00:01
All right, so in this question, we are dealing with a specific point on a ball or a wheel as it rotates.
00:10
We have equations for the x and y coordinates of this point as it rotates.
00:15
Those are x is equal to r, capital r times the quantity of 2 pi times time divided by, capital t minus sign of that same fraction two pi times time divided by capital t and then close the parentheses and then the y equation is capital r times uh one minus cosine of that same fraction as before divided by capital t.
01:10
So these are extra x and my equations and in part a we're asked to make a graph of the x of my coordinates over time using these two equations.
01:24
So we'll make a parametric curve and so let's go ahead and get started on that.
01:31
So i think the easiest way to do this is to just go ahead and set up a table with t x and y.
01:36
So three columns and in order to get like a good shape, it'd be helpful to get like eight or so points.
01:47
So if we set up our table, we have t on the left, and then we have our x and y values.
01:59
So t, time, we'll go ahead and do this in fractions of capital t.
02:06
Because if we look at our functions, you know, we have signs and cosines.
02:10
So it'd be nice if we can get those sign and cosine functions to go through a full period, which will happen when the argument goes from zero to two pi.
02:23
And since we are dividing by t, if we make time multiplied by capital t, that'll get rid of the denominator and it'll make the arguments a lot of the sign and cosine functions a lot simpler.
02:38
So basically we want to go, we want what's in the number.
02:42
Inside the sign and cosine parentheses to go from 0 to 2 pi.
02:47
And that'll happen if we have t time go from 0 to capital t.
02:55
So and we'll go ahead and split that up into eighths.
03:00
So we'll go, we'll start with 0, then we'll go to 1 eighth t and then 1 4th t, then 3 eighths, and so on.
03:14
And i will, in order to get like good values for the points, for the x and y coordinates, you should go through the whole thing from zero to t with every eighth.
03:29
But i'll just show you how to do the first few coordinates up to one half t.
03:35
And then you'll kind of get the idea of how to do the rest of them.
03:41
So, we'll go ahead and we'll evaluate x and y at each of these values of t, and we'll fill them in on our table, and that'll give us the coordinates that we can put in our graph, which i made before i started the video, or the axes that i made before i started the video.
03:57
So if we look at t is equal to zero for our x coordinate, we look at the first term inside the parentheses, the pi, 2 pi, t over capital t.
04:10
Well if little t is equal to zero then that whole fraction is going to be equal to zero and then that's minus sign of two pi times zero over t well that'll be sign of zero um sign of zero is also equal to zero so you have zero minus zero r times zero is zero so our first x coordinate is zero and then if we look at y um we'll have cosine of two pi times zero over t which will make a cosine of zero cosine of zero is one so within the brackets we'll have one minus one which is zero you'll have arm times zero which is also zero so at time equal zero our x and y coordinates are zero zero then at one eighth of t we'll have something a little bit different so for the x coordinate we'll have um two pi times one eighth t over t.
05:12
If you simplify that, you'll end up with 2 pi over 8, which is the same as pi over 4.
05:19
And then that fraction, the 2 pi t over t, will end up evaluating as pi over 4 for every time that happens.
05:30
So the equation for the x corner becomes r times pi over 4 minus sine of pi over 4.
05:38
So sine of pi over 4 is of course root 2 over 2.
05:41
So you'll end up with r times pi over 4 minus root 2 over 2.
05:46
And if you calculate that out, you'll actually get that a decimal value of 0 .07 times r.
05:54
So our x coordinate here is 0 .07 times r.
05:58
Now we do the same thing for y with 1 eighth of t.
06:02
So remember when we, so at 1 eighth the t, when we evaluate 2 pi t over t, we'll get pi over 4.
06:10
So our y equation is r times 1 minus cosine pi over 4.
06:16
Well, cosine of pi over 4 is also root 2 over 2.
06:20
So we'll have r times 1 minus root 2 over 2.
06:24
If you calculate that, you'll get that y is going to be equal to 0 .29 r.
06:32
And now we do the same thing for 1 4th t.
06:35
Well, so let's go ahead and see what the fraction 2 pi t over capital t is for 1 4th t.
06:44
We'll end up with the t's canceling again, and then we'll end up with just 2 pi over 4, which is pi over 2.
06:52
So for the x coordinate, we'll have r times pi over 2 minus sine of pi over 2.
06:59
So pi over 2, well, sine of pi over 2 is 1, so we'll have pi over 2 minus 1 times r.
07:08
If you convert that to decimal, that means that our x coordinate is going to be 0 .57r.
07:14
We do the same thing for y.
07:17
So we have r times 1 minus cosine pi over 2.
07:21
Well, cosine of pi over 2 is 0.
07:25
So we have r times 1, which is just r.
07:30
Then for 3 .8st t, we do the same thing.
07:34
You can kind of, you might be starting to see a pattern here.
07:36
If we evaluate the fraction 2 pi t over t at 3 .8s t, capital t is cancel.
07:42
So you end up with 6 pi over 8, which is 3 pi over 3.
07:47
4.
07:48
So our x becomes r times 3 pi over 4 minus sine of 3 pi over 4.
07:55
Well, sine of 3 pi over 4 is root 2 over 2.
07:57
So you end up with 3 pi over 4 minus 2 over 2.
08:01
And if you evaluate that as decimal, that ends up being 1 .6.
08:05
And of course, it's multiplied by r.
08:07
Now we'll do the same thing for y.
08:09
So our y value is r times 1 minus cosine of 3 pi over 4.
08:19
And cosine 3 pi over 4 is negative root 2 over 2.
08:23
So for a y coordinate, we end up with r times 1 plus root 2 over 2.
08:29
If you evaluate that, you'll get that the y -a coordinate is 1 .7 times r.
08:36
And then for 1 half t, the fraction 2 pi t over t ends up being just pi.
08:48
Because you end up with 2 pi over 2.
08:50
So when we evaluate our x, we end up with r times pi minus sine of pi.
09:04
Well, sine of pi is zero, so you end up with r times pi.
09:08
So our x coordinate becomes a pi times r, which if you evaluate as decimal, of course, is 3 .14r.
09:17
Now, if we do the same thing for y, we end up with r times 1 minus cosine of pi.
09:24
Well, cosine of pi is negative 1, so you end up with 1 minus negative 1 in the brackets, so you end up with 2 times r for your y, for your y coordinate.
09:35
So we get 2r there.
09:39
And so you'll go ahead and go through that, and you should go at least until you get to t.
09:44
If you want to, you can do a couple points after that, but you may start, you may suspect already that after you get to t, the values are going to start repeating, and we're going to have a new point.
09:55
Period and you're going to be correct in that assumption.
09:58
So go at least until t.
10:00
Go, you know, one or two points after that if you want to confirm that it's periodic.
10:06
But i'll go ahead and show you what the coordinate points look like when we graph them.
10:13
So i'll do this in another color.
10:17
Do this in red.
10:19
So our first point is zero zero.
10:21
So that goes right here.
10:24
And then our next point at 1 8th t is 0 .07r and 0 .29r for the x and y coordinates.
10:32
So we barely move to the right at all, but then we go up a little past a quarter to get our next point.
10:41
So that's about here.
10:43
Then our next point at 1 4th t is 0 .57r for the x coordinate and 1r for the y coordinate.
10:49
So that's going to be about here.
10:51
And then for the next point, we have 1 .6r and 1 .7 r.
10:56
Or sorry, the third point will be about here.
11:05
And then for 1 .6r and 1 .7 r, our next point will be around here.
11:16
And then for 1 half t, we have pi times r and then 2r for a y -cordid.
11:23
So pi is going to make it, pi times r is going to make it at 3 .14, which is, you know, little past three and then 2r puts us right about here.
11:35
And then i'll go ahead and read out the coordinates that i had for the next ones through time equals capital t.
11:42
So at 5 eighths t i had 4 .6r for the x and 1 .7 r for y.
11:48
4 .6 and 1 .7 puts us at about here.
11:54
For 3 4 4ths t i had 5 .7 r and for the x and 1 r for the y for the y.
12:02
So that would put us at about here.
12:06
And then for 7 .8th t, i had 6 .2r for the x and 0 .29r for the y, which puts us at about here.
12:16
And then for 1t, the x coordinate was 2 pi r and the y coordinate was 0, and that puts us about here.
12:26
And then after this, it starts repeating because it is a periodic function.
12:30
And if we connect these dots as smoothly as we can we get a shape that is something like this and then it would repeat so it would continue along there something like that um so that is our sketch for part a and that's what we need to do for part a and so now we'll move on to part b in part b we're just asked what capital r and what capital t represent and as we've noticed you know after we get to time equals capital t, the motion starts repeating again, which indicates that t is going to be equal to the period.
13:10
And that is, in fact, what it is.
13:12
And you may guess that r is going to be equal to the radius.
13:16
And this makes sense because the y coordinate varies between zero and two r.
13:24
And then the x coordinate starts at zero and doesn't return, it starts at zero and then at the beginning of the next period we've moved 2 pi r ahead in the x direction and that would be you know one full revolution 2 pi r one full revolution around the circle which would make sense so we get to the conclusion that in part b that capital r is the radius of the ball and capital t is the period and so that's all we need to do for part b in part c we need to develop equations for the x and y components of the velocity and the um acceleration of this point on the ball and so we you know to do this we just take time derivatives of the x and y equations, once for the velocity equations, and once for the acceleration equations.
14:33
So if we take the derivative of the x equation, we'll go ahead and write that here.
14:38
So i'll say that x is equal to r times 2 pi time divided by capital t, minus sign of the same fraction, 2 pi t over t.
14:59
So if we take the first derivative of this, we'll have vx is equal to r and i'll go ahead and we'll leave a little bit of space for that and then this first term is proportional to t and so if we take the derivative of that that'll just be two pi over t so we end up with uh two pi over t there and if we move on to the next term we have negative sign of something which will turn into negative cosine but because there is you know time is multiplied by 2 pi over t that's going to come out of the sign function and so we'll have 2 pi over t here times cosine of 2 pi time divided by t and so we can simplify this equation by fact factoring out the 2 pi over t from the brackets.
16:13
And so we get that vx is equal to 2 pi r over capital t times one minus cosine of 2 pi times time over capital t.
16:31
And so that's our equation for v of x.
16:36
And so for a of x, we'll just take the next derivative of that.
16:40
So we have a 2 pi r over t at the front.
16:45
And then we have 1 minus cosine.
16:52
So the 1, of course, is just a constant.
16:55
So when we take the derivative of that, that'll reduce to 0.
16:59
And then the derivative of negative cosine is sign of whatever that is...