going at</font> <font color="#ffffff">the speed of light, so we're not going at the speed of light, so really we're down</font> <font color="#ffffff">to two options: 33 degrees or 31 degrees. </font> <font color="#ffffff">Did anybody punch it in and get an answer for this?</font> <font color="#ffffff">>> (student speaking) 31 degrees.</font> <font color="#ffffff">>> 31 degrees.</font>