Length of Earth's Orbit around the Sun is 299 million km wide

Transcript

the solution to Problem 3, and look at length of Earth's orbit. So the period of the Earth's orbit is approximately 365.25 days, and we're told to use this fact, and Kepler's third law, to find the length of the major axis of the Earth's orbit, and you will need the mass of the Sun, capital M equals 1 .99 times 10 to the power of 30 kilograms. and the gravitational constant, capital G equals 6 .67 times 10 to the negative 11 newtons times meters squared divided by kilograms squared. Those are the units. All right, so let's take a look at the solution. So from Problem 2, and that's basically this part right here, this is the third law, this right here. So what I'll do is I'll just copy this, actually. We're going to need that. I'll just copy that. So from a problem to end converting the period of T of the Earth to second since the unit Newton, in the end, there's a Wikipedia link, in the gravitational concept has units of kilogram meters per second. In other words, mass times acceleration. This is just... Yeah, acceleration is meters, distance divided by seconds, squared or second per, yeah, basically meters per second per second, and that's how the rate of velocity is changing. So that's this Newton right here, and that involves seconds. So, and since we're given this period is in days, we want to convert that to seconds, and then throw that in there. And so I'm going to put this right here. So t squared equals to four. pi squared over gm, a squared, and again, m has kilograms, but then now this Newton has seconds. So that's the thing we need to convert it over there, and what we'll do is, well, T equals 2. I'll also put it right here, actually. I'll put it down, I'll put this in a circle, I'll move this around here, so that is our, from Problem 2, this is Law, so that's Law 3 from Problem 2. All right, and then we know that T is equal to, for the Earth 365 days, 3.25, yeah, 365.25 days. Now we need to convert that to second. So we know that there's 24 hours in a day, so 24 hours in a day. So then again, we just cancel the units, so then this would cancel. with this, again, and then left with hours, but then there is also 60 minutes. Yeah, 60 minutes, I'll put, yeah, I'll just put 60 minutes in one hour. Put an hour like that. And then there's also 60 seconds. Now let's put 60 seconds over one minute. One minute. So then basically what we end up having is, 365.25 times 24 times 60 times 60, and this is actually going to be approximately equals to 1, I mean, 3.1558 times 10 to the 7 seconds, or there's a lot of zeros there, that's how long it takes for the earth to rotate about the sun, so it's period. So that's how many seconds it is. And I'm just going to box this in, actually. All right. All right, so that's what we have. And now we're asked for, again, we want to find the length of Earth's orbit, to find the major axis of Earth's orbit, so how big it is. And that's going to be, well, we just solve for a cube there. So we basically do this, move it over all there. So we would move everything over to the other side. So we get a cubed. is equal to, this is going to be, well, we're going to have to have a t squared times by gm, and, yeah, and divided by, instead of writing t square, I'm going to write this as actually a GM, t, change order, and then t squared, and then divide this by 4 pi squared. All right, so that's what we have. Now, this is going to be approximately equal to all those figures there. Because those ones aren't exact value. So G is going to be, G is 6.67 times 10 to the negative 11. Newton. So 6.67 times 10 to the negative 11. And this is going to be, what is it, Newton's meters. This is going to be Newton meters squared over kilograms squared. And so that's the gravitational constant. And then this is going to be now mass of the sun is 1.99 times 10 to the 30 kilograms, right? So just so we know all the units have to all line up, so this is going to put a bracket like that. And then t squared. is going to be what we just solved. So this is going to be 3.1558. And again, I will do a calculation check. So we plug that in the calculator, you get that, and 3.1558 times 10 to the 7s, like that. And then this is squared. All right, and then divide this all by 4 pi squared. All right, here I just shrunk it all the way inside here. And again, notice the units, we have this Newton's. Remember, Newton's is going to be, it's just kilogram mass times acceleration, so it's going to be KG. Yeah, it's going to be KG times by meters per second squared. So in other words, we're going to get this KG, and then there's a KG here. and then that becomes, well, cancels with this KG squared, and then we have a meters square times meter, we're going to have a meters cubed, and then we have a second square, this cancels with the second squared there. So again, that's the units, and these other ones don't have any units, those are just random constants. All right, related to the circle, again, whenever you have 4 pi, a circle, or ellipse, and so on. So this is going to equal to roughly, and when you plug this into the calculator, you're going to get 3.34. 3, 4, 8, around there, and I'll do a calculation check soon, times 10 to the power of 33, and then now we have the units of meters cubed, like that. And then what that means is, well, we take the cube root of it, so A is going to be, well, the cube root, and that's going to be equals to 1 .496 times 10 to the power of 11, like that, meters. And yeah, so we get the cube root, gets the meter, so that's the length of the major axis. Alright, and yeah, basically now what we get is, well, this is A, but again, notice that, and recall that the length of the major axis is actually 2A. So again, if you recall above here, so the major axis is 2A. So that was just from origin to the end. So yeah, it's actually 2A, so we'd have to multiply that by 2. So, and again, we're asked to get the length of the major axis. Yeah, so it's not just a. So this is just half, one half major axis. Alright, so thus the length of the major axis of Earth's orbit, that is two ways, approximately 2.99, so multiplied everything by 2, times 10 the power of 11, which is equal to 2.99 times 10 the power of 8. And again, when you divide this by 1 ,000. kilograms, I mean, kilometers, so there's 1,000 meters in a kilometer, so divided by 1 ,000, in other words, divided by 3 zeros, so we get 8 instead of 11. All right, and here's a calculation check, so that I did above here, so 365, and again my OneNote calculator, my OneNote thingy, this Microsoft OneNote has a built-in calculator, so 365.25 times 24 times 60, so just plug that in, and all you do is write. Let's go, I mean, we can just erase this, and then write space, equals space, and it automatically solves it. And again, 3.1, they have a 31 ,500,000, 5,000, 600, and then we could just use the dots there. So we have dot, 3.155 round us up to the 8, and then there's going to be 6 zeros plus 7. Those are power of 7. Now, yeah, basically, yeah, that's how many orders of magnitude there is. a how many digits there. And then this one right here, 1 over 4 pi squared, that's this whole setup here, times it by 6.67 times 10 to power of negative 11, times 1.99 times center power of 30, and then times 3 .1558 times the power of 7, and then all squared, again, put this in the one -hut calculator, it automatically solves this is actually... Quite remarkable. So there's 3.34840 all the way, and then it says E33, that's the same thing as writing, times 10 to power of 33. So that's just the exponent, 13. All right. So 3.348, and then that's what we have, 3.348 times 10 to power of 3. And that's meters cubed. And then we need to take the cube root, and again, you just take the cube like that, and automatically solves it. Space, that's 1.49. 6, and then times, and then you would have to get, there's E11, same thing as 10 to 11, and then when you multiply by 2, so there is this number, 1 .496 times 10 to power 11, times by 2, we get 2.99, or, and then I'll do it to the E11, and again, you can just use this, I'll do this just for completeness, and per space, automatically calculates, or 2.99 times 10, power of 11, and divided by 1,000, you get the setup here. which is what we just did. All right, so that's what we get. So, yes, fascinating stuff there.

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