03 - Inverse Variation & Joint Variation - Part 1 (Hyperbolas & Inverse Square Law) - Free Educational videos for Students in K-12 | Lumos Learning

03 - Inverse Variation & Joint Variation - Part 1 (Hyperbolas & Inverse Square Law) - Free Educational videos for Students in k-12


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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