03 - Inverse Variation & Joint Variation - Part 1 (Hyperbolas & Inverse Square Law) - By Math and Science
Transcript
00:00 | Hello . Welcome back to algebra and Jason with math | |
00:02 | and science dot com . The title of this lesson | |
00:04 | is called inverse and joint variation . This is part | |
00:08 | one of two . Now , if you remember in | |
00:09 | the last lesson we introduced the very important concept of | |
00:12 | what we call direct variation . So in a direct | |
00:15 | variation , in contrast to what we're learning here , | |
00:17 | direct variations when two variables are related to one another | |
00:20 | directly , which means one variable goes up . The | |
00:23 | partner variable also goes up with it . Right . | |
00:26 | We talked about that . That's called direct variation here | |
00:28 | inverse variation . What do you think the word in | |
00:31 | verse means ? Um some people might say in verse | |
00:34 | might be the opposite of something or if something is | |
00:36 | turned inside out , it's kind of inverse when you | |
00:40 | flip it over . Maybe that's the inverse . There's | |
00:42 | different ways of thinking about it . But in verse | |
00:44 | in this context kind of does mean opposite . So | |
00:47 | instead of the two variables both going up together , | |
00:50 | like they do with a direct variation for an inverse | |
00:53 | variation , one variable goes up , the other partner | |
00:56 | variable goes the opposite direction going down . That's why | |
00:59 | it's called an inverse variation . So we're going to | |
01:01 | explore that and we're going to actually , it's very | |
01:04 | soon here in the lesson , we're gonna talk about | |
01:06 | some examples . And the most famous one that's going | |
01:08 | to have an inverse variation is the law of gravitation | |
01:10 | . So I'm actually gonna introduce and show you the | |
01:13 | law of gravity that has some direct variation and also | |
01:16 | some inverse variation . So you can you understand how | |
01:19 | we use this stuff in the real world . So | |
01:22 | let's recall for just a second because direct variation and | |
01:26 | inverse , they're kind of like peanut butter and jelly | |
01:28 | , they go together . So we're going to recall | |
01:30 | the direct variation and we did this extensively . If | |
01:36 | you haven't looked at that , you probably should go | |
01:37 | back and look at that before getting too far down | |
01:40 | the road here . Now we said in the last | |
01:43 | lesson that the concept of direct variation always really has | |
01:45 | the same form . Some variable is related to some | |
01:49 | other variable in a direct fashion with what we call | |
01:51 | a constant of variation . That was the variable in | |
01:54 | . Now I like to use the letter M . | |
01:56 | When we were talking about direct variation because it was | |
01:58 | basically a line we talked about this , this looks | |
02:00 | like a line that goes through the origin . But | |
02:02 | I also told you that oftentimes in books , you'll | |
02:05 | see it written like this . Why is equal to | |
02:07 | K times X ? It's exactly the same equation here | |
02:10 | , we're seeing the constant of variation as M . | |
02:13 | Here we're saying the constant variation is something called K | |
02:16 | . It's exactly the same thing . And this is | |
02:18 | often used as well . Now , what we mean | |
02:21 | here is that for this type of direct variation , | |
02:23 | if you did a plot of Y and X , | |
02:25 | it would be a line that goes directly through the | |
02:27 | origin . The slope of this line is going to | |
02:29 | govern how Y and X are related to one another | |
02:33 | . Um And so the slope of the line determines | |
02:35 | how fast Y goes up whenever X goes up . | |
02:38 | And so we call that a direct variation . Uh | |
02:42 | It's a line that goes basically through the origin like | |
02:45 | that . What it basically means is that when the | |
02:47 | X variable goes up , the why variable also goes | |
02:50 | up . So if X goes up , why also | |
02:52 | goes up now , we need to contrast that with | |
02:56 | what we're gonna learn today , which is just as | |
02:58 | important . And it's called inverse variation . Just like | |
03:05 | I gave examples of direct variation in that lesson . | |
03:07 | There are many , many , many , many , | |
03:09 | many examples of inverse variation in real life . Everything | |
03:12 | from gravitation . We'll talk about it today to electricity | |
03:15 | , magnetism . I can go on and on any | |
03:18 | branch of science or engineering or math has inverse variation | |
03:22 | uh type of uh of relationships you can find . | |
03:26 | So whereas direct looks like something like this . Why | |
03:29 | is equal to K times X ? Inverse variation looks | |
03:33 | like this . Why is equal to K divided by | |
03:36 | X . Right . So look , first of all | |
03:38 | , how these things are related to one another . | |
03:40 | This one is a direct variation as X increases . | |
03:43 | Why must also then increase ? How much does why | |
03:46 | increase ? Well , it depends on what K is | |
03:48 | . If K is 100 then why will go up | |
03:51 | a pretty large amount compared to X ? If y | |
03:53 | is 25 million , then of course , as why | |
03:57 | is gonna go up a whole lot more ? And | |
03:58 | the slope of this line will be very , very | |
04:00 | steep . For inverse variation has all the same parts | |
04:03 | , you have Y and X . That are related | |
04:05 | to one another through a constant variation . So we | |
04:07 | say that the constant variation is right here , the | |
04:11 | constant of variation . All right . And whereas before | |
04:17 | we would say why varies directly as X . We | |
04:20 | did that in the last lesson , what we would | |
04:22 | say here is why varies inversely as X . Yes | |
04:32 | . Right . So , this word inversely tells you | |
04:34 | something very , very important . What does it tell | |
04:36 | you ? It tells me that whenever I look at | |
04:39 | this guy , if the X variable increases , if | |
04:44 | this denominator of this fraction goes bigger , bigger , | |
04:47 | bigger , let's make it 100 . Let's make it | |
04:49 | 1000 . Let's make it 25 million . Let's make | |
04:51 | it 75 million , that I'm dividing by a bigger | |
04:53 | and bigger and bigger numbers . So the Y variable | |
04:56 | is then has to be forced down , Right ? | |
04:58 | So that means that when X goes up , the | |
05:02 | Y variable goes down in the opposite direction , That's | |
05:05 | why we call it an inverse variation . If you | |
05:07 | flip it around the other way and see what happens | |
05:08 | . If the X variable goes down , then what | |
05:11 | am I doing here ? This variable goes down . | |
05:14 | Let's make it 10 . Let's make it five . | |
05:16 | Let's make it one . Let's make it point . | |
05:18 | Oh one . Let's make it 10.1 You see what | |
05:22 | I'm dividing by a really small number . Then that | |
05:25 | makes the Y variable get really , really big . | |
05:27 | So no matter what X is doing , if X | |
05:30 | is going up , why is going down effects is | |
05:32 | going down ? Why is going up ? How much | |
05:35 | does why respond to the change in X ? Well | |
05:38 | , that depends on the constant variation which we call | |
05:40 | K . If k is bigger or smaller is going | |
05:43 | to affect exactly how much why responds in relation to | |
05:46 | X . But no matter what , if it has | |
05:49 | this form , you can see from the equation , | |
05:51 | that why always has to go in the opposite direction | |
05:53 | of X . From what we just talked about here | |
05:55 | . Now , I'm gonna do a more detailed plot | |
05:59 | in just a second . But a lot of students | |
06:01 | , a lot of students think that an equation like | |
06:05 | this , why is equal to K over X . | |
06:10 | They know that it kind of looks like this one | |
06:12 | , and if this one is a straight line like | |
06:14 | this , then they think , well , maybe it's | |
06:16 | a straight line that looks like this , and you | |
06:18 | kind of trick yourself into thinking that this is this | |
06:21 | is a straight line that goes in the opposite direction | |
06:23 | . This in fact , I'm gonna put it in | |
06:25 | big letters , so you don't get confused here , | |
06:27 | This is not true . The shape of this thing | |
06:32 | does not look like this . Let me show you | |
06:33 | what it does look like . And then we're gonna | |
06:35 | do a more detailed plot here in just a minute | |
06:37 | . An inverse variation looks like this . If this | |
06:39 | is X , and this is why is equal to | |
06:41 | some K . Divided by X . Then the shape | |
06:44 | of this curve looks something like this . It has | |
06:49 | a curve to it notice it does start high over | |
06:51 | here , and it does end low over here , | |
06:54 | but it is not a straight line that goes between | |
06:56 | these points , it goes down , it bends over | |
06:59 | and it goes something like this . And if you | |
07:01 | think about it , it actually makes sense that it | |
07:03 | cannot be a straight line , because if you think | |
07:06 | about it , let's let's look at a a regular | |
07:09 | equation of a line that we know , um Let's | |
07:11 | say that y is equal to two X plus one | |
07:14 | . We know what this guy looks like . Two | |
07:17 | X plus one means that we have a Y intercept | |
07:20 | over here at one somewhere , right ? And the | |
07:23 | slope is positive . So I'm not gonna do rise | |
07:25 | over run , but , you know , you rise | |
07:26 | to and you run one or something like that . | |
07:28 | So , you know , this line is going to | |
07:29 | look something like this . Of course , I could | |
07:31 | continue my axes or whatever , but you know , | |
07:33 | it's going to go in a positive sense like this | |
07:35 | . All right . What is going to happen if | |
07:38 | you if you try to to take a line , | |
07:41 | I'm running out of space here . If you try | |
07:43 | to take a line that goes in , like in | |
07:46 | the opposite sense that slopes negatively , what would something | |
07:49 | like that look look like that would be something like | |
07:51 | why is equal to negative two X . Plus whatever | |
07:54 | ? Why intercept it is three or whatever ? I'm | |
07:57 | just making it up . It doesn't it doesn't really | |
07:59 | matter . What I'm trying to say is that we | |
08:00 | know what the equation of a line looks like when | |
08:03 | it's slanted this way , it looks like the regular | |
08:06 | equation of a line . It's just a slope is | |
08:07 | negative . So the y intercept is going to govern | |
08:10 | where the thing crosses and the negative slope is gonna | |
08:12 | make it slant this way . So a lot of | |
08:14 | students will look at this equation in the in in | |
08:16 | the moment and they'll think , well this this one | |
08:19 | slant like this way , then this one's gonna slant | |
08:20 | this way . But you know that this cannot be | |
08:22 | the equation of a line . It can't be the | |
08:25 | equation of the line because in this equation the X | |
08:28 | . Is not on the bottom . It's not downstairs | |
08:30 | . We know what equations of the line look like | |
08:32 | when they slant like this . It's MX plus B | |
08:34 | . But M . Is just negative . So I'm | |
08:37 | showing you without proving to you that it cannot be | |
08:40 | that this equation cannot be a line slanted in the | |
08:42 | opposite direction . In fact , it actually looks like | |
08:45 | this . It's not a straight line at all . | |
08:47 | It bends and it does go from a high value | |
08:49 | to a low value but it bends over like that | |
08:51 | . All right . So , I want to give | |
08:53 | you a concrete example as to what an inverse variation | |
08:57 | really is . Something that you can wrap your brain | |
08:59 | around . I'm gonna talk about gravity in a few | |
09:01 | minutes , but before we get to that , I | |
09:03 | want you to think about speed . You're driving down | |
09:06 | the road in a car or a bicycle , whatever | |
09:09 | you want . Let's say , you're going 10 km | |
09:12 | , that's the distance of your journey from your house | |
09:15 | to your school , let's say , and it's 10 | |
09:17 | kilometers , right ? And you want to figure out | |
09:20 | what the speed is , you know , kilometers per | |
09:22 | second or kilometers per hour . You know , it's | |
09:24 | a distance divided by time , that's what speed is | |
09:27 | , right ? So if you're driving down the road | |
09:30 | , if you travel 10 km , this is the | |
09:34 | fixed distance , it doesn't change . Then the speed | |
09:38 | here Is distance divided by time , but the distance | |
09:41 | already gave you , it's just a 10 km , | |
09:44 | Right ? 10 km . And I have to divide | |
09:46 | it by the time . A lot of times we | |
09:49 | talk about ours , but let's talk about in terms | |
09:51 | of seconds , so this is gonna be kilometers per | |
09:53 | second . So it's distance divided by time . So | |
09:56 | in other words the speed s Is going to be | |
09:59 | equal to 10 divided by some tea sometime . T | |
10:03 | and you see what happens here . This equation for | |
10:05 | the speed that you're traveling to the grocery store or | |
10:09 | whatever is inversely related to the time . It has | |
10:13 | exactly the same form of this . There's a number | |
10:15 | on the top which is the constant variation . There's | |
10:18 | a variable on one side of the equal sign and | |
10:20 | there's a variable on the downstairs underneath the number . | |
10:23 | And what this means here is that as the as | |
10:27 | the speed increases or I should say let's do it | |
10:30 | this way as the as the time increases . In | |
10:34 | other words , as it takes you long time let's | |
10:36 | say it takes you six hours to make that journey | |
10:39 | . You convert that two seconds really long time you | |
10:41 | must be traveling really slow . If it takes me | |
10:43 | six hours to get there then I must be walking | |
10:46 | really really slow . So as the time it takes | |
10:49 | goes very very high , that means that my speed | |
10:52 | must have been really really low . Let's take the | |
10:55 | alternative . Let's say that it takes me hardly any | |
10:57 | time at all my time . As I decrease the | |
11:00 | time , let's say , it only takes me six | |
11:02 | seconds to make that journey . Like maybe I'm fire | |
11:05 | a bullet or going on a super fast rocket or | |
11:07 | something like only takes me six seconds . If I | |
11:10 | go 10 kilometers in six seconds , that means I | |
11:13 | must be traveling really really fast . So as the | |
11:15 | time goes down , that must mean that the speed | |
11:18 | has gone up . Why is that the case ? | |
11:20 | Because you see what happens here ? If I put | |
11:22 | a really big number for the time in here , | |
11:24 | I divide by a big number and I get a | |
11:26 | small number over here . So that's what this means | |
11:28 | . If I put a really small number in here | |
11:31 | , tiny , let's make it half a second or | |
11:32 | something . Then I'm dividing by a small number and | |
11:35 | my speed goes really , really fast . So whereas | |
11:38 | in direct variation , everything is on the same level | |
11:41 | here as the X goes up , the Y goes | |
11:44 | up . But for the inverse variation as the X | |
11:47 | goes up , the why goes down as the ex | |
11:49 | go down goes down to why goes up . And | |
11:51 | that's why it's called inversely related to one another inversely | |
11:54 | . Uh You don't use the word proportional , You | |
11:56 | say they're inversely . They vary inversely with respect to | |
11:59 | one another . Okay , um now what I wanna | |
12:03 | do is I want to get a little more detail | |
12:06 | is to the shape of this curve right here , | |
12:08 | I told you hey , the shape of this curve | |
12:09 | looks like a bent over kind of like a bent | |
12:12 | line . Kind of , it's not a line at | |
12:13 | all , but I want to talk to you a | |
12:15 | little bit more about um where that shape of that | |
12:21 | line comes from . So let's talk about our example | |
12:24 | where we have the speed is equal to the distance | |
12:27 | divided by the time . Right ? So we said | |
12:29 | that the speed here that we're calculating , we're only | |
12:32 | going 10 km is 10 km divided by the time | |
12:35 | in seconds . Right ? So let's make a table | |
12:37 | . So let's say if journey takes , we're gonna | |
12:46 | make a little table of values here . Uh Let's | |
12:49 | say it takes one second , two seconds , three | |
12:52 | seconds , four seconds or five seconds like this . | |
12:58 | Then we want to calculate on the speed . The | |
13:01 | speed is equal to 10 km divided by however long | |
13:04 | the thing took . All right , so , we're | |
13:06 | gonna go and calculate this . So the speed is | |
13:08 | 10 divided by the one second . So this is | |
13:11 | 10 and it depends on your I guess I did | |
13:15 | km and let's do second . So this will be | |
13:17 | kilometers per second . That's a speed , right , | |
13:20 | kilometers per second . It doesn't matter what the units | |
13:22 | are , but whatever S here for this guy , | |
13:25 | I still went 10 km , but it took me | |
13:26 | two seconds . So it took me longer , right | |
13:29 | ? And so when I take 10 divided by two | |
13:31 | , it actually takes me five km or my velocity | |
13:33 | . My speed turns into five km/s . Notice that | |
13:36 | the speed is actually half because it took me twice | |
13:39 | as long to go that distance . So , that | |
13:41 | makes sense . Now , if I it takes me | |
13:43 | even longer . Mhm . Three seconds , 10 kilometers | |
13:47 | divided by three seconds and I have to stick that | |
13:49 | in the calculator , I'm gonna get 3.33 I'm rounding | |
13:52 | here uh kilometers per second and then if I go | |
13:56 | into four seconds you see the pattern 10 divided by | |
13:59 | four , you're gonna get exactly 2.5 kilometers /s and | |
14:03 | at five seconds Mhm . What you're gonna get is | |
14:07 | two km per whoops kilometers per second . So you | |
14:13 | can see what happens that as the time goes up | |
14:17 | here , right here , the time Uh huh goes | |
14:23 | up Then what happens here is the speed goes down | |
14:33 | but it doesn't go down in a line , you | |
14:35 | might think it's a line because we talked about earlier | |
14:37 | , but it doesn't go down the line so we | |
14:39 | need to actually draw a plot of this to show | |
14:40 | you what it really looks like . So this is | |
14:42 | not going to be exact but it's going to be | |
14:44 | enough to get the point across . So this is | |
14:46 | the um plotting the speed and this is the time | |
14:53 | . All right ? So now we need some tick | |
14:54 | marks we have points . Right ? So I'm gonna | |
14:56 | go uh notice I went from 1 to 5 seconds | |
14:59 | . So I'll say one second two seconds three seconds | |
15:01 | four seconds five seconds . So I'll say 12345 seconds | |
15:08 | and my speed went from 2 to 10 . So | |
15:11 | I'll do something like 123456789 10 . So this is | |
15:17 | 123456789 10 . Okay so now I have enough information | |
15:24 | and of course the distance between the tick marks isn't | |
15:26 | quite the same but it's still going to show me | |
15:27 | that basically what I'm trying to find and that is | |
15:30 | that if it takes me one second to go on | |
15:32 | this journey then my speed must be 10 . So | |
15:35 | one comma 10 . That's what I'm gonna plot for | |
15:37 | the first point two comma five . If I'm plotting | |
15:41 | a speed versus time , kind of a relationship between | |
15:44 | the two variables , two seconds would be five kilometers | |
15:47 | per second , which cuts it in half , which | |
15:49 | is right around here . Three comma 333 So here's | |
15:52 | 123 and I'm gonna go just a little bit up | |
15:56 | from that . I'm trying to get 3.33 it's not | |
15:58 | exact four comma 2.5 , this is 1 to 2.5 | |
16:03 | is gonna be somewhere around this and then five comma | |
16:06 | 25 comma here is number two right down here so | |
16:09 | you can see clearly this is not a straight line | |
16:12 | and if you were to connect it with a smooth | |
16:13 | curve , I'm not gonna probably do the best job | |
16:15 | freehand but you can see what's basically going on here | |
16:20 | . That's not too bad . It goes something like | |
16:22 | this . So basically the curve gets really really close | |
16:25 | to the axis but it never really touches it and | |
16:28 | it gets really close to the axis over here and | |
16:29 | it never really touches it because if I put a | |
16:33 | time of zero in here , zero if literally the | |
16:36 | journey took zero seconds . If I take from going | |
16:39 | 10 km it took me zero seconds , You divide | |
16:41 | by zero . It's like you got infinity speed , | |
16:44 | you can't really ever get there in zero seconds . | |
16:46 | So it's like infinity speeds of the speeds like off | |
16:48 | the chart . And if it takes me 100 million | |
16:51 | years in in terms of seconds then the speed is | |
16:54 | going to basically be zero or so close to zero | |
16:57 | . That's why it gets close to zero this direction | |
16:59 | . But this is what the curve looks like this | |
17:01 | curve here . Um It does go from high to | |
17:05 | low as time goes on but it is not a | |
17:07 | straight line . We know what the equation of the | |
17:08 | line looks like . This is not the equation of | |
17:10 | the line . This is actually what we call a | |
17:12 | hyper Bella and we're going to study . Hyperbole is | |
17:15 | in the next unit of algebra here when we talk | |
17:18 | about comic sections , it's a very special curve called | |
17:21 | a hyperbole . Um But in this case I just | |
17:24 | want you to know that the shape of the inverse | |
17:25 | variation goes down like this in the smooth fashion . | |
17:29 | Now what I want to do for the rest of | |
17:30 | the lesson is I want to pause the idea of | |
17:33 | the inverse variation and I wanna talk to you about | |
17:35 | something which we call joint variation . And then we're | |
17:39 | gonna talk a little bit about gravity because gravity is | |
17:41 | a good equation that has example of direct variation along | |
17:45 | with inverse variation . And then we'll call it a | |
17:47 | day and we'll solve some problems . So the final | |
17:50 | thing I want to talk to you about . Let | |
17:51 | me see . What am I going to do it | |
17:53 | ? Um Yeah , I'm gonna do it right here | |
17:57 | . I want to talk to you about actually . | |
18:01 | I want to do it over here . I want | |
18:03 | to talk to you about what we call joint variation | |
18:08 | . Don't worry , it's very easy to understand . | |
18:11 | We just have a new word for it . But | |
18:13 | basically what it is , joint variation happens is when | |
18:16 | uh we have variation or varies directly as the product | |
18:28 | of two or more variables . Yes , let me | |
18:36 | give you a couple of examples of that , Let's | |
18:39 | say that . I gave you the equation S . | |
18:41 | Whatever S . Is is equal to some constant of | |
18:43 | variation times X times Y squared . Now , if | |
18:48 | the y squared wasn't there and it was just Kay | |
18:52 | times X , you would say that that K varies | |
18:54 | directly as X . Right ? And that's correct . | |
18:58 | Ok . Does vary directly as X . Because as | |
19:00 | X gets bigger than S also gets bigger . Now | |
19:03 | , if I were to cover up , I'll put | |
19:05 | my thumb over . If I cover up . Just | |
19:06 | the ex parte , you would say S varies directly | |
19:09 | as Y squared . Now , this does not mean | |
19:12 | that S . And Y very directly . You can | |
19:14 | see that why has a square on it , but | |
19:16 | still why squared is sitting over here next to this | |
19:19 | constant here and it's multiplied . So the relationship between | |
19:23 | S and X is direct and the relationship with S | |
19:27 | and Y squared as an entire term . If you | |
19:30 | want , I can put little princes around it as | |
19:32 | an entire term is a direct relationship to the relationship | |
19:36 | between S . And just the variable why , of | |
19:37 | course , is a square is a square variation , | |
19:40 | but if I look at the entire term , the | |
19:42 | whole thing , it varies directly as S when you | |
19:45 | have two things that very directly and they're multiplied together | |
19:48 | , you call it a joint variation . So you | |
19:50 | say that S varies um jointly as X . And | |
20:00 | the square of why ? So it does not vary | |
20:06 | directly as X . And directly as why ? Because | |
20:09 | we have a square there . It varies directly or | |
20:12 | jointly , I should say as X . And the | |
20:14 | square of why ? Another thing you might want to | |
20:17 | say is you could say s uh is jointly yes | |
20:23 | , proportional , uh two X and y squared . | |
20:32 | So the same sort of the same sort of thing | |
20:34 | . So a lot of this stuff is terminology . | |
20:36 | When you have a joint variation , it just means | |
20:39 | you have a direct variation between two different things . | |
20:41 | We have a direct variation between S and X . | |
20:44 | We have another direct variation between S and the quantity | |
20:48 | Y squared . But since they're all together in the | |
20:50 | numerator and they're multiplied , we say that we have | |
20:53 | a joint variation . Joint variation means it varies directly | |
20:57 | as the product of two or more variables . So | |
20:58 | we have a joint variation between X and Y squared | |
21:02 | . So we've talked about direct variation , we've talked | |
21:05 | about inverse variation and now we've talked about joint variation | |
21:08 | which is basically just direct variation with two or more | |
21:11 | variables . Right ? So now I want to give | |
21:14 | you an example of when you have an equation that | |
21:16 | can have a direct variation and an inverse variation in | |
21:19 | the same equation . And the most famous example I | |
21:22 | can give you that is the law of gravitation . | |
21:26 | The law of gravity . Now , what I should | |
21:33 | say is this is Newton's law of gravitation . So | |
21:35 | we're gonna write the equation on the board . Newton's | |
21:37 | law of gravity uh is accurate enough for most uses | |
21:42 | in in everyday life . We used Newton's law of | |
21:44 | gravity to send the astronauts to the moon , to | |
21:47 | calculate the trajectory to go to the moon's accurate enough | |
21:50 | for almost anything we're going to do around the neighborhood | |
21:53 | of the Earth and the sun . However , we | |
21:55 | now know that Einstein's theory of gravity is more accurate | |
21:58 | and it's much more complicated than Newton's theory , but | |
22:01 | for most purposes , the law of gravitation . Newton | |
22:03 | came up with is perfectly fine . So we're just | |
22:05 | gonna talk about this one , even though we know | |
22:07 | Einstein's theory of gravity is a little more accurate than | |
22:09 | this . So the law of gravity says the force | |
22:12 | that exists between two objects is going to be something | |
22:16 | we call the gravitational constant right times the mass of | |
22:21 | body number one times the mass of body number two | |
22:24 | . Because you have to have two things to attract | |
22:25 | each other . So this is a massive body number | |
22:27 | one . The massive body number two . And you're | |
22:29 | gonna divide that by the distance between them . But | |
22:33 | the distance between them squared . Now , I'm gonna | |
22:36 | draw a picture and I'm gonna show you what this | |
22:37 | means in just a second . But I want you | |
22:39 | to realize that this G on here is just a | |
22:42 | number . This G out here is exactly the same | |
22:46 | thing when we have , why is equal to K | |
22:48 | . X . Or Y is equal to K over | |
22:49 | X . It's just a constant variation . It's a | |
22:52 | number that Newton calculated based on , you know based | |
22:56 | on his theory based on the experiments . So for | |
22:58 | the purpose of this class I'm gonna rename this G | |
23:01 | . I'm gonna rename it . Kay right . And | |
23:04 | so it's M . One times M . Two divided | |
23:07 | by our squared . And just to make it even | |
23:11 | more clear , I'm gonna basically right at one final | |
23:14 | way and say that it's K . I'm gonna bring | |
23:16 | this K . I'm gonna bring it upstairs because it's | |
23:18 | K over one . I can multiply that times infraction | |
23:21 | . So it's K . Uh times M . One | |
23:24 | M . Two all divided by R . Squared . | |
23:26 | So it's a little easier for us to see what's | |
23:28 | going on . I'm just moving the K upstairs here | |
23:30 | . This is the law of gravity . I mean | |
23:32 | just think about what you're looking at . This is | |
23:33 | like hundreds of years of people wondering what happens , | |
23:36 | How is the moon , you know , going around | |
23:39 | us and all this ? How does it work ? | |
23:40 | This equation describes the force between any two objects in | |
23:44 | the universe . It describes the force between you and | |
23:48 | the door . It describes the force between the sun | |
23:51 | and the earth . It describes the force between this | |
23:53 | marker and my nose . It describes the force between | |
23:56 | a proton and you know , an elephant . It | |
23:59 | describes the force between any two objects . All you | |
24:02 | have to know is the masses of the two objects | |
24:06 | , and you have to know the distance between those | |
24:09 | two objects . Right ? So for instance , if | |
24:13 | I was going to uh do this in terms of | |
24:16 | the Earth and the sun , I would say , | |
24:18 | well , okay , I have a son here and | |
24:21 | then way far away I have a smaller ball and | |
24:24 | that's going to be called the Earth . In reality | |
24:26 | the sun is way bigger than the earth . So | |
24:28 | this is not really to scale . But the distance | |
24:31 | between these guys were just going to call it r | |
24:33 | it's just the distance , how many meters it is | |
24:35 | . The Son is away from the Earth , that's | |
24:37 | all it is . And the Son , of course | |
24:39 | , has some mass , and we're gonna call it | |
24:42 | the mass of the sun . Now , that could | |
24:43 | be um won I'm going to relabel it ? S | |
24:45 | because it's the mass of the sun , and we | |
24:47 | could call this the mass of the Earth , so | |
24:49 | that could be M two in this equation . So | |
24:51 | then if I was going to calculate , you know | |
24:53 | , the force between the Sun and the Earth , | |
24:55 | all I would really do is I would say , | |
24:57 | well , the force between these things is this constant | |
25:00 | variation . It's a it's a gravitational constant times the | |
25:03 | mass of the sun , times the mass of the | |
25:06 | earth . And kilograms literally just if you could weigh | |
25:09 | it in so many , you know , trillion kilograms | |
25:12 | or whatever it is , the sun and the earth | |
25:14 | . You put those numbers in and you would divide | |
25:15 | not by Are you divide by r squared ? So | |
25:19 | you would figure out how many million meters it was | |
25:21 | . I have to go look up the number , | |
25:23 | I know it's 93 million miles away but convert that | |
25:25 | two m is gonna be a bigger number . You | |
25:27 | would square that . And you would divide the gravitational | |
25:30 | constant times the masses and divide by this guy . | |
25:33 | So notice what you have here in wrapped up in | |
25:38 | all of this stuff as the mass of the sun | |
25:43 | increases . In other words , let's say the mass | |
25:45 | gets bigger , bigger , bigger , bigger , bigger | |
25:47 | then this number , if everything else is the same | |
25:50 | , this number getting bigger , bigger , bigger is | |
25:51 | going to mean that the numerator is bigger , bigger | |
25:53 | , bigger . Which means the force has to be | |
25:56 | bigger . So you see how because it's in the | |
25:59 | numerator is it gets larger , it makes the force | |
26:02 | go larger . And this is what we called a | |
26:04 | direct variation . So we say that the force the | |
26:08 | gravitational force is directly varies directly as the mass of | |
26:12 | the sun . We can also see from this equation | |
26:15 | that as the mass of the earth increases , we | |
26:19 | can also see that the force also increases because it | |
26:21 | also lives upstairs . So if we make the earth | |
26:23 | bigger , bigger , bigger , bigger and everything else | |
26:25 | we hold constant . Like we leave everything else alone | |
26:28 | , then the force must be bigger . And that | |
26:29 | is what we call a direct variation right now , | |
26:33 | remember we just said a joint variation is when things | |
26:37 | vary directly as the product of two or more variables | |
26:40 | . So forget about the bottom here , two or | |
26:42 | more variables . It seems that that's what's happening . | |
26:44 | We have direct variation , two or more variables . | |
26:46 | So these two on the top , we call joint | |
26:50 | variation . It's just a label . It means we | |
26:54 | have two or more direct variations , that's all . | |
26:56 | And then finally to bring it home . Let's look | |
26:58 | at the bottom as this radius or the distance between | |
27:03 | uh the sun and the earth goes up . If | |
27:06 | I make this distance bigger , bigger , bigger . | |
27:07 | If I make everything farther and farther apart , you | |
27:09 | would expect the gravitational force to go down . And | |
27:12 | it does because if this number gets bigger , I'm | |
27:14 | dividing by a large large number , which makes this | |
27:18 | go down . So the force then goes down . | |
27:20 | This means that we have an opposite direction of the | |
27:23 | movement of everything and we call this in verse variation | |
27:29 | . So I wanted to bring it home to make | |
27:31 | sure I caught everything on my previous notes here . | |
27:32 | So what we basically figured out is that we've used | |
27:36 | the very famous law of gravitation to really bring what | |
27:40 | we're learning , you know , down to earth , | |
27:42 | so to speak to to show that it actually has | |
27:44 | a practical relationship . We know that the gravitational force | |
27:47 | between I did the sun and the earth here , | |
27:48 | but between any two objects varies directly as the massive | |
27:52 | object , number one . It also varies directly as | |
27:55 | the massive object number two , and it varies inversely | |
27:58 | as the square of the uh inversely as the square | |
28:03 | of the radius . And it turns out that this | |
28:05 | inverse variation thing . It happens for gravity . It | |
28:08 | also happens for electricity and magnetism to grab the electromagnetic | |
28:12 | force also is an inverse square kind of distance kind | |
28:16 | of relationship . There's lots of examples in nature when | |
28:19 | you have direct variation , you also have inverse variation | |
28:21 | . Of course this is not just a straight inverse | |
28:23 | , it's it's an R . Squared . So it's | |
28:25 | a little more uh it's a little more uh steeply | |
28:30 | of a steep of a drop off because you're not | |
28:32 | just having the thing in the bottom , you're having | |
28:33 | the thing in the bottom square , but it's still | |
28:35 | an inverse relationship . So bottom line direct variation means | |
28:39 | increased variable A . Then variable B also goes up | |
28:43 | , inverse means increased variable . A The other variable | |
28:46 | then goes down . We looked at the shape of | |
28:48 | the curves , we've looked at an example from real | |
28:51 | life from , from physics , but I could pull | |
28:53 | examples from any any branch of science . And so | |
28:56 | you should now know that inverse and direct variation are | |
28:58 | basically part of our uh something that we need to | |
29:00 | learn because it exists in the real world . Now | |
29:02 | in the next lesson , we're gonna solve some more | |
29:04 | problems . This is more of a lesson here and | |
29:06 | we're gonna solve some problems with inverse variation . So | |
29:08 | make sure you understand this . Follow me on to | |
29:09 | the next lesson and we'll conquer a few problems with | |
29:12 | inverse variation . |
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