Polar Coordinates Lesson
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[{"Name":"Polar Coordinates Lesson","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Connection Between Polar and Cartesian Variables","Duration":"3m 39s","ChapterTopicVideoID":8985,"CourseChapterTopicPlaylistID":5399,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:04.650","Text":"In this lecture, I want to talk about polar or cylindrical coordinates."},{"Start":"00:04.650 ","End":"00:07.230","Text":"Polar coordinates are basically a different way of"},{"Start":"00:07.230 ","End":"00:10.530","Text":"describing a point or object or equation in space."},{"Start":"00:10.530 ","End":"00:14.850","Text":"The way we\u0027re going to address this is by looking at our Cartesian coordinate system."},{"Start":"00:14.850 ","End":"00:18.930","Text":"We have here, we have our x-axis and our y-axis,"},{"Start":"00:18.930 ","End":"00:21.540","Text":"and we can put a random point in the middle here,"},{"Start":"00:21.540 ","End":"00:23.175","Text":"let\u0027s say around here."},{"Start":"00:23.175 ","End":"00:25.200","Text":"Now, we know from before that one of"},{"Start":"00:25.200 ","End":"00:28.080","Text":"the easier ways to describe this point is by dropping"},{"Start":"00:28.080 ","End":"00:33.015","Text":"an orthogonal line down to the x-axis to give us our x value,"},{"Start":"00:33.015 ","End":"00:38.925","Text":"we\u0027ll call it x, and another orthogonal line over to the y-axis to give us our y value,"},{"Start":"00:38.925 ","End":"00:41.130","Text":"which we\u0027ll call y."},{"Start":"00:41.420 ","End":"00:46.300","Text":"We can describe this point as having the value x, y,"},{"Start":"00:46.300 ","End":"00:50.270","Text":"and we can also describe this point in terms of a vector,"},{"Start":"00:50.270 ","End":"00:53.090","Text":"we call it the position vector or the r vector,"},{"Start":"00:53.090 ","End":"00:58.415","Text":"and this r vector goes from the origin here out to the object itself over here,"},{"Start":"00:58.415 ","End":"01:01.445","Text":"and describes its position like this."},{"Start":"01:01.445 ","End":"01:08.610","Text":"We can describe this r vector as the vector going through the point x,"},{"Start":"01:08.610 ","End":"01:13.700","Text":"y or we can describe it as a vector with length and direction,"},{"Start":"01:13.700 ","End":"01:16.580","Text":"saying that it has the magnitude or length of x,"},{"Start":"01:16.580 ","End":"01:21.875","Text":"meaning this length in the direction of x plus the length of y,"},{"Start":"01:21.875 ","End":"01:25.500","Text":"or this length in the direction of y."},{"Start":"01:26.860 ","End":"01:31.145","Text":"Now keep in mind that any position vector can be described this way."},{"Start":"01:31.145 ","End":"01:33.935","Text":"It can be described as the position vector that goes to the point x,"},{"Start":"01:33.935 ","End":"01:36.230","Text":"y or is the position vector with"},{"Start":"01:36.230 ","End":"01:41.065","Text":"the magnitude x and the direction of x and the magnitude of y in the direction of y."},{"Start":"01:41.065 ","End":"01:46.220","Text":"We\u0027ve seen that we can describe this point using the variables x and y."},{"Start":"01:46.220 ","End":"01:49.220","Text":"But we can also describe this point using 2 other variables"},{"Start":"01:49.220 ","End":"01:54.485","Text":"interchangeably using the variables r and Theta."},{"Start":"01:54.485 ","End":"01:57.230","Text":"Now, r and Theta, just so you know,"},{"Start":"01:57.230 ","End":"02:00.170","Text":"are not the same thing as the r vector,"},{"Start":"02:00.170 ","End":"02:02.330","Text":"and we\u0027re not going to use them as vectors."},{"Start":"02:02.330 ","End":"02:03.440","Text":"In fact, r and Theta,"},{"Start":"02:03.440 ","End":"02:05.065","Text":"just like x and y,"},{"Start":"02:05.065 ","End":"02:07.820","Text":"are just magnitudes, they\u0027re only variables that only"},{"Start":"02:07.820 ","End":"02:11.310","Text":"we are talking about lengths, magnitudes, distances here."},{"Start":"02:11.930 ","End":"02:14.475","Text":"What are r and Theta?"},{"Start":"02:14.475 ","End":"02:18.730","Text":"Well, r is the length of the position vector r that we talked about before."},{"Start":"02:18.730 ","End":"02:22.100","Text":"If you look for the length of this position vector,"},{"Start":"02:22.100 ","End":"02:24.335","Text":"that equals r here,"},{"Start":"02:24.335 ","End":"02:26.885","Text":"and Theta is the angle,"},{"Start":"02:26.885 ","End":"02:31.025","Text":"the measure of the angle of that same position vector."},{"Start":"02:31.025 ","End":"02:34.940","Text":"If I have an equation set out in terms of x and y,"},{"Start":"02:34.940 ","End":"02:37.370","Text":"and I want to move it into terms of r and Theta,"},{"Start":"02:37.370 ","End":"02:40.280","Text":"I can do that pretty easily and I can do the same vice versa."},{"Start":"02:40.280 ","End":"02:43.490","Text":"The reason is, that there is a fixed relationship between"},{"Start":"02:43.490 ","End":"02:47.755","Text":"these 2 sets of variables between r and Theta and x and y."},{"Start":"02:47.755 ","End":"02:51.340","Text":"The fixed relationship means you can switch back and forth between"},{"Start":"02:51.340 ","End":"02:56.180","Text":"coordinate systems really easily as long as you know the formula to get you there."},{"Start":"02:56.340 ","End":"03:00.100","Text":"Let\u0027s say my equation is in terms of r and Theta,"},{"Start":"03:00.100 ","End":"03:02.725","Text":"but I wanted to have my equation in terms of x and y."},{"Start":"03:02.725 ","End":"03:08.065","Text":"Well, x=r times the cosine of Theta,"},{"Start":"03:08.065 ","End":"03:10.270","Text":"and you can see why if you look up here,"},{"Start":"03:10.270 ","End":"03:15.380","Text":"and y=r times the sine of Theta,"},{"Start":"03:15.380 ","End":"03:18.490","Text":"and we should put this in your formula sheet."},{"Start":"03:18.490 ","End":"03:23.200","Text":"We also have formulas to switch from x and y in terms of r and Theta."},{"Start":"03:23.200 ","End":"03:24.730","Text":"As you can see here,"},{"Start":"03:24.730 ","End":"03:33.145","Text":"r equals the square root of x^2 plus y^2 and tangent of Theta equals y/x."},{"Start":"03:33.145 ","End":"03:35.630","Text":"I want you to make an effort to remember"},{"Start":"03:35.630 ","End":"03:39.450","Text":"these formula going forward because we\u0027re going to use them a lot."}],"ID":9279},{"Watched":false,"Name":"Hat Theta-Hat and Their Dependence on Time","Duration":"8m 17s","ChapterTopicVideoID":8986,"CourseChapterTopicPlaylistID":5399,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"Now that we\u0027ve covered the relationship between x and"},{"Start":"00:03.240 ","End":"00:06.315","Text":"y and r and Theta in terms of their magnitude,"},{"Start":"00:06.315 ","End":"00:07.770","Text":"we can move on to the next subject,"},{"Start":"00:07.770 ","End":"00:12.105","Text":"which are the unit vectors r-hat and Theta-hat."},{"Start":"00:12.105 ","End":"00:19.170","Text":"The first thing to mention is that r-hat and Theta-hat are not equal to r and Theta,"},{"Start":"00:19.170 ","End":"00:22.260","Text":"r and Theta are scalars, their magnitudes,"},{"Start":"00:22.260 ","End":"00:25.410","Text":"whereas r-hat and Theta-hat are unit vectors."},{"Start":"00:25.410 ","End":"00:28.410","Text":"The vectors, have direction and they have magnitude,"},{"Start":"00:28.410 ","End":"00:30.030","Text":"and they\u0027re not the same thing."},{"Start":"00:30.030 ","End":"00:31.500","Text":"This can be very confusing."},{"Start":"00:31.500 ","End":"00:34.005","Text":"It\u0027s important to remember that r-hat"},{"Start":"00:34.005 ","End":"00:38.410","Text":"and Theta-hat are not the same thing as r and Theta."},{"Start":"00:39.290 ","End":"00:42.110","Text":"Starting with r-hat,"},{"Start":"00:42.110 ","End":"00:43.520","Text":"r-hat is a unit vector,"},{"Start":"00:43.520 ","End":"00:45.545","Text":"meaning that it has a magnitude of 1,"},{"Start":"00:45.545 ","End":"00:48.620","Text":"and like our r vector from before,"},{"Start":"00:48.620 ","End":"00:51.770","Text":"r-hat points towards our object up here,"},{"Start":"00:51.770 ","End":"00:53.360","Text":"and it starts at the origin so,"},{"Start":"00:53.360 ","End":"00:56.040","Text":"it looks something like this."},{"Start":"00:57.280 ","End":"01:01.010","Text":"Similarly, Theta-hat is also a unit vector and it"},{"Start":"01:01.010 ","End":"01:04.100","Text":"points at a 90 degree angle with respect to r-hat."},{"Start":"01:04.100 ","End":"01:07.685","Text":"It can either go down into the right or up and towards the left."},{"Start":"01:07.685 ","End":"01:09.995","Text":"In this case, it goes up and towards the left."},{"Start":"01:09.995 ","End":"01:14.450","Text":"The angle between Theta-hat and r-hat is 90 degrees, it\u0027s a right angle."},{"Start":"01:14.450 ","End":"01:16.850","Text":"We can use them as a substitute for y-hat"},{"Start":"01:16.850 ","End":"01:21.325","Text":"and x-hat in order to understand what\u0027s going on on our plane."},{"Start":"01:21.325 ","End":"01:24.200","Text":"Ultimately I can use Theta-hat and r-hat to"},{"Start":"01:24.200 ","End":"01:27.695","Text":"describe any point such as this point on my coordinate system."},{"Start":"01:27.695 ","End":"01:30.470","Text":"They are replacement for x hat and y hat."},{"Start":"01:30.470 ","End":"01:34.850","Text":"The reason that Theta-hat points up into the left in this case is"},{"Start":"01:34.850 ","End":"01:39.455","Text":"because it\u0027s representing the direction in which the angle Theta opens."},{"Start":"01:39.455 ","End":"01:42.020","Text":"In this particular problem,"},{"Start":"01:42.020 ","End":"01:45.886","Text":"the Theta angle opens in a counterclockwise direction,"},{"Start":"01:45.886 ","End":"01:50.675","Text":"r develops in a counterclockwise direction."},{"Start":"01:50.675 ","End":"01:53.990","Text":"Theta has to represent how the angle is getting bigger."},{"Start":"01:53.990 ","End":"01:55.700","Text":"It gets smaller in a clockwise direction,"},{"Start":"01:55.700 ","End":"01:58.295","Text":"it gets larger in a counterclockwise direction."},{"Start":"01:58.295 ","End":"02:02.585","Text":"Therefore, Theta is on the counterclockwise or left side of"},{"Start":"02:02.585 ","End":"02:07.790","Text":"r. We have our unit vectors, r-hat and Theta-hat."},{"Start":"02:07.790 ","End":"02:09.260","Text":"Just to remember that they\u0027re unit vectors,"},{"Start":"02:09.260 ","End":"02:10.850","Text":"we should really write this out,"},{"Start":"02:10.850 ","End":"02:13.320","Text":"r-hat the magnitude of it,"},{"Start":"02:13.320 ","End":"02:18.700","Text":"equals the magnitude of Theta-hat equals 1."},{"Start":"02:19.750 ","End":"02:22.910","Text":"This is a very unique coordinate system."},{"Start":"02:22.910 ","End":"02:25.115","Text":"It\u0027s different than Cartesian coordinates."},{"Start":"02:25.115 ","End":"02:28.730","Text":"Why? Let\u0027s say that this body we have here, this object at x,"},{"Start":"02:28.730 ","End":"02:31.250","Text":"y moves in about 2 seconds later,"},{"Start":"02:31.250 ","End":"02:33.110","Text":"it\u0027s at the 2nd point here."},{"Start":"02:33.110 ","End":"02:36.455","Text":"By nature of r-hat and Theta-hat,"},{"Start":"02:36.455 ","End":"02:38.720","Text":"r-hat still has to point to the object,"},{"Start":"02:38.720 ","End":"02:42.620","Text":"so it has to move down into the right in a clockwise direction and still point at"},{"Start":"02:42.620 ","End":"02:47.750","Text":"the object and Theta-hat moves with respect to r-hat also in a clockwise direction."},{"Start":"02:47.750 ","End":"02:50.266","Text":"This is really different from x-hat and y-hat,"},{"Start":"02:50.266 ","End":"02:52.580","Text":"x-hat and y-hat are stationary."},{"Start":"02:52.580 ","End":"02:54.785","Text":"They don\u0027t move over time, they\u0027re fixed."},{"Start":"02:54.785 ","End":"02:59.915","Text":"But the takeaway from this is that r-hat and Theta-hat are not fixed in time."},{"Start":"02:59.915 ","End":"03:03.710","Text":"1 of the best ways to describe r-hat and Theta-hat is with"},{"Start":"03:03.710 ","End":"03:07.550","Text":"the aid of x-hat and y-hat are Cartesian coordinates."},{"Start":"03:07.550 ","End":"03:13.490","Text":"We can think of r-hat and Theta-hat as unit vectors, total normal vectors."},{"Start":"03:13.490 ","End":"03:15.800","Text":"That is saying that they have a magnitude of 1."},{"Start":"03:15.800 ","End":"03:18.080","Text":"We know the magnitude and we know the direction."},{"Start":"03:18.080 ","End":"03:19.235","Text":"Starting with r-hat,"},{"Start":"03:19.235 ","End":"03:22.250","Text":"we know the direction is dictated by the angle Theta,"},{"Start":"03:22.250 ","End":"03:24.450","Text":"not the vector Theta-hat,"},{"Start":"03:24.450 ","End":"03:25.785","Text":"rather the angle Theta."},{"Start":"03:25.785 ","End":"03:27.030","Text":"Remember the difference."},{"Start":"03:27.030 ","End":"03:30.680","Text":"If we want to find the x element in the direction of x,"},{"Start":"03:30.680 ","End":"03:35.840","Text":"we know that we can use a little bit of trigonometry and that if we multiply the length"},{"Start":"03:35.840 ","End":"03:41.165","Text":"of the angle 1 times cos(Theta), then we\u0027ll get that."},{"Start":"03:41.165 ","End":"03:49.150","Text":"We have 1 which is the length times the cos(Theta) x-hat."},{"Start":"03:49.150 ","End":"03:51.260","Text":"In the direction of y,"},{"Start":"03:51.260 ","End":"03:54.500","Text":"we have the same thing except using sine."},{"Start":"03:54.500 ","End":"04:00.345","Text":"1 times the sin(Theta) in the direction of y-hat."},{"Start":"04:00.345 ","End":"04:03.355","Text":"When we\u0027re talking about Theta-hat,"},{"Start":"04:03.355 ","End":"04:08.435","Text":"we can do a very similar procedure and the result is in the direction of y,"},{"Start":"04:08.435 ","End":"04:12.660","Text":"we get cos(Theta) in the direction"},{"Start":"04:12.660 ","End":"04:18.365","Text":"of y-hat and in the direction of x because it\u0027s going in a negative angle."},{"Start":"04:18.365 ","End":"04:20.735","Text":"Remember this is the same angle Theta."},{"Start":"04:20.735 ","End":"04:22.880","Text":"We just added 90 degrees to it."},{"Start":"04:22.880 ","End":"04:31.940","Text":"Then we can say that it is negative 1 times sin(Theta) in the direction of x."},{"Start":"04:31.940 ","End":"04:35.840","Text":"We can simplify this and I\u0027ll put it in tight for you,"},{"Start":"04:35.840 ","End":"04:38.060","Text":"is that you can put it in your formula sheet."},{"Start":"04:38.060 ","End":"04:40.505","Text":"You see that we\u0027ve used Theta,"},{"Start":"04:40.505 ","End":"04:42.385","Text":"the magnitude from before."},{"Start":"04:42.385 ","End":"04:46.520","Text":"This means that our vectors are dependent on time."},{"Start":"04:46.520 ","End":"04:49.820","Text":"Why? Because Theta changes with time as our object changes,"},{"Start":"04:49.820 ","End":"04:52.180","Text":"therefore, these vectors change over time."},{"Start":"04:52.180 ","End":"04:57.235","Text":"The next thing I want to do is explore how these 2 vectors change over time."},{"Start":"04:57.235 ","End":"05:05.750","Text":"What we\u0027re going to do is we\u0027re going to take a derivative of r-hat with respect to time."},{"Start":"05:05.750 ","End":"05:12.365","Text":"This may seem odd because we never before have done a derivative with our axes, x or y."},{"Start":"05:12.365 ","End":"05:14.945","Text":"But that\u0027s because they don\u0027t change in relation to time,"},{"Start":"05:14.945 ","End":"05:18.485","Text":"whereas r-hat and Theta-hat do change with time."},{"Start":"05:18.485 ","End":"05:23.570","Text":"We\u0027re going to take dr over dt."},{"Start":"05:23.570 ","End":"05:26.000","Text":"The way that we\u0027re going to do that is we\u0027re going to"},{"Start":"05:26.000 ","End":"05:28.925","Text":"find the variable that changes with respect to time."},{"Start":"05:28.925 ","End":"05:30.040","Text":"In our case it\u0027s Theta."},{"Start":"05:30.040 ","End":"05:31.550","Text":"Using the chain rule,"},{"Start":"05:31.550 ","End":"05:36.640","Text":"we\u0027re going to first take dr in relation to d Theta,"},{"Start":"05:36.640 ","End":"05:44.710","Text":"and then we\u0027ll take that times dot d Theta over dt."},{"Start":"05:44.830 ","End":"05:49.775","Text":"What does this look like? Well, first we\u0027re going to take dr over d Theta."},{"Start":"05:49.775 ","End":"05:56.950","Text":"What we end up with is negative sin(Theta) in the direction of x,"},{"Start":"05:56.950 ","End":"05:59.075","Text":"and in the direction of y,"},{"Start":"05:59.075 ","End":"06:01.430","Text":"we get cosine of Theta."},{"Start":"06:01.430 ","End":"06:08.360","Text":"I\u0027m going to take all of this and we\u0027re going to multiply it by d Theta over dt,"},{"Start":"06:08.360 ","End":"06:11.160","Text":"and we can just call that Theta-dot."},{"Start":"06:11.180 ","End":"06:15.530","Text":"What we\u0027ll see is that this whole bit here that\u0027s"},{"Start":"06:15.530 ","End":"06:19.580","Text":"in parenthesis actually equals Theta-hat."},{"Start":"06:19.580 ","End":"06:28.360","Text":"What we get is we have Theta-dot as our magnitude and Theta-hat as our direction."},{"Start":"06:30.080 ","End":"06:36.875","Text":"What this means is that r is actually changing in the direction of Theta over time."},{"Start":"06:36.875 ","End":"06:40.790","Text":"It might be easier to understand this using our graph over here."},{"Start":"06:40.790 ","End":"06:44.435","Text":"Let me clean this up a minute and then we can look at it."},{"Start":"06:44.435 ","End":"06:49.415","Text":"Looking at our graph, let\u0027s assume the point moves in a counterclockwise direction,"},{"Start":"06:49.415 ","End":"06:50.855","Text":"maybe to somewhere here."},{"Start":"06:50.855 ","End":"06:56.135","Text":"That\u0027s a positive direction for us because the angle Theta has increased and now"},{"Start":"06:56.135 ","End":"07:01.370","Text":"our whole system has to move along with this point in a counterclockwise direction."},{"Start":"07:01.370 ","End":"07:04.295","Text":"That means that we\u0027re going to have a new r vector."},{"Start":"07:04.295 ","End":"07:06.900","Text":"It\u0027s going to look something like this."},{"Start":"07:07.090 ","End":"07:11.270","Text":"If our blue line is the old r-hat,"},{"Start":"07:11.270 ","End":"07:13.250","Text":"and the green line is the new r-hat,"},{"Start":"07:13.250 ","End":"07:14.930","Text":"then the difference between the 2,"},{"Start":"07:14.930 ","End":"07:17.570","Text":"the change over time is this red line here."},{"Start":"07:17.570 ","End":"07:20.630","Text":"Look at that red line and imagine we\u0027re talking about"},{"Start":"07:20.630 ","End":"07:25.190","Text":"an infinitesimally small distance as opposed to this slightly larger distance."},{"Start":"07:25.190 ","End":"07:30.910","Text":"You\u0027ll notice it\u0027s going exactly in the same direction as r Theta-hat."},{"Start":"07:30.910 ","End":"07:34.090","Text":"This solves for dr-hat,"},{"Start":"07:34.090 ","End":"07:36.395","Text":"and we can do the same thing for Theta-hat."},{"Start":"07:36.395 ","End":"07:43.025","Text":"That is, take d Theta-hat over dt and you get a similar answer."},{"Start":"07:43.025 ","End":"07:46.460","Text":"I\u0027m not going to do the whole development of the equation for you."},{"Start":"07:46.460 ","End":"07:50.015","Text":"I would suggest that you in fact stop the video now and try it on your own."},{"Start":"07:50.015 ","End":"07:51.230","Text":"It\u0027s a very good exercise."},{"Start":"07:51.230 ","End":"07:53.495","Text":"It\u0027s a good way to get a feel review"},{"Start":"07:53.495 ","End":"07:57.260","Text":"understanding the work here and understanding the theory."},{"Start":"07:57.260 ","End":"07:59.975","Text":"But for those who just want the answer,"},{"Start":"07:59.975 ","End":"08:02.660","Text":"in fact is negative Theta-dot,"},{"Start":"08:02.660 ","End":"08:07.090","Text":"this Theta dot reappears and that multiplied by r-hat."},{"Start":"08:07.090 ","End":"08:14.855","Text":"From here on, I\u0027m going to use negative Theta-dot r-hat and Theta-hat, Theta-dot a lot."},{"Start":"08:14.855 ","End":"08:18.330","Text":"Make sure you understand why we got these answers."}],"ID":9280},{"Watched":false,"Name":"Position Vectors Velocity and Acceleration","Duration":"11m 19s","ChapterTopicVideoID":8987,"CourseChapterTopicPlaylistID":5399,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.915","Text":"Now what I want to do is find my position vector,"},{"Start":"00:03.915 ","End":"00:08.460","Text":"my velocity vector, and most importantly, my acceleration vector."},{"Start":"00:08.460 ","End":"00:13.710","Text":"I want to use math to define all 3 of these in our polar coordinates."},{"Start":"00:13.710 ","End":"00:16.575","Text":"Let\u0027s start with my position vector,"},{"Start":"00:16.575 ","End":"00:18.420","Text":"r. Now, if you recall,"},{"Start":"00:18.420 ","End":"00:23.295","Text":"we did that over here and the formula we used was that in Cartesian coordinates,"},{"Start":"00:23.295 ","End":"00:26.895","Text":"r(x,y)=x in the direction of x,"},{"Start":"00:26.895 ","End":"00:29.700","Text":"plus y in the direction of y."},{"Start":"00:29.700 ","End":"00:33.075","Text":"The reason for that is if we were to add 2 vectors,"},{"Start":"00:33.075 ","End":"00:36.915","Text":"1 vector is of a magnitude x in the direction of the x-axis,"},{"Start":"00:36.915 ","End":"00:39.500","Text":"and the other is in a magnitude of y,"},{"Start":"00:39.500 ","End":"00:41.705","Text":"this y, in the direction of y."},{"Start":"00:41.705 ","End":"00:42.980","Text":"We would get to this point."},{"Start":"00:42.980 ","End":"00:44.075","Text":"We would get the vector,"},{"Start":"00:44.075 ","End":"00:46.470","Text":"r. In polar coordinates,"},{"Start":"00:46.470 ","End":"00:47.870","Text":"this is actually a lot simpler."},{"Start":"00:47.870 ","End":"00:53.990","Text":"All we need to do is go a length of r in the direction of the r hat vector."},{"Start":"00:53.990 ","End":"00:56.945","Text":"If we go in the direction of the r hat unit vector,"},{"Start":"00:56.945 ","End":"01:00.635","Text":"r distance, we get to the point x,y from before."},{"Start":"01:00.635 ","End":"01:05.175","Text":"Now, we don\u0027t have to add any Theta in the direction of Theta here,"},{"Start":"01:05.175 ","End":"01:06.630","Text":"like we did with x and y."},{"Start":"01:06.630 ","End":"01:07.845","Text":"That\u0027s not correct."},{"Start":"01:07.845 ","End":"01:10.010","Text":"In fact, it doesn\u0027t make much sense because"},{"Start":"01:10.010 ","End":"01:12.335","Text":"we can\u0027t talk about Theta in terms of a length."},{"Start":"01:12.335 ","End":"01:15.335","Text":"It has magnitude but not a length, so to speak."},{"Start":"01:15.335 ","End":"01:19.430","Text":"Now we can move on to the velocity vector, v. Now,"},{"Start":"01:19.430 ","End":"01:23.935","Text":"the definition of the velocity vector is the differential of the position vector,"},{"Start":"01:23.935 ","End":"01:26.230","Text":"r. That is r vector dot."},{"Start":"01:26.230 ","End":"01:27.580","Text":"The velocity vector,"},{"Start":"01:27.580 ","End":"01:29.630","Text":"it\u0027s a common mistake that students make,"},{"Start":"01:29.630 ","End":"01:31.300","Text":"but it\u0027s not true."},{"Start":"01:31.300 ","End":"01:37.355","Text":"People will say the velocity vector is the differential of just r. That\u0027s not the case,"},{"Start":"01:37.355 ","End":"01:38.840","Text":"r is simply a magnitude."},{"Start":"01:38.840 ","End":"01:40.805","Text":"It\u0027s a scalar. It is not a vector."},{"Start":"01:40.805 ","End":"01:43.130","Text":"Therefore, we want to not take this."},{"Start":"01:43.130 ","End":"01:47.640","Text":"Rather we\u0027re looking for the differential of the r vector."},{"Start":"01:52.790 ","End":"01:59.590","Text":"Now, I know that my r vector can also be written as r r hat."},{"Start":"01:59.590 ","End":"02:03.620","Text":"What I can do is take a derivative of this whole thing together."},{"Start":"02:03.620 ","End":"02:06.230","Text":"I put a dot above those in parentheses."},{"Start":"02:06.230 ","End":"02:09.575","Text":"The best way to do this is to break it down into 2 parts."},{"Start":"02:09.575 ","End":"02:11.555","Text":"First, we have r dot,"},{"Start":"02:11.555 ","End":"02:14.785","Text":"that\u0027s the derivative of r times r hat."},{"Start":"02:14.785 ","End":"02:20.750","Text":"Then we add to that r times r hat dot."},{"Start":"02:20.750 ","End":"02:22.925","Text":"Now, r data I can use;"},{"Start":"02:22.925 ","End":"02:24.470","Text":"r hat, I can use, r,"},{"Start":"02:24.470 ","End":"02:28.415","Text":"I can use, but I don\u0027t really like this r hat dot thing here."},{"Start":"02:28.415 ","End":"02:30.020","Text":"It\u0027s not very proper thing to use."},{"Start":"02:30.020 ","End":"02:32.485","Text":"We need to break that down a little bit further."},{"Start":"02:32.485 ","End":"02:36.965","Text":"Luckily, I can look over here and I\u0027ve already done the work for r hat dot."},{"Start":"02:36.965 ","End":"02:39.830","Text":"What I can do is rewrite it as the following."},{"Start":"02:39.830 ","End":"02:42.260","Text":"I can say that I have r dot,"},{"Start":"02:42.260 ","End":"02:45.820","Text":"r hat, plus r times,"},{"Start":"02:45.820 ","End":"02:47.375","Text":"and instead of r hat dot,"},{"Start":"02:47.375 ","End":"02:52.700","Text":"I can fill that in with Theta dot Theta hat."},{"Start":"02:52.700 ","End":"02:55.950","Text":"Now I have this in units that I can use."},{"Start":"02:56.030 ","End":"02:59.990","Text":"Now we\u0027ve typed this out and you should put it in your formula sheet."},{"Start":"02:59.990 ","End":"03:02.150","Text":"Let me show you a drawing that might give you"},{"Start":"03:02.150 ","End":"03:07.645","Text":"a better intuitive understanding of what this velocity means in polar coordinates."},{"Start":"03:07.645 ","End":"03:10.415","Text":"Here I have some circle with a radius of"},{"Start":"03:10.415 ","End":"03:14.120","Text":"r and an object moving along it in a circular path."},{"Start":"03:14.120 ","End":"03:19.255","Text":"What I need to find are my r variables and my Theta variables, and then I can solve."},{"Start":"03:19.255 ","End":"03:23.840","Text":"First of all, r is going to be the position relative to the origin."},{"Start":"03:23.840 ","End":"03:26.810","Text":"Let\u0027s say the origin is in the middle of the circle here."},{"Start":"03:26.810 ","End":"03:29.450","Text":"In that case, we can say that the distance from"},{"Start":"03:29.450 ","End":"03:33.860","Text":"the origin in terms of magnitude is R, the radius."},{"Start":"03:33.860 ","End":"03:42.290","Text":"Little r equals R. Now we know that R is a constant."},{"Start":"03:42.290 ","End":"03:43.880","Text":"It doesn\u0027t change over time."},{"Start":"03:43.880 ","End":"03:46.310","Text":"Therefore, when we take r dot,"},{"Start":"03:46.310 ","End":"03:49.245","Text":"that is the derivative of R,"},{"Start":"03:49.245 ","End":"03:50.850","Text":"it\u0027s going to equal 0."},{"Start":"03:50.850 ","End":"03:55.525","Text":"That means that this whole first part of the equation will equal 0."},{"Start":"03:55.525 ","End":"03:58.400","Text":"As for the second part of the equation,"},{"Start":"03:58.400 ","End":"03:59.975","Text":"we can find Theta dot,"},{"Start":"03:59.975 ","End":"04:02.540","Text":"and we can find it in the following way; R,"},{"Start":"04:02.540 ","End":"04:06.740","Text":"the radius times Theta dot is the change in the angle over time,"},{"Start":"04:06.740 ","End":"04:10.460","Text":"and that is our angular velocity or Omega."},{"Start":"04:10.460 ","End":"04:15.055","Text":"We can say that r Theta dot equals r Omega."},{"Start":"04:15.055 ","End":"04:16.685","Text":"For our Omega element,"},{"Start":"04:16.685 ","End":"04:19.760","Text":"we have R times Omega."},{"Start":"04:19.760 ","End":"04:22.820","Text":"That means that the velocity of our object,"},{"Start":"04:22.820 ","End":"04:29.325","Text":"the velocity vector equals r Omega in the direction of Theta."},{"Start":"04:29.325 ","End":"04:32.390","Text":"That is something you may recognize from high school physics."},{"Start":"04:32.390 ","End":"04:35.065","Text":"This is where that equation comes from."},{"Start":"04:35.065 ","End":"04:37.410","Text":"This is your velocity vector."},{"Start":"04:37.410 ","End":"04:41.615","Text":"Now we can use this velocity vector to find the acceleration vector."},{"Start":"04:41.615 ","End":"04:44.210","Text":"Just like we did with the velocity vector,"},{"Start":"04:44.210 ","End":"04:49.010","Text":"where we took the derivative of the position vector to find our acceleration vector,"},{"Start":"04:49.010 ","End":"04:52.420","Text":"we\u0027re going to find the derivative of the velocity vector."},{"Start":"04:52.420 ","End":"04:54.365","Text":"In the same way that we did above,"},{"Start":"04:54.365 ","End":"04:57.920","Text":"we can use our new definition of the velocity vector and"},{"Start":"04:57.920 ","End":"05:01.950","Text":"take a derivative of that. Here\u0027s what we get."},{"Start":"05:01.950 ","End":"05:03.905","Text":"First, we need to write this up properly."},{"Start":"05:03.905 ","End":"05:05.675","Text":"We have dv,"},{"Start":"05:05.675 ","End":"05:07.625","Text":"the vector of the velocity,"},{"Start":"05:07.625 ","End":"05:10.745","Text":"not just the magnitude, over dt."},{"Start":"05:10.745 ","End":"05:14.030","Text":"We\u0027re going to end up with 5 different derivative terms."},{"Start":"05:14.030 ","End":"05:17.810","Text":"Because again, we can break this down just like we did here into different parts."},{"Start":"05:17.810 ","End":"05:19.525","Text":"I\u0027m going to do the same thing again."},{"Start":"05:19.525 ","End":"05:21.625","Text":"You end up with the following;"},{"Start":"05:21.625 ","End":"05:30.675","Text":"r double dot, r hat plus r dot r hat dot."},{"Start":"05:30.675 ","End":"05:33.060","Text":"That accounts for our first bit here."},{"Start":"05:33.060 ","End":"05:35.100","Text":"We\u0027re going to add to that,"},{"Start":"05:35.100 ","End":"05:42.750","Text":"r dot Theta dot Theta hat plus"},{"Start":"05:42.750 ","End":"05:53.800","Text":"r Theta double dot Theta hat plus r Theta dot Theta hat dot."},{"Start":"05:53.880 ","End":"05:57.620","Text":"The first thing to do here is simplify a little bit."},{"Start":"05:57.620 ","End":"05:59.935","Text":"Anything that has the r hat dot,"},{"Start":"05:59.935 ","End":"06:02.380","Text":"we\u0027re going to replace like we did before with"},{"Start":"06:02.380 ","End":"06:05.590","Text":"Theta hat Theta dot or Theta dot Theta hat,"},{"Start":"06:05.590 ","End":"06:11.685","Text":"and anything that has Theta hat dot will be replaced with negative Theta dot r hat."},{"Start":"06:11.685 ","End":"06:15.070","Text":"The other thing we can do is also split this into directions."},{"Start":"06:15.070 ","End":"06:16.645","Text":"What\u0027s going in the direction of r,"},{"Start":"06:16.645 ","End":"06:20.610","Text":"that is r hat, and what\u0027s going in the direction of Theta, that is Theta hat."},{"Start":"06:20.610 ","End":"06:26.100","Text":"Now, remember, this Theta hat dot is actually be going in the direction of r. First,"},{"Start":"06:26.100 ","End":"06:27.795","Text":"let\u0027s do our r elements."},{"Start":"06:27.795 ","End":"06:32.835","Text":"In the parentheses, here we have our first element, r double dot."},{"Start":"06:32.835 ","End":"06:34.910","Text":"Then we\u0027re going to take our last element and we\u0027re going to"},{"Start":"06:34.910 ","End":"06:37.070","Text":"subtract because it turns into a negative."},{"Start":"06:37.070 ","End":"06:41.340","Text":"We have negative r Theta"},{"Start":"06:41.340 ","End":"06:43.530","Text":"dot^2 because we end up with this Theta dot and"},{"Start":"06:43.530 ","End":"06:46.490","Text":"a Theta dot from down here multiplied by each other."},{"Start":"06:46.490 ","End":"06:49.535","Text":"These both go in the direction of r,"},{"Start":"06:49.535 ","End":"06:51.020","Text":"so that\u0027s r hat."},{"Start":"06:51.020 ","End":"06:54.020","Text":"Now, in the direction of Theta or Theta hat,"},{"Start":"06:54.020 ","End":"06:59.680","Text":"we have in our parentheses 2r dot Theta dot,"},{"Start":"06:59.680 ","End":"07:03.645","Text":"and that is r dot Theta dot here."},{"Start":"07:03.645 ","End":"07:07.050","Text":"We have r dot Theta dot here."},{"Start":"07:07.050 ","End":"07:12.270","Text":"Then we\u0027re going to add to that r Theta double dot."},{"Start":"07:12.270 ","End":"07:15.891","Text":"I forgot here, a line my Theta."},{"Start":"07:15.891 ","End":"07:17.955","Text":"r Theta double dot here."},{"Start":"07:17.955 ","End":"07:23.140","Text":"Both of those go in the direction of Theta or Theta hat."},{"Start":"07:23.140 ","End":"07:26.810","Text":"This is your formula and you can put it into your formula sheet."},{"Start":"07:26.810 ","End":"07:29.495","Text":"To break this down a little bit further,"},{"Start":"07:29.495 ","End":"07:33.520","Text":"everything here that\u0027s in the direction of r or r hat,"},{"Start":"07:33.520 ","End":"07:36.500","Text":"we can call this the radial acceleration."},{"Start":"07:36.500 ","End":"07:41.810","Text":"It\u0027s all of the acceleration in the direction that r hat vector outwards."},{"Start":"07:41.810 ","End":"07:48.920","Text":"Everything over here, we can call that the acceleration in the direction of Theta."},{"Start":"07:48.920 ","End":"07:50.240","Text":"Everything that\u0027s in the direction of Theta,"},{"Start":"07:50.240 ","End":"07:52.145","Text":"that is the direction of the turning."},{"Start":"07:52.145 ","End":"07:56.015","Text":"By simplifying, we can get a clear mathematical understanding."},{"Start":"07:56.015 ","End":"07:59.060","Text":"First of all, r double dot equals 0."},{"Start":"07:59.060 ","End":"08:01.130","Text":"We know that because of R dot equals 0,"},{"Start":"08:01.130 ","End":"08:03.335","Text":"then r double dot must equal 0."},{"Start":"08:03.335 ","End":"08:05.390","Text":"If we\u0027re looking at our r hat element,"},{"Start":"08:05.390 ","End":"08:12.855","Text":"then what we\u0027re left with for our acceleration vector is r Theta dot squared r hat."},{"Start":"08:12.855 ","End":"08:13.980","Text":"Now, Theta dot,"},{"Start":"08:13.980 ","End":"08:15.900","Text":"we can write as Omega."},{"Start":"08:15.900 ","End":"08:22.140","Text":"We\u0027re left with negative r Omega squared r hat."},{"Start":"08:22.140 ","End":"08:24.105","Text":"For our Theta element,"},{"Start":"08:24.105 ","End":"08:25.890","Text":"that going in the direction of Theta,"},{"Start":"08:25.890 ","End":"08:28.320","Text":"we know that our r dot again equals 0."},{"Start":"08:28.320 ","End":"08:34.005","Text":"This part will fall out and we\u0027re left with r Theta double dot Theta hat."},{"Start":"08:34.005 ","End":"08:36.015","Text":"Now, we know that Theta double dot,"},{"Start":"08:36.015 ","End":"08:37.425","Text":"we can call that Alpha,"},{"Start":"08:37.425 ","End":"08:40.740","Text":"Alpha is the angular acceleration."},{"Start":"08:40.740 ","End":"08:42.285","Text":"We can write Alpha here,"},{"Start":"08:42.285 ","End":"08:46.115","Text":"and that multiplied by Theta hat."},{"Start":"08:46.115 ","End":"08:50.735","Text":"What we\u0027re left with here actually can explain some things from circular motion before."},{"Start":"08:50.735 ","End":"08:55.380","Text":"First of all, our radial acceleration equals Omega"},{"Start":"08:55.380 ","End":"09:01.455","Text":"squared r. We can also write that as v^2/r."},{"Start":"09:01.455 ","End":"09:08.585","Text":"Remember v=Omega r. Why is this negative here as opposed to positive over here?"},{"Start":"09:08.585 ","End":"09:09.875","Text":"Well, if you recall,"},{"Start":"09:09.875 ","End":"09:12.020","Text":"over in these equations for circular motion,"},{"Start":"09:12.020 ","End":"09:16.595","Text":"we generally chose a positive direction to be inwards towards the center of the circle."},{"Start":"09:16.595 ","End":"09:18.335","Text":"Whereas in this equation,"},{"Start":"09:18.335 ","End":"09:21.905","Text":"we\u0027ve chosen r hat as going outwards from the origin,"},{"Start":"09:21.905 ","End":"09:25.045","Text":"so that accounts for the minus or the negative value."},{"Start":"09:25.045 ","End":"09:27.875","Text":"Now if we look at our Theta acceleration,"},{"Start":"09:27.875 ","End":"09:29.255","Text":"we can really rename this,"},{"Start":"09:29.255 ","End":"09:32.915","Text":"our tangential acceleration, as we called it in circular motion."},{"Start":"09:32.915 ","End":"09:36.485","Text":"Now, Theta acceleration includes this element which really zeros out."},{"Start":"09:36.485 ","End":"09:39.320","Text":"But if we\u0027re talking about our tangential acceleration,"},{"Start":"09:39.320 ","End":"09:41.210","Text":"what we\u0027re left with is Alpha r,"},{"Start":"09:41.210 ","End":"09:44.300","Text":"which you may remember from our circular motion unit."},{"Start":"09:44.300 ","End":"09:48.040","Text":"At this point, we\u0027ve really developed everything we need to get out of this."},{"Start":"09:48.040 ","End":"09:50.960","Text":"Just to give you a quick reminder, as we showed before,"},{"Start":"09:50.960 ","End":"09:53.320","Text":"the position in terms of Cartesian coordinates,"},{"Start":"09:53.320 ","End":"09:56.615","Text":"we can also write the velocity in terms of Cartesian coordinates."},{"Start":"09:56.615 ","End":"09:59.330","Text":"That\u0027ll be x dot in the direction of x,"},{"Start":"09:59.330 ","End":"10:03.620","Text":"that is x hat, and y dot in the direction of y."},{"Start":"10:03.620 ","End":"10:07.955","Text":"We can talk about this as the velocity in the direction of x,"},{"Start":"10:07.955 ","End":"10:11.165","Text":"and this is the velocity in the direction of y."},{"Start":"10:11.165 ","End":"10:14.810","Text":"Over here we can talk about the velocity in the direction of r,"},{"Start":"10:14.810 ","End":"10:19.895","Text":"the radial velocity, and the velocity in the direction of Theta, the Theta velocity."},{"Start":"10:19.895 ","End":"10:22.655","Text":"Below, we can do the same thing with acceleration."},{"Start":"10:22.655 ","End":"10:28.775","Text":"The acceleration in Cartesian coordinates equals x double dot in the direction of x,"},{"Start":"10:28.775 ","End":"10:34.255","Text":"x hat, plus y double dot in the direction of y, y hat."},{"Start":"10:34.255 ","End":"10:37.160","Text":"The same holds true as above in terms of the acceleration,"},{"Start":"10:37.160 ","End":"10:39.845","Text":"we can call this the acceleration in the direction of x,"},{"Start":"10:39.845 ","End":"10:41.495","Text":"or the x acceleration."},{"Start":"10:41.495 ","End":"10:45.740","Text":"This is the y acceleration or acceleration in the direction of y."},{"Start":"10:45.740 ","End":"10:48.140","Text":"In the same way that we have the acceleration in"},{"Start":"10:48.140 ","End":"10:50.914","Text":"the direction of r or the radial acceleration,"},{"Start":"10:50.914 ","End":"10:53.975","Text":"and the acceleration in the direction of Theta,"},{"Start":"10:53.975 ","End":"10:56.650","Text":"or the tangential acceleration."},{"Start":"10:56.650 ","End":"10:59.630","Text":"Again, if you\u0027re working in Cartesian coordinates,"},{"Start":"10:59.630 ","End":"11:02.480","Text":"you want to use rx, ry,"},{"Start":"11:02.480 ","End":"11:05.725","Text":"vx, vy, ax, ay."},{"Start":"11:05.725 ","End":"11:08.880","Text":"If you\u0027re working in polar coordinates,"},{"Start":"11:08.880 ","End":"11:11.865","Text":"you want to be using vr and v Theta,"},{"Start":"11:11.865 ","End":"11:14.445","Text":"obviously r and Theta."},{"Start":"11:14.445 ","End":"11:19.030","Text":"For acceleration, you want to be using ar and a Theta."}],"ID":9281}],"Thumbnail":null,"ID":5399},{"Name":"Exercise Man Riding a Motorbike","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise","Duration":"7m 56s","ChapterTopicVideoID":8988,"CourseChapterTopicPlaylistID":5400,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:04.154","Text":"In our 1st example problem for polar coordinates,"},{"Start":"00:04.154 ","End":"00:05.325","Text":"we\u0027re given a biker,"},{"Start":"00:05.325 ","End":"00:09.510","Text":"and this biker is standing at the origin right here."},{"Start":"00:09.510 ","End":"00:11.850","Text":"We\u0027re given that r,"},{"Start":"00:11.850 ","End":"00:16.230","Text":"The biker\u0027s distance from the origin over time equals ct,"},{"Start":"00:16.230 ","End":"00:18.659","Text":"c being some constant."},{"Start":"00:18.659 ","End":"00:22.440","Text":"We\u0027re also given another variable and that is Omega,"},{"Start":"00:22.440 ","End":"00:27.640","Text":"the angular velocity equals Omega_0, another constant."},{"Start":"00:27.640 ","End":"00:31.190","Text":"These are given data points for the problem."},{"Start":"00:31.190 ","End":"00:35.495","Text":"The question asks us to find the maximal distance"},{"Start":"00:35.495 ","End":"00:40.580","Text":"reached by the bike rider given the static friction coefficient Mu_s."},{"Start":"00:40.580 ","End":"00:42.095","Text":"What\u0027s happening here,"},{"Start":"00:42.095 ","End":"00:44.690","Text":"and it can help you understand a little better maybe,"},{"Start":"00:44.690 ","End":"00:47.315","Text":"is that as time goes forward,"},{"Start":"00:47.315 ","End":"00:50.600","Text":"ct or r as a result gets larger."},{"Start":"00:50.600 ","End":"00:56.345","Text":"You have the distance from the origin getting greater as there\u0027s some turning motion."},{"Start":"00:56.345 ","End":"00:59.480","Text":"You end up with some spiral motion like this."},{"Start":"00:59.480 ","End":"01:07.790","Text":"What we\u0027re looking for in our problem is the maximal distance reached."},{"Start":"01:07.790 ","End":"01:09.965","Text":"We\u0027re looking for some r distance,"},{"Start":"01:09.965 ","End":"01:15.560","Text":"r_max, which is the maximum distance the rider gets from the origin."},{"Start":"01:15.560 ","End":"01:19.860","Text":"What we\u0027re looking for is r_max."},{"Start":"01:20.240 ","End":"01:24.380","Text":"You should know that as the bike rider is going around the whole way,"},{"Start":"01:24.380 ","End":"01:27.200","Text":"he\u0027s being pushed by static friction,"},{"Start":"01:27.200 ","End":"01:29.870","Text":"not kinetic friction, even though he\u0027s moving."},{"Start":"01:29.870 ","End":"01:32.240","Text":"At the moment you may not understand why that is"},{"Start":"01:32.240 ","End":"01:34.790","Text":"the case and we\u0027ll talk about it in later chapters,"},{"Start":"01:34.790 ","End":"01:39.610","Text":"but for now just accept it as a given assumption and it will help us solve this problem."},{"Start":"01:39.610 ","End":"01:44.900","Text":"This static friction is a force really that\u0027s pushing the bike rider along the whole way."},{"Start":"01:44.900 ","End":"01:47.810","Text":"We\u0027re going to call this force f_s."},{"Start":"01:47.810 ","End":"01:54.845","Text":"F_s is always equal to or less than Mu_sN."},{"Start":"01:54.845 ","End":"02:00.170","Text":"N is the normal force or the force that\u0027s pushing the rider up from the street."},{"Start":"02:00.170 ","End":"02:04.190","Text":"In that case, the normal has to equal mg,"},{"Start":"02:04.190 ","End":"02:05.360","Text":"the force of gravity."},{"Start":"02:05.360 ","End":"02:08.705","Text":"If N equals mg,"},{"Start":"02:08.705 ","End":"02:11.090","Text":"we can then replace it in the equation above."},{"Start":"02:11.090 ","End":"02:13.095","Text":"But when we\u0027re looking for the f_s,"},{"Start":"02:13.095 ","End":"02:16.970","Text":"we\u0027re looking for the maximal f_s so we can find the r_max."},{"Start":"02:16.970 ","End":"02:21.600","Text":"We\u0027re looking for f_smax and"},{"Start":"02:21.600 ","End":"02:27.545","Text":"that should equal Mu_smg."},{"Start":"02:27.545 ","End":"02:33.335","Text":"We know that Mu_smg is the only force acting upon our object."},{"Start":"02:33.335 ","End":"02:35.735","Text":"According to Newton\u0027s 2nd law,"},{"Start":"02:35.735 ","End":"02:38.725","Text":"we know that it has to equal ma."},{"Start":"02:38.725 ","End":"02:40.340","Text":"Specifically in our situation,"},{"Start":"02:40.340 ","End":"02:42.395","Text":"we\u0027re talking about a maximal point."},{"Start":"02:42.395 ","End":"02:48.415","Text":"We\u0027re looking for ma_max and the m\u0027s will drop out here."},{"Start":"02:48.415 ","End":"02:51.050","Text":"What we\u0027re left with is that a_max,"},{"Start":"02:51.050 ","End":"02:57.205","Text":"that\u0027s the maximum acceleration, equals Mu_sg."},{"Start":"02:57.205 ","End":"03:00.905","Text":"Our maximum acceleration equals Mu_sg."},{"Start":"03:00.905 ","End":"03:03.260","Text":"If you recall, in 1 of our last videos,"},{"Start":"03:03.260 ","End":"03:07.235","Text":"we talked about another way to represent our acceleration as a vector."},{"Start":"03:07.235 ","End":"03:09.500","Text":"The acceleration vector equals the fall."},{"Start":"03:09.500 ","End":"03:16.655","Text":"A vector equals r-double dot minus rTheta-dot^2 in the direction of"},{"Start":"03:16.655 ","End":"03:20.450","Text":"r-hat plus 2r-dot Theta-dot"},{"Start":"03:20.450 ","End":"03:25.580","Text":"plus rTheta-double dot in the direction of Theta or Theta-hat."},{"Start":"03:25.580 ","End":"03:29.030","Text":"At this point, we can add in some of our given data points from"},{"Start":"03:29.030 ","End":"03:33.020","Text":"the left side to help us understand this a little better and simplify things."},{"Start":"03:33.020 ","End":"03:37.770","Text":"We know that r equals ct. We can fill in all of our r\u0027s with"},{"Start":"03:37.770 ","End":"03:42.840","Text":"ct. We also know that Theta-dot equals Omega."},{"Start":"03:42.840 ","End":"03:48.050","Text":"In our case from over here, Omega equals Omega_0."},{"Start":"03:48.050 ","End":"03:52.100","Text":"Remember both c and Omega_0 are constants."},{"Start":"03:52.100 ","End":"03:55.220","Text":"When you start filling in these pieces of information,"},{"Start":"03:55.220 ","End":"03:58.985","Text":"you get the following; in the direction of r or r-hat,"},{"Start":"03:58.985 ","End":"04:02.810","Text":"r-double dot goes to 0 because we\u0027re taking the 2nd derivative,"},{"Start":"04:02.810 ","End":"04:05.525","Text":"which 1st drops out t and then drops out the c,"},{"Start":"04:05.525 ","End":"04:07.010","Text":"so we\u0027re left with 0."},{"Start":"04:07.010 ","End":"04:09.955","Text":"What we have left is negative rTheta-dot."},{"Start":"04:09.955 ","End":"04:13.535","Text":"R, we\u0027re going to keep the same because this is really what we\u0027re solving for."},{"Start":"04:13.535 ","End":"04:17.389","Text":"The Theta-dot, we can take and turn into Omega_0."},{"Start":"04:17.389 ","End":"04:19.145","Text":"From the direction of r,"},{"Start":"04:19.145 ","End":"04:21.950","Text":"we\u0027re left with negative r"},{"Start":"04:21.950 ","End":"04:29.945","Text":"Omega_0^2 in the direction of r. In the direction of Theta,"},{"Start":"04:29.945 ","End":"04:34.420","Text":"we have 2r-dot will equals c. Theta-dot, again,"},{"Start":"04:34.420 ","End":"04:41.150","Text":"is Omega_0 and rTheta-double dot will equal 0 because the derivative of Omega_0,"},{"Start":"04:41.150 ","End":"04:42.890","Text":"a constant, equals 0."},{"Start":"04:42.890 ","End":"04:50.040","Text":"In the direction of Theta, we have 2c Omega_0."},{"Start":"04:52.230 ","End":"04:55.800","Text":"I\u0027m going to put that in a type so it\u0027s a little easier to read."},{"Start":"04:55.800 ","End":"04:59.185","Text":"Our next step is if we look above,"},{"Start":"04:59.185 ","End":"05:00.625","Text":"notice that a max,"},{"Start":"05:00.625 ","End":"05:02.815","Text":"we\u0027re talking about a magnitude or a scalar,"},{"Start":"05:02.815 ","End":"05:05.635","Text":"not a vector like we have written out here."},{"Start":"05:05.635 ","End":"05:10.540","Text":"We can actually take this and turn it into our scalar by eliminating the direction."},{"Start":"05:10.540 ","End":"05:13.075","Text":"They equal the same thing in terms of magnitude,"},{"Start":"05:13.075 ","End":"05:16.840","Text":"just our vector has incorporated directions r-hat and Theta-hat."},{"Start":"05:16.840 ","End":"05:20.845","Text":"The way we eliminate those is by taking the square of"},{"Start":"05:20.845 ","End":"05:23.380","Text":"each element and taking the square root of all of"},{"Start":"05:23.380 ","End":"05:26.170","Text":"that to give us the magnitude of our vector."},{"Start":"05:26.170 ","End":"05:27.595","Text":"Let\u0027s do that real quick."},{"Start":"05:27.595 ","End":"05:33.105","Text":"The magnitude of the a vector equals,"},{"Start":"05:33.105 ","End":"05:35.210","Text":"we\u0027re going to take out our negative because it doesn\u0027t matter,"},{"Start":"05:35.210 ","End":"05:36.701","Text":"we\u0027re doing a square,"},{"Start":"05:36.701 ","End":"05:40.085","Text":"(Omega_0^2 times"},{"Start":"05:40.085 ","End":"05:49.565","Text":"r)^2 plus (2c Omega_0)^2."},{"Start":"05:49.565 ","End":"05:52.345","Text":"I\u0027m going to take the square root of this whole thing."},{"Start":"05:52.345 ","End":"05:55.310","Text":"Notice above the magnitude of the a vector,"},{"Start":"05:55.310 ","End":"05:58.475","Text":"that is a_max equals Mu_sg."},{"Start":"05:58.475 ","End":"06:03.080","Text":"This also will equal Mu_sg."},{"Start":"06:03.080 ","End":"06:05.240","Text":"Also above, we\u0027re looking for"},{"Start":"06:05.240 ","End":"06:08.870","Text":"the maximum acceleration when we\u0027re talking about equaling Mu_sg,"},{"Start":"06:08.870 ","End":"06:10.940","Text":"and not being less than Mu_sg."},{"Start":"06:10.940 ","End":"06:13.010","Text":"We\u0027re talking about maximum acceleration,"},{"Start":"06:13.010 ","End":"06:16.025","Text":"and based on the relationship between r and a,"},{"Start":"06:16.025 ","End":"06:20.190","Text":"we know that we\u0027re also looking for max_r or r_max."},{"Start":"06:20.190 ","End":"06:22.610","Text":"We can really solve for this r_max,"},{"Start":"06:22.610 ","End":"06:24.950","Text":"which is what we were looking for from the beginning."},{"Start":"06:24.950 ","End":"06:27.680","Text":"The way we\u0027re going to do that, I\u0027ll spare you the headache,"},{"Start":"06:27.680 ","End":"06:30.335","Text":"but it\u0027s going to equal the following."},{"Start":"06:30.335 ","End":"06:33.485","Text":"All I\u0027ve done here is isolate my r_max,"},{"Start":"06:33.485 ","End":"06:35.405","Text":"and if the math doesn\u0027t make sense to you,"},{"Start":"06:35.405 ","End":"06:37.310","Text":"you can do it yourself to make sure"},{"Start":"06:37.310 ","End":"06:40.145","Text":"that you\u0027re understanding how we\u0027re getting to this answer."},{"Start":"06:40.145 ","End":"06:41.780","Text":"This is going to be your answer though,"},{"Start":"06:41.780 ","End":"06:43.550","Text":"and you can use it moving forward."},{"Start":"06:43.550 ","End":"06:46.685","Text":"1 more note I want to make before we end is,"},{"Start":"06:46.685 ","End":"06:48.920","Text":"let\u0027s look at our spiral over here again."},{"Start":"06:48.920 ","End":"06:51.590","Text":"Let\u0027s assume it keeps going so we find a clean spot."},{"Start":"06:51.590 ","End":"06:55.715","Text":"At this point I\u0027m going to have 2 kinds of acceleration."},{"Start":"06:55.715 ","End":"06:57.620","Text":"1 is going to be towards the center,"},{"Start":"06:57.620 ","End":"07:00.515","Text":"and that\u0027s going to a be my radial acceleration, ar."},{"Start":"07:00.515 ","End":"07:04.640","Text":"I\u0027m also going to have acceleration in the direction of Theta."},{"Start":"07:04.640 ","End":"07:06.499","Text":"That\u0027s going to be Theta acceleration."},{"Start":"07:06.499 ","End":"07:09.260","Text":"It\u0027s similar to our tangential acceleration,"},{"Start":"07:09.260 ","End":"07:10.400","Text":"if you recall earlier."},{"Start":"07:10.400 ","End":"07:14.060","Text":"Our total acceleration is not going to be either of these things,"},{"Start":"07:14.060 ","End":"07:16.930","Text":"rather, it\u0027s going to be something in the middle, something like this."},{"Start":"07:16.930 ","End":"07:21.290","Text":"This total acceleration is what we\u0027re talking about when we talk about a_max,"},{"Start":"07:21.290 ","End":"07:23.090","Text":"not in 1 of these 2 directions."},{"Start":"07:23.090 ","End":"07:26.510","Text":"Similarly, when we\u0027re talking about f_s,"},{"Start":"07:26.510 ","End":"07:29.890","Text":"we\u0027re talking about something going in the same direction as our at,"},{"Start":"07:29.890 ","End":"07:32.430","Text":"as our total acceleration."},{"Start":"07:33.320 ","End":"07:35.870","Text":"Just to reiterate, f_s,"},{"Start":"07:35.870 ","End":"07:38.495","Text":"is going in the direction of the total acceleration;"},{"Start":"07:38.495 ","End":"07:40.280","Text":"not towards the middle of the circle,"},{"Start":"07:40.280 ","End":"07:42.175","Text":"not towards the outside."},{"Start":"07:42.175 ","End":"07:43.955","Text":"That\u0027s it for this problem."},{"Start":"07:43.955 ","End":"07:47.150","Text":"Once you realize you need to use our formulas from before,"},{"Start":"07:47.150 ","End":"07:49.835","Text":"it\u0027s really a matter of some simple mathematical work,"},{"Start":"07:49.835 ","End":"07:51.455","Text":"and then you can find your answer."},{"Start":"07:51.455 ","End":"07:52.820","Text":"If any of this was unclear,"},{"Start":"07:52.820 ","End":"07:56.790","Text":"you should go back and do the work again though. Thanks for listening."}],"ID":9282}],"Thumbnail":null,"ID":5400},{"Name":"Exercise Carousel","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Question","Duration":"2m 15s","ChapterTopicVideoID":8996,"CourseChapterTopicPlaylistID":5401,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8996.jpeg","UploadDate":"2021-11-18T10:50:36.2570000","DurationForVideoObject":"PT2M15S","Description":null,"MetaTitle":"The Question: Video + Workbook | Proprep","MetaDescription":"Polar Coordinates - Exercise Carousel. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/physics-1-mechanics-waves-and-thermodynamics/polar-coordinates/exercise-carousel/vid9283","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"Let\u0027s move on to our second example,"},{"Start":"00:02.490 ","End":"00:04.650","Text":"the example of the carousel."},{"Start":"00:04.650 ","End":"00:08.640","Text":"Essentially we have a carousel which is a large circle."},{"Start":"00:08.640 ","End":"00:12.555","Text":"On that carousel, we have a bug."},{"Start":"00:12.555 ","End":"00:18.420","Text":"The bug is sitting on the outside of the carousel and moving towards the center,"},{"Start":"00:18.420 ","End":"00:22.110","Text":"at a constant velocity of V sub 0."},{"Start":"00:22.110 ","End":"00:25.170","Text":"This is the center of our circle,"},{"Start":"00:25.170 ","End":"00:34.081","Text":"and the radius is given as R. Now the carousel is also moving with an angular velocity,"},{"Start":"00:34.081 ","End":"00:39.085","Text":"and that angular velocity is given to us as Omega sub 0."},{"Start":"00:39.085 ","End":"00:41.135","Text":"We have 2, motion."},{"Start":"00:41.135 ","End":"00:45.020","Text":"First, the bug is moving towards the center of the carousel and second,"},{"Start":"00:45.020 ","End":"00:47.225","Text":"the carousel itself is turning."},{"Start":"00:47.225 ","End":"00:49.865","Text":"Now we have 4 observers in this problem."},{"Start":"00:49.865 ","End":"00:51.610","Text":"First we have observer A."},{"Start":"00:51.610 ","End":"00:54.560","Text":"Observer A is standing at the same point as"},{"Start":"00:54.560 ","End":"00:59.120","Text":"the bug and stays on its same point along the carousel as it moves."},{"Start":"00:59.120 ","End":"01:03.785","Text":"Secondly, we have observer B standing in the middle of the carousel,"},{"Start":"01:03.785 ","End":"01:06.465","Text":"in the center here and moving with the carousel."},{"Start":"01:06.465 ","End":"01:11.720","Text":"His or her point of view is also rotating with the carousel."},{"Start":"01:11.720 ","End":"01:16.835","Text":"We then have the observers C who\u0027s standing in the same point as observer B,"},{"Start":"01:16.835 ","End":"01:18.440","Text":"but isn\u0027t moving at all."},{"Start":"01:18.440 ","End":"01:22.685","Text":"You can imagine that there\u0027s a fly hovering above the carousel as it moves."},{"Start":"01:22.685 ","End":"01:26.510","Text":"Lastly, we have observer D. Observer D is"},{"Start":"01:26.510 ","End":"01:30.455","Text":"standing outside of the carousel and also is not spinning,"},{"Start":"01:30.455 ","End":"01:33.620","Text":"but observes the carousel spinning from the outside."},{"Start":"01:33.620 ","End":"01:39.860","Text":"The question asks us to find the position of the bug in terms of x and y,"},{"Start":"01:39.860 ","End":"01:42.890","Text":"and in terms of r and Theta."},{"Start":"01:42.890 ","End":"01:46.715","Text":"That is, we need to find the r vector in terms of x and y,"},{"Start":"01:46.715 ","End":"01:48.335","Text":"and in terms of r and Theta."},{"Start":"01:48.335 ","End":"01:52.325","Text":"For the bug from the perspective of all 4 of our observers."},{"Start":"01:52.325 ","End":"01:55.100","Text":"We then need to find the velocity from each of"},{"Start":"01:55.100 ","End":"01:59.180","Text":"those 4 perspectives in terms of both x and y,"},{"Start":"01:59.180 ","End":"02:00.560","Text":"and r and Theta."},{"Start":"02:00.560 ","End":"02:05.345","Text":"We also need to find the acceleration in terms of x and y and r and Theta."},{"Start":"02:05.345 ","End":"02:09.290","Text":"We need to find rv vector,"},{"Start":"02:09.290 ","End":"02:12.695","Text":"and our a vector in terms of both x and y,"},{"Start":"02:12.695 ","End":"02:15.600","Text":"and in terms of r and Theta."}],"ID":9283},{"Watched":false,"Name":"Observer A","Duration":"4m 43s","ChapterTopicVideoID":8997,"CourseChapterTopicPlaylistID":5401,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Let\u0027s start with observer A."},{"Start":"00:02.580 ","End":"00:05.655","Text":"Now as observer A rotates around the circle,"},{"Start":"00:05.655 ","End":"00:07.320","Text":"the bug will move with her,"},{"Start":"00:07.320 ","End":"00:09.765","Text":"but will move towards the center of the circle."},{"Start":"00:09.765 ","End":"00:12.495","Text":"First let\u0027s assign some coordinates."},{"Start":"00:12.495 ","End":"00:18.860","Text":"Let\u0027s say that the x coordinate moves in a positive direction to the right as normal,"},{"Start":"00:18.860 ","End":"00:24.120","Text":"and let\u0027s say that the y-coordinate also moves upwards in a positive direction."},{"Start":"00:24.120 ","End":"00:29.039","Text":"Now let\u0027s turn the carousel little to see what it looks like after some rotation."},{"Start":"00:29.039 ","End":"00:32.864","Text":"After the carousel has rotated 45 degrees,"},{"Start":"00:32.864 ","End":"00:35.775","Text":"our observer A is at its new location,"},{"Start":"00:35.775 ","End":"00:39.210","Text":"the bug has moved slightly closer to the center,"},{"Start":"00:39.210 ","End":"00:42.890","Text":"but in relation to one another they\u0027re still on the same x-axis"},{"Start":"00:42.890 ","End":"00:47.200","Text":"and haven\u0027t moved relative to each other in terms of the y component."},{"Start":"00:47.200 ","End":"00:50.810","Text":"We can describe the position of the bug in"},{"Start":"00:50.810 ","End":"00:55.565","Text":"relation to observer A in terms of x and y coordinates."},{"Start":"00:55.565 ","End":"00:56.885","Text":"The x value,"},{"Start":"00:56.885 ","End":"01:00.645","Text":"x in relation to point A and this is how we usually write that,"},{"Start":"01:00.645 ","End":"01:09.200","Text":"is with large X and a small a above it equals negative V_0 times t time because it\u0027s,"},{"Start":"01:09.200 ","End":"01:13.610","Text":"the bug was going at a velocity of V_0 towards the left over"},{"Start":"01:13.610 ","End":"01:18.425","Text":"time t. Now in terms of y in relation to A,"},{"Start":"01:18.425 ","End":"01:21.695","Text":"there is no change, y equals 0."},{"Start":"01:21.695 ","End":"01:25.460","Text":"If we\u0027re going to talk about this in terms of an r vector,"},{"Start":"01:25.460 ","End":"01:32.600","Text":"the r vector equals x in the direction of x plus y in the direction of y."},{"Start":"01:32.600 ","End":"01:38.300","Text":"In this case, it equals negative V_0t x-hat,"},{"Start":"01:38.300 ","End":"01:40.115","Text":"and there is no y-value."},{"Start":"01:40.115 ","End":"01:45.010","Text":"Now our next step is to transpose this into terms of r and Theta."},{"Start":"01:45.010 ","End":"01:47.780","Text":"The way we do that is if you recall,"},{"Start":"01:47.780 ","End":"01:49.565","Text":"we have our formulas from before,"},{"Start":"01:49.565 ","End":"01:59.050","Text":"x equals r cosine Theta and y equals r sine Theta."},{"Start":"01:59.050 ","End":"02:03.410","Text":"For the opposite transposition to find r and Theta,"},{"Start":"02:03.410 ","End":"02:05.180","Text":"which we need to do in this case,"},{"Start":"02:05.180 ","End":"02:10.775","Text":"r^2 equals x^2 plus y^2 and"},{"Start":"02:10.775 ","End":"02:16.825","Text":"tangent Theta equals y over x."},{"Start":"02:16.825 ","End":"02:20.645","Text":"In our case to find r in relation to A,"},{"Start":"02:20.645 ","End":"02:24.545","Text":"we take x^2 plus y^2 and find the square root."},{"Start":"02:24.545 ","End":"02:29.480","Text":"There is no y^2 so we take x^2 and take the square root of that,"},{"Start":"02:29.480 ","End":"02:32.630","Text":"which equals V sub 0t."},{"Start":"02:32.630 ","End":"02:34.535","Text":"Now as for Theta,"},{"Start":"02:34.535 ","End":"02:37.190","Text":"y over x equals 0,"},{"Start":"02:37.190 ","End":"02:40.505","Text":"so the tangent of Theta equals 0."},{"Start":"02:40.505 ","End":"02:46.680","Text":"That means that Theta can either equal 0 or 180."},{"Start":"02:46.720 ","End":"02:49.910","Text":"To find out which of these two options,"},{"Start":"02:49.910 ","End":"02:52.685","Text":"0 or 180 is the correct option,"},{"Start":"02:52.685 ","End":"02:55.940","Text":"we need to go back to our chart over here and look at"},{"Start":"02:55.940 ","End":"03:00.045","Text":"the vector for the position of the bug."},{"Start":"03:00.045 ","End":"03:02.480","Text":"If we look at the position of the bug,"},{"Start":"03:02.480 ","End":"03:04.895","Text":"we can draw this vector as follows."},{"Start":"03:04.895 ","End":"03:11.065","Text":"We\u0027re going from the origin to the left along the x-axis."},{"Start":"03:11.065 ","End":"03:13.690","Text":"This is the vector rA,"},{"Start":"03:13.690 ","End":"03:17.390","Text":"the position of the bug in relation to the point A."},{"Start":"03:17.390 ","End":"03:22.920","Text":"In order to find the angle of this position vector,"},{"Start":"03:22.920 ","End":"03:28.815","Text":"you measure from the right where x is positive all the way over to where our vector lies."},{"Start":"03:28.815 ","End":"03:32.495","Text":"This whole arc here is the angle of the vector,"},{"Start":"03:32.495 ","End":"03:38.402","Text":"meaning that this vector is at an angle of 180 degrees not at an angle of 0 degrees,"},{"Start":"03:38.402 ","End":"03:43.600","Text":"so Theta equals 180 degrees and not 0 degrees."},{"Start":"03:45.410 ","End":"03:51.075","Text":"We can write out our r vector in terms of"},{"Start":"03:51.075 ","End":"03:56.150","Text":"r and Theta as r in the direction of r. In our case,"},{"Start":"03:56.150 ","End":"03:59.900","Text":"that equals V_0t in the direction"},{"Start":"03:59.900 ","End":"04:03.680","Text":"of r. In order to find your velocity and your acceleration,"},{"Start":"04:03.680 ","End":"04:06.200","Text":"you\u0027ll have to apply our formulas from before,"},{"Start":"04:06.200 ","End":"04:10.730","Text":"just to give you an example for the velocity in terms of r and Theta,"},{"Start":"04:10.730 ","End":"04:17.365","Text":"we would need r.r hat plus rTheta.Theta hat."},{"Start":"04:17.365 ","End":"04:19.480","Text":"I\u0027m not going to do the hard work for you here,"},{"Start":"04:19.480 ","End":"04:24.920","Text":"but once you\u0027ve found a way to describe any point be that XA, XB, XC,"},{"Start":"04:24.920 ","End":"04:28.790","Text":"XD, or in terms of r and Theta you can then go back and apply"},{"Start":"04:28.790 ","End":"04:33.215","Text":"your formulas for before to find your velocity and also your acceleration."},{"Start":"04:33.215 ","End":"04:35.255","Text":"But for now I hope you\u0027ve gotten the point."},{"Start":"04:35.255 ","End":"04:39.320","Text":"We found a way to describe how the bug moves in terms"},{"Start":"04:39.320 ","End":"04:44.490","Text":"of observer A\u0027s perspective and now we can move on to observer B."}],"ID":9284},{"Watched":false,"Name":"Observer B","Duration":"3m 46s","ChapterTopicVideoID":8998,"CourseChapterTopicPlaylistID":5401,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.480","Text":"Let\u0027s move on to Observer B and the first thing we want to do with Observer B is find"},{"Start":"00:06.480 ","End":"00:09.045","Text":"some axes that we can use to measure"},{"Start":"00:09.045 ","End":"00:13.570","Text":"the motion of the bug from the perspective of Observer B."},{"Start":"00:13.610 ","End":"00:17.730","Text":"Now it just so happens that our x and y-axis for"},{"Start":"00:17.730 ","End":"00:22.455","Text":"Observer B are parallel to the x and y axes for Observer A."},{"Start":"00:22.455 ","End":"00:26.040","Text":"The x-axis looks like this with a positive direction to"},{"Start":"00:26.040 ","End":"00:31.113","Text":"the right and the y-axis looks like this."},{"Start":"00:31.113 ","End":"00:39.140","Text":"Because our x-axis and our y-axis are mirrors of the x and y-axis for Observer A,"},{"Start":"00:39.140 ","End":"00:41.870","Text":"we can then come up with a general equation that"},{"Start":"00:41.870 ","End":"00:45.005","Text":"describes how any object like our bug observed from"},{"Start":"00:45.005 ","End":"00:47.510","Text":"the point of Observer A can then be"},{"Start":"00:47.510 ","End":"00:51.890","Text":"transposed to a description in terms of what observer B see."},{"Start":"00:51.890 ","End":"00:57.350","Text":"We can say, is that any given object observed from the perspective of"},{"Start":"00:57.350 ","End":"01:04.130","Text":"Observer B equals the same object observed from point A plus R,"},{"Start":"01:04.130 ","End":"01:05.899","Text":"the distance between the 2."},{"Start":"01:05.899 ","End":"01:10.370","Text":"Now, from our work with relative motion and relative measurements,"},{"Start":"01:10.370 ","End":"01:14.090","Text":"we know that it could be plus or minus R. In this case,"},{"Start":"01:14.090 ","End":"01:18.365","Text":"we can look at our example to show whether it should be plus or minus."},{"Start":"01:18.365 ","End":"01:21.830","Text":"Let\u0027s assume we\u0027re talking about time t=0 when"},{"Start":"01:21.830 ","End":"01:25.790","Text":"the bug is sharing the same location as Observer A,"},{"Start":"01:25.790 ","End":"01:28.623","Text":"the location described by Observer A would be 0,"},{"Start":"01:28.623 ","End":"01:32.330","Text":"0, in terms of x, it would just be 0."},{"Start":"01:32.330 ","End":"01:34.445","Text":"In terms of Observer B,"},{"Start":"01:34.445 ","End":"01:35.690","Text":"its location would be r,"},{"Start":"01:35.690 ","End":"01:37.295","Text":"the distance between the 2 of them,"},{"Start":"01:37.295 ","End":"01:38.930","Text":"the radius of the circle."},{"Start":"01:38.930 ","End":"01:41.490","Text":"So we know that when X^a=0,"},{"Start":"01:41.490 ","End":"01:44.610","Text":"X^b has to equal R. In fact over here,"},{"Start":"01:44.610 ","End":"01:47.565","Text":"that is the case so we know we\u0027ve done this correctly."},{"Start":"01:47.565 ","End":"01:49.760","Text":"Now that we have our x value,"},{"Start":"01:49.760 ","End":"01:51.455","Text":"we can find our y value."},{"Start":"01:51.455 ","End":"01:54.140","Text":"Now Y^b in relation to Y^a?"},{"Start":"01:54.140 ","End":"01:55.865","Text":"If we look over to the left again,"},{"Start":"01:55.865 ","End":"01:57.364","Text":"the 2 are parallel,"},{"Start":"01:57.364 ","End":"02:00.992","Text":"so they\u0027re in sync; there\u0027s no need to transpose or transform,"},{"Start":"02:00.992 ","End":"02:02.953","Text":"in fact, they\u0027re the same thing."},{"Start":"02:02.953 ","End":"02:07.250","Text":"Y^b=Y^a, and in this case, that equals 0."},{"Start":"02:07.250 ","End":"02:11.915","Text":"Of course, our X^b=negative V_0 t,"},{"Start":"02:11.915 ","End":"02:17.660","Text":"which is X^a plus R. Now that we have these 2 values,"},{"Start":"02:17.660 ","End":"02:21.995","Text":"we can find our R-value so R^b, if you recall,"},{"Start":"02:21.995 ","End":"02:30.840","Text":"equals the square root of (X^b)^2 plus (Y^b)^2."},{"Start":"02:30.840 ","End":"02:40.645","Text":"In our case, that equals V_0 t plus R. Now the last thing remaining is Theta B."},{"Start":"02:40.645 ","End":"02:46.525","Text":"We know that tangent of Theta equals y over x and so when we come over here,"},{"Start":"02:46.525 ","End":"02:48.970","Text":"we know that we\u0027re either going to get an answer of 0 or"},{"Start":"02:48.970 ","End":"02:52.060","Text":"180 because the bug is on the same axis,"},{"Start":"02:52.060 ","End":"02:57.160","Text":"the x-axis from observer B\u0027s perspective and from observer A\u0027s perspective."},{"Start":"02:57.160 ","End":"03:00.775","Text":"But what we need to do first is draw the new R vector,"},{"Start":"03:00.775 ","End":"03:02.620","Text":"that is the position vector."},{"Start":"03:02.620 ","End":"03:08.680","Text":"Now the position vector from observer B\u0027s perspective is something like this."},{"Start":"03:08.680 ","End":"03:15.440","Text":"That means that our bug is on the right of the origin as opposed to being on the left."},{"Start":"03:15.440 ","End":"03:18.470","Text":"Whereas from observer A\u0027s perspective,"},{"Start":"03:18.470 ","End":"03:20.825","Text":"the bug was at a 180-degree angle."},{"Start":"03:20.825 ","End":"03:22.715","Text":"From observer B\u0027s perspective,"},{"Start":"03:22.715 ","End":"03:24.596","Text":"the bug is at a 0-degree angle."},{"Start":"03:24.596 ","End":"03:30.430","Text":"Therefore, in this case, Theta B=0."},{"Start":"03:30.430 ","End":"03:33.770","Text":"I\u0027m not going to do the heavy lifting again to find"},{"Start":"03:33.770 ","End":"03:37.160","Text":"the velocity vectors and acceleration vectors for this."},{"Start":"03:37.160 ","End":"03:40.130","Text":"But now that you have X^b, Y^b, R^b,"},{"Start":"03:40.130 ","End":"03:43.010","Text":"and Theta b, it shouldn\u0027t be too difficult"},{"Start":"03:43.010 ","End":"03:46.830","Text":"to use the proper formulas and find your answers."}],"ID":9285},{"Watched":false,"Name":"Observer C","Duration":"2m 46s","ChapterTopicVideoID":8999,"CourseChapterTopicPlaylistID":5401,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.500 ","End":"00:03.705","Text":"Let\u0027s move on to observer C,"},{"Start":"00:03.705 ","End":"00:05.520","Text":"and with observer C,"},{"Start":"00:05.520 ","End":"00:07.455","Text":"just like with observer B."},{"Start":"00:07.455 ","End":"00:13.440","Text":"The first thing we need to do is establish the axes that it\u0027s using to observe."},{"Start":"00:13.440 ","End":"00:16.155","Text":"If we go back to our diagram on the left,"},{"Start":"00:16.155 ","End":"00:19.590","Text":"you\u0027ll remember that observer C does not move at all,"},{"Start":"00:19.590 ","End":"00:22.800","Text":"so it\u0027s axes don\u0027t move either."},{"Start":"00:22.800 ","End":"00:30.997","Text":"It\u0027s x-axis will still be at a 90 degree angle with respect to the ground,"},{"Start":"00:30.997 ","End":"00:35.445","Text":"and its y-axis will still be completely vertical."},{"Start":"00:35.445 ","End":"00:38.880","Text":"There are no obvious similarities between the axes"},{"Start":"00:38.880 ","End":"00:42.105","Text":"used by observer C and those used by observer B."},{"Start":"00:42.105 ","End":"00:46.110","Text":"But they do share an origin and that means that are R have to"},{"Start":"00:46.110 ","End":"00:50.325","Text":"be equal because R measures the distance from the origin."},{"Start":"00:50.325 ","End":"00:59.470","Text":"Therefore, r^c equals RB and that equals V_0t plus R,"},{"Start":"00:59.470 ","End":"01:03.095","Text":"that is capital R. In terms of measuring the angle Theta,"},{"Start":"01:03.095 ","End":"01:05.390","Text":"we\u0027re going to have to do something different."},{"Start":"01:05.390 ","End":"01:10.130","Text":"We know that the bug has rotated in relation to the origin,"},{"Start":"01:10.130 ","End":"01:15.065","Text":"and that C has not rotated with it and we know that that rotation is defined by Omega."},{"Start":"01:15.065 ","End":"01:19.715","Text":"In this case, the Omega is set as the constant Omega_0."},{"Start":"01:19.715 ","End":"01:24.945","Text":"We can say is that the rotational constant equals"},{"Start":"01:24.945 ","End":"01:31.325","Text":"Omega_0 and we also know that Theta dot equals Omega,"},{"Start":"01:31.325 ","End":"01:33.770","Text":"or in our case, Omega_0."},{"Start":"01:33.770 ","End":"01:40.175","Text":"Therefore, we can say that Theta equals Omega_0t."},{"Start":"01:40.175 ","End":"01:46.490","Text":"All we\u0027ve done is taken an integral of Theta dot and an integral of Omega_0."},{"Start":"01:46.490 ","End":"01:51.730","Text":"The result is that Theta equals Omega_0t."},{"Start":"01:51.730 ","End":"02:00.800","Text":"Now we found r^c and we found Theta C. If we want to find X^c or Y^c it\u0027s rather simple."},{"Start":"02:00.800 ","End":"02:03.290","Text":"We need to use the same equations for"},{"Start":"02:03.290 ","End":"02:06.455","Text":"transformation and transposition that we used before."},{"Start":"02:06.455 ","End":"02:13.085","Text":"If you recall, X^c equals R cosine Theta and if we plug in our new points,"},{"Start":"02:13.085 ","End":"02:21.865","Text":"we have X^c equals V_0t plus R cosine"},{"Start":"02:21.865 ","End":"02:28.215","Text":"Omega_0t and Y^C equals"},{"Start":"02:28.215 ","End":"02:37.290","Text":"V_0t plus R sine Omega_0t."},{"Start":"02:37.290 ","End":"02:39.990","Text":"Now, that you have r^c Theta C,"},{"Start":"02:39.990 ","End":"02:41.665","Text":"X^c and Y^c,"},{"Start":"02:41.665 ","End":"02:47.219","Text":"it should be easy to find your velocity and acceleration vectors."}],"ID":9286},{"Watched":false,"Name":"Observer D","Duration":"5m 22s","ChapterTopicVideoID":12196,"CourseChapterTopicPlaylistID":5401,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"Now let\u0027s move on to our last observer,"},{"Start":"00:02.730 ","End":"00:06.600","Text":"observer D. Now we need to clean some things up because in fact,"},{"Start":"00:06.600 ","End":"00:10.589","Text":"observer D stayed down here and did not move with the carousel."},{"Start":"00:10.589 ","End":"00:13.140","Text":"We\u0027re also going to change the color of"},{"Start":"00:13.140 ","End":"00:16.935","Text":"the c-axis so they\u0027re clear because we\u0027ll be using them soon."},{"Start":"00:16.935 ","End":"00:18.600","Text":"Now that we\u0027ve cleaned this up,"},{"Start":"00:18.600 ","End":"00:23.400","Text":"we can draw out the axes for the observer at the point D. These axes"},{"Start":"00:23.400 ","End":"00:25.770","Text":"should look just like they did it the first moment"},{"Start":"00:25.770 ","End":"00:28.470","Text":"because the observer at the point D does not move."},{"Start":"00:28.470 ","End":"00:36.015","Text":"You\u0027ll have a completely horizontal x-axis and a completely vertical y-axis."},{"Start":"00:36.015 ","End":"00:39.685","Text":"As you may have noticed with A and B,"},{"Start":"00:39.685 ","End":"00:44.405","Text":"C and D also have mirroring x and y-axis,"},{"Start":"00:44.405 ","End":"00:45.680","Text":"they\u0027re parallel to each other."},{"Start":"00:45.680 ","End":"00:50.360","Text":"The x-axis is point in the same direction and the y-axis is pointing the same direction."},{"Start":"00:50.360 ","End":"00:56.030","Text":"Now there seems to be no real relationship between the point D and the bug as it moves,"},{"Start":"00:56.030 ","End":"01:01.055","Text":"it\u0027s going in some spiraling motion that\u0027s really hard to define from this vantage point."},{"Start":"01:01.055 ","End":"01:03.715","Text":"However, using the parallelism,"},{"Start":"01:03.715 ","End":"01:05.205","Text":"with observer C,"},{"Start":"01:05.205 ","End":"01:10.210","Text":"we can do the same transposition that we did between observer A and observer B,"},{"Start":"01:10.210 ","End":"01:13.670","Text":"you could say that the axes for the observer at point D are"},{"Start":"01:13.670 ","End":"01:19.595","Text":"the same axes from the observer at point C moved over R distance,"},{"Start":"01:19.595 ","End":"01:23.560","Text":"the size or length of the radius of the circle."},{"Start":"01:23.560 ","End":"01:26.395","Text":"We can do is when we go down here,"},{"Start":"01:26.395 ","End":"01:29.435","Text":"and we\u0027re talking about observer D,"},{"Start":"01:29.435 ","End":"01:36.469","Text":"we can say that X^D equals"},{"Start":"01:36.469 ","End":"01:43.370","Text":"X^C plus or minus R. To verify whether it\u0027s plus or minus,"},{"Start":"01:43.370 ","End":"01:46.880","Text":"we\u0027re going to go back up and look at the relationship again."},{"Start":"01:46.880 ","End":"01:51.995","Text":"It seems that if the bug starts at point D,"},{"Start":"01:51.995 ","End":"01:56.670","Text":"and that means that X^D=0,"},{"Start":"01:56.670 ","End":"02:02.800","Text":"then X^C has to equal R. That would mean here that X^C=R,"},{"Start":"02:02.800 ","End":"02:05.160","Text":"R plus R=2R,"},{"Start":"02:05.160 ","End":"02:08.190","Text":"that doesn\u0027t work, so therefore it must be minus."},{"Start":"02:08.190 ","End":"02:14.220","Text":"We have X^D=X^C minus"},{"Start":"02:14.220 ","End":"02:20.600","Text":"R. Now I can plug in X of C and see what the equation gives me."},{"Start":"02:20.600 ","End":"02:26.450","Text":"Over here, we have the X(C)=V_O t plus R"},{"Start":"02:26.450 ","End":"02:32.174","Text":"cosine Omega_0 t. We have,"},{"Start":"02:32.174 ","End":"02:37.990","Text":"over here, X^D=v_0 t"},{"Start":"02:37.990 ","End":"02:46.965","Text":"plus r cosine Omega_0 t,"},{"Start":"02:46.965 ","End":"02:50.280","Text":"and we have to have minus R on the end."},{"Start":"02:50.280 ","End":"02:52.735","Text":"The same for y^D."},{"Start":"02:52.735 ","End":"03:00.390","Text":"We can look above and see that y^D=y^C."},{"Start":"03:03.530 ","End":"03:09.425","Text":"If y^D=y^C, then all we need to do is move to our right,"},{"Start":"03:09.425 ","End":"03:13.760","Text":"find our value for y^C right here,"},{"Start":"03:13.760 ","End":"03:15.620","Text":"and bring it over to our left."},{"Start":"03:15.620 ","End":"03:24.545","Text":"If y^D=y^C, then y^D=V_0"},{"Start":"03:24.545 ","End":"03:28.515","Text":"t plus R sine"},{"Start":"03:28.515 ","End":"03:34.815","Text":"Omega_0 t. Now that I have my x^D and my y^D,"},{"Start":"03:34.815 ","End":"03:36.675","Text":"I can find r^D."},{"Start":"03:36.675 ","End":"03:46.830","Text":"R^D equals the square root of (x^D)^2 plus (y^D)^2."},{"Start":"03:46.830 ","End":"03:49.275","Text":"I can also find Theta."},{"Start":"03:49.275 ","End":"03:58.780","Text":"We know that tangent of Theta^D=y^D over x^D."},{"Start":"03:59.840 ","End":"04:03.390","Text":"Lastly, I can find my r vector,"},{"Start":"04:03.390 ","End":"04:06.375","Text":"and the r vector of D, of course,"},{"Start":"04:06.375 ","End":"04:09.995","Text":"equals x, x hat plus y,"},{"Start":"04:09.995 ","End":"04:13.625","Text":"y hat, or equals r, r hat,"},{"Start":"04:13.625 ","End":"04:17.510","Text":"that is x and the direction of x plus y in the direction of y,"},{"Start":"04:17.510 ","End":"04:19.235","Text":"or perhaps more simply put,"},{"Start":"04:19.235 ","End":"04:24.575","Text":"r in the direction of r. Now we\u0027ve solved all the different parts of the problem."},{"Start":"04:24.575 ","End":"04:28.100","Text":"But what we\u0027ve really done is use transformations to go from"},{"Start":"04:28.100 ","End":"04:32.570","Text":"the point of view of the bug itself at point A over to point B,"},{"Start":"04:32.570 ","End":"04:35.870","Text":"then the point C and over to point D. That if"},{"Start":"04:35.870 ","End":"04:39.935","Text":"the original question only asked us something about the observer D,"},{"Start":"04:39.935 ","End":"04:45.245","Text":"say asked us about the force that the observer D observed."},{"Start":"04:45.245 ","End":"04:51.740","Text":"We could say that the force of D equals maD,"},{"Start":"04:51.740 ","End":"04:56.135","Text":"and do all these calculations to find the answer going from"},{"Start":"04:56.135 ","End":"05:01.400","Text":"A to B to C to D. One last note,"},{"Start":"05:01.400 ","End":"05:03.560","Text":"I think there\u0027s a little typo over here."},{"Start":"05:03.560 ","End":"05:07.700","Text":"This r should actually be lowercase, not uppercase."},{"Start":"05:07.700 ","End":"05:10.430","Text":"I assume that you understood that throughout the problem,"},{"Start":"05:10.430 ","End":"05:11.840","Text":"but I didn\u0027t want to be confusing."},{"Start":"05:11.840 ","End":"05:13.325","Text":"Now we\u0027ve done it."},{"Start":"05:13.325 ","End":"05:16.430","Text":"We\u0027ve solved for observers A, B,"},{"Start":"05:16.430 ","End":"05:20.075","Text":"C, and D. Now that we\u0027ve done that,"},{"Start":"05:20.075 ","End":"05:22.740","Text":"we can move on to the next lecture."}],"ID":12671}],"Thumbnail":null,"ID":5401}]

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