[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.08,0:00:01.59,Default,,0000,0000,0000,,- [Instructor] What we're\Ngoing to do in this video
Dialogue: 0,0:00:01.59,0:00:03.33,Default,,0000,0000,0000,,is look at the tangible example
Dialogue: 0,0:00:03.33,0:00:05.81,Default,,0000,0000,0000,,where we calculate angular velocity,
Dialogue: 0,0:00:05.81,0:00:08.32,Default,,0000,0000,0000,,but then we're going to\Nsee if we can connect that
Dialogue: 0,0:00:08.32,0:00:10.55,Default,,0000,0000,0000,,to the notion of speed.
Dialogue: 0,0:00:10.55,0:00:13.08,Default,,0000,0000,0000,,So let's start with this\Nexample where once again
Dialogue: 0,0:00:13.08,0:00:15.61,Default,,0000,0000,0000,,we have some type of a ball tethered
Dialogue: 0,0:00:15.61,0:00:18.69,Default,,0000,0000,0000,,to some type of center of\Nrotation right over here.
Dialogue: 0,0:00:18.69,0:00:21.16,Default,,0000,0000,0000,,Let's say this is connected with a string.
Dialogue: 0,0:00:21.16,0:00:22.78,Default,,0000,0000,0000,,And so if you were to\Nmove the ball around,
Dialogue: 0,0:00:22.78,0:00:27.36,Default,,0000,0000,0000,,it would move along this blue\Ncircle in either direction.
Dialogue: 0,0:00:27.36,0:00:28.96,Default,,0000,0000,0000,,Let's just say for the sake of argument,
Dialogue: 0,0:00:28.96,0:00:32.15,Default,,0000,0000,0000,,the length of the string is seven meters.
Dialogue: 0,0:00:32.15,0:00:36.07,Default,,0000,0000,0000,,We know that at time is\Nequal to three seconds,
Dialogue: 0,0:00:37.22,0:00:39.06,Default,,0000,0000,0000,,our angle is equal to,
Dialogue: 0,0:00:41.23,0:00:44.87,Default,,0000,0000,0000,,theta is equal to pi over two radians,
Dialogue: 0,0:00:44.87,0:00:46.29,Default,,0000,0000,0000,,which we've seen in previous videos.
Dialogue: 0,0:00:46.29,0:00:51.06,Default,,0000,0000,0000,,We would measure from the\Npositive x-axis, just like that.
Dialogue: 0,0:00:51.06,0:00:55.89,Default,,0000,0000,0000,,And let's say that at T\Nis equal to six seconds,
Dialogue: 0,0:00:55.89,0:01:00.06,Default,,0000,0000,0000,,T is equal to six seconds,\Ntheta is equal to pi radians.
Dialogue: 0,0:01:01.63,0:01:05.80,Default,,0000,0000,0000,,And so after three seconds, the\Nball is now right over here.
Dialogue: 0,0:01:08.27,0:01:12.16,Default,,0000,0000,0000,,And so if we wanted to actually\Nvisualize how that happens,
Dialogue: 0,0:01:12.16,0:01:15.89,Default,,0000,0000,0000,,let me see if I can rotate\Nthis ball in three seconds.
Dialogue: 0,0:01:15.89,0:01:17.57,Default,,0000,0000,0000,,So it would look like this.
Dialogue: 0,0:01:17.57,0:01:20.90,Default,,0000,0000,0000,,One Mississippi, two\NMississippi, three Mississippi.
Dialogue: 0,0:01:20.90,0:01:22.26,Default,,0000,0000,0000,,Let's do that again.
Dialogue: 0,0:01:22.26,0:01:23.43,Default,,0000,0000,0000,,It would be...
Dialogue: 0,0:01:24.50,0:01:28.68,Default,,0000,0000,0000,,One Mississippi, two\NMississippi, three Mississippi.
Dialogue: 0,0:01:28.68,0:01:30.82,Default,,0000,0000,0000,,So now that we can visualize
Dialogue: 0,0:01:30.82,0:01:33.18,Default,,0000,0000,0000,,or conceptualize what's going on,
Dialogue: 0,0:01:33.18,0:01:37.02,Default,,0000,0000,0000,,see if you can pause this\Nvideo and calculate two things.
Dialogue: 0,0:01:37.02,0:01:40.01,Default,,0000,0000,0000,,So the first thing that I\Nwant you to calculate is
Dialogue: 0,0:01:40.01,0:01:43.42,Default,,0000,0000,0000,,what is the angular velocity of the ball?
Dialogue: 0,0:01:44.32,0:01:45.39,Default,,0000,0000,0000,,And actually, it would be the ball
Dialogue: 0,0:01:45.39,0:01:47.37,Default,,0000,0000,0000,,and every point on that string.
Dialogue: 0,0:01:47.37,0:01:51.44,Default,,0000,0000,0000,,What is that angular velocity\Nwhich we denote with omega?
Dialogue: 0,0:01:51.44,0:01:53.20,Default,,0000,0000,0000,,And then I want you to figure out
Dialogue: 0,0:01:53.20,0:01:56.32,Default,,0000,0000,0000,,what is the speed of the ball?
Dialogue: 0,0:01:56.32,0:01:58.39,Default,,0000,0000,0000,,So what is the speed?
Dialogue: 0,0:01:58.39,0:02:00.41,Default,,0000,0000,0000,,See if you can figure\Nout both of those things,
Dialogue: 0,0:02:00.41,0:02:03.00,Default,,0000,0000,0000,,and for extra points,\Nsee if you can figure out
Dialogue: 0,0:02:03.00,0:02:05.56,Default,,0000,0000,0000,,a relationship between the two.
Dialogue: 0,0:02:05.56,0:02:07.91,Default,,0000,0000,0000,,Alright, well let's go\Nangular velocity first.
Dialogue: 0,0:02:07.91,0:02:09.90,Default,,0000,0000,0000,,I'm assuming you've had a go at it.
Dialogue: 0,0:02:09.90,0:02:12.74,Default,,0000,0000,0000,,Angular velocity, you might remember
Dialogue: 0,0:02:12.74,0:02:16.97,Default,,0000,0000,0000,,is just going to be equal\Nto our angular displacement,
Dialogue: 0,0:02:16.97,0:02:19.42,Default,,0000,0000,0000,,which we could say is delta theta,
Dialogue: 0,0:02:19.42,0:02:21.89,Default,,0000,0000,0000,,and it is a vector quantity.
Dialogue: 0,0:02:21.89,0:02:26.06,Default,,0000,0000,0000,,And we are going to divide\Nthat by our change in time.
Dialogue: 0,0:02:27.94,0:02:29.58,Default,,0000,0000,0000,,So delta T.
Dialogue: 0,0:02:29.58,0:02:30.87,Default,,0000,0000,0000,,And so what is this going to be?
Dialogue: 0,0:02:30.87,0:02:34.10,Default,,0000,0000,0000,,Well this is going to be\Nour angular displacement.
Dialogue: 0,0:02:34.10,0:02:35.94,Default,,0000,0000,0000,,Our final angle is pi.
Dialogue: 0,0:02:36.98,0:02:41.82,Default,,0000,0000,0000,,Pi radians minus our initial\Nangle, pi over two radians.
Dialogue: 0,0:02:41.82,0:02:45.02,Default,,0000,0000,0000,,And then all of that's going\Nto be over our change in time,
Dialogue: 0,0:02:45.02,0:02:47.53,Default,,0000,0000,0000,,which is six seconds,\Nwhich is our final time
Dialogue: 0,0:02:47.53,0:02:49.98,Default,,0000,0000,0000,,minus our initial time,\Nminus three seconds.
Dialogue: 0,0:02:49.98,0:02:53.08,Default,,0000,0000,0000,,And so we are going to\Nget in the numerator,
Dialogue: 0,0:02:53.08,0:02:56.14,Default,,0000,0000,0000,,we have been rotated in\Nthe positive direction,
Dialogue: 0,0:02:56.14,0:02:58.07,Default,,0000,0000,0000,,Pi over two radians.
Dialogue: 0,0:02:58.07,0:03:01.63,Default,,0000,0000,0000,,Because it's positive, we\Nknow it's counterclockwise.
Dialogue: 0,0:03:01.63,0:03:04.71,Default,,0000,0000,0000,,And that happened over three seconds.
Dialogue: 0,0:03:05.56,0:03:08.20,Default,,0000,0000,0000,,And so we could rewrite this as
Dialogue: 0,0:03:08.20,0:03:11.46,Default,,0000,0000,0000,,this is going to be equal to pi over six.
Dialogue: 0,0:03:11.46,0:03:13.74,Default,,0000,0000,0000,,And let's remind\Nourselves about the units.
Dialogue: 0,0:03:13.74,0:03:17.16,Default,,0000,0000,0000,,Our change in angle, that's\Ngoing to be in radians,
Dialogue: 0,0:03:17.16,0:03:20.100,Default,,0000,0000,0000,,and then that is going to be per second.
Dialogue: 0,0:03:20.100,0:03:23.60,Default,,0000,0000,0000,,So we're going pi over\Nsix radians per second,
Dialogue: 0,0:03:23.60,0:03:25.40,Default,,0000,0000,0000,,and if you do that over three seconds,
Dialogue: 0,0:03:25.40,0:03:29.48,Default,,0000,0000,0000,,well then you're going to\Ngo pi over two radians.
Dialogue: 0,0:03:31.04,0:03:32.93,Default,,0000,0000,0000,,Now with that out of the way,
Dialogue: 0,0:03:32.93,0:03:34.70,Default,,0000,0000,0000,,let's see if we can calculate speed.
Dialogue: 0,0:03:34.70,0:03:36.96,Default,,0000,0000,0000,,If you haven't done so\Nalready, pause this video
Dialogue: 0,0:03:36.96,0:03:39.03,Default,,0000,0000,0000,,and see if you can calculate it.
Dialogue: 0,0:03:39.03,0:03:43.00,Default,,0000,0000,0000,,Well speed is going to\Nbe equal to the distance
Dialogue: 0,0:03:43.00,0:03:45.69,Default,,0000,0000,0000,,the ball travels, and we've\Ntouched on that in other videos.
Dialogue: 0,0:03:45.69,0:03:48.26,Default,,0000,0000,0000,,I encourage you to watch\Nthose if you haven't already.
Dialogue: 0,0:03:48.26,0:03:52.82,Default,,0000,0000,0000,,The distance that we travel\Nwe could denote with S.
Dialogue: 0,0:03:52.82,0:03:55.06,Default,,0000,0000,0000,,S is sometimes used to denote arc length,
Dialogue: 0,0:03:55.06,0:03:57.34,Default,,0000,0000,0000,,or the distance traveled right over here.
Dialogue: 0,0:03:57.34,0:03:59.84,Default,,0000,0000,0000,,So the speed is going to be our arc length
Dialogue: 0,0:03:59.84,0:04:02.84,Default,,0000,0000,0000,,divided by our change in time.
Dialogue: 0,0:04:02.84,0:04:05.45,Default,,0000,0000,0000,,Divided by our change in time.
Dialogue: 0,0:04:05.45,0:04:08.15,Default,,0000,0000,0000,,But what is our arc length going to be?
Dialogue: 0,0:04:08.15,0:04:09.90,Default,,0000,0000,0000,,Well we saw in a previous video
Dialogue: 0,0:04:09.90,0:04:13.25,Default,,0000,0000,0000,,where we related angular\Ndisplacement to arc length,
Dialogue: 0,0:04:13.25,0:04:16.14,Default,,0000,0000,0000,,or distance, that our arc\Nlength is nothing more
Dialogue: 0,0:04:16.14,0:04:20.31,Default,,0000,0000,0000,,than the absolute value of\Nour angular displacement,
Dialogue: 0,0:04:23.78,0:04:27.58,Default,,0000,0000,0000,,of our angular displacement,\Ntimes the radius.
Dialogue: 0,0:04:27.58,0:04:29.12,Default,,0000,0000,0000,,Times the radius.
Dialogue: 0,0:04:29.12,0:04:31.81,Default,,0000,0000,0000,,And in this case, our radius\Nwould be seven meters.
Dialogue: 0,0:04:31.81,0:04:34.76,Default,,0000,0000,0000,,So if we substitute all of this up here,
Dialogue: 0,0:04:34.76,0:04:36.31,Default,,0000,0000,0000,,what are we going to get?
Dialogue: 0,0:04:36.31,0:04:39.23,Default,,0000,0000,0000,,We are going to get that our speed,
Dialogue: 0,0:04:40.20,0:04:42.46,Default,,0000,0000,0000,,I'm writing it out because\NI don't want to overuse,
Dialogue: 0,0:04:42.46,0:04:43.66,Default,,0000,0000,0000,,well I am overusing S,
Dialogue: 0,0:04:43.66,0:04:45.26,Default,,0000,0000,0000,,but I don't want people to get confused.
Dialogue: 0,0:04:45.26,0:04:48.15,Default,,0000,0000,0000,,Our speed is going to be equal\Nto the distance we travel,
Dialogue: 0,0:04:48.15,0:04:50.79,Default,,0000,0000,0000,,which we just wrote is the magnitude
Dialogue: 0,0:04:50.79,0:04:52.18,Default,,0000,0000,0000,,of our angular displacement.
Dialogue: 0,0:04:52.18,0:04:53.55,Default,,0000,0000,0000,,And this is all fancy notation,
Dialogue: 0,0:04:53.55,0:04:56.43,Default,,0000,0000,0000,,but when you actually apply it,\Nit's pretty straightforward.
Dialogue: 0,0:04:56.43,0:04:59.02,Default,,0000,0000,0000,,Times the radius of the circle.
Dialogue: 0,0:04:59.89,0:05:03.92,Default,,0000,0000,0000,,I guess you can say we\Nare traveling along.
Dialogue: 0,0:05:03.92,0:05:06.14,Default,,0000,0000,0000,,Let me write that in a different color.
Dialogue: 0,0:05:06.14,0:05:08.63,Default,,0000,0000,0000,,So times the radius.
Dialogue: 0,0:05:08.63,0:05:11.63,Default,,0000,0000,0000,,All of that over our change in time.
Dialogue: 0,0:05:12.66,0:05:15.07,Default,,0000,0000,0000,,All of that over our change in time.
Dialogue: 0,0:05:15.07,0:05:17.48,Default,,0000,0000,0000,,Well we could put in the\Nnumbers right over here.
Dialogue: 0,0:05:17.48,0:05:21.03,Default,,0000,0000,0000,,We know that this is\Ngoing to be pi over two.
Dialogue: 0,0:05:21.03,0:05:22.21,Default,,0000,0000,0000,,You take the absolute value of that.
Dialogue: 0,0:05:22.21,0:05:23.98,Default,,0000,0000,0000,,It's still going to be pi over two.
Dialogue: 0,0:05:23.98,0:05:26.41,Default,,0000,0000,0000,,We know that our radius in this case
Dialogue: 0,0:05:26.41,0:05:27.76,Default,,0000,0000,0000,,is the length of that string.
Dialogue: 0,0:05:27.76,0:05:29.29,Default,,0000,0000,0000,,It is seven meters.
Dialogue: 0,0:05:29.29,0:05:31.63,Default,,0000,0000,0000,,And we know that our change in time here,
Dialogue: 0,0:05:31.63,0:05:34.84,Default,,0000,0000,0000,,we know that this over here\Nis going to be three seconds,
Dialogue: 0,0:05:34.84,0:05:36.68,Default,,0000,0000,0000,,and we can calculate everything.
Dialogue: 0,0:05:36.68,0:05:39.66,Default,,0000,0000,0000,,But what's even more\Ninteresting is to recognize
Dialogue: 0,0:05:39.66,0:05:42.50,Default,,0000,0000,0000,,that what is this right over here?
Dialogue: 0,0:05:45.80,0:05:48.50,Default,,0000,0000,0000,,What is the absolute value\Nof our angular displacement
Dialogue: 0,0:05:48.50,0:05:50.17,Default,,0000,0000,0000,,over change in time?
Dialogue: 0,0:05:51.06,0:05:53.07,Default,,0000,0000,0000,,Well, this is just the absolute value
Dialogue: 0,0:05:53.07,0:05:55.57,Default,,0000,0000,0000,,of our angular velocity.
Dialogue: 0,0:05:55.57,0:05:57.74,Default,,0000,0000,0000,,So we could say that speed
Dialogue: 0,0:05:59.92,0:06:04.72,Default,,0000,0000,0000,,is equal to the absolute\Nvalue of our angular velocity.
Dialogue: 0,0:06:04.72,0:06:09.60,Default,,0000,0000,0000,,Absolute value of our angular\Nvelocity, times our radius.
Dialogue: 0,0:06:09.60,0:06:11.02,Default,,0000,0000,0000,,Times our radius.
Dialogue: 0,0:06:12.30,0:06:14.43,Default,,0000,0000,0000,,And now so this is super useful.
Dialogue: 0,0:06:14.43,0:06:19.28,Default,,0000,0000,0000,,Our speed in this case,\Nis going to be pi over six
Dialogue: 0,0:06:19.28,0:06:20.86,Default,,0000,0000,0000,,radians per second.
Dialogue: 0,0:06:22.37,0:06:23.29,Default,,0000,0000,0000,,So pi over,
Dialogue: 0,0:06:25.97,0:06:26.97,Default,,0000,0000,0000,,pi over six.
Dialogue: 0,0:06:28.27,0:06:30.15,Default,,0000,0000,0000,,Times the radius.
Dialogue: 0,0:06:30.15,0:06:32.40,Default,,0000,0000,0000,,Times seven meters.
Dialogue: 0,0:06:32.40,0:06:34.47,Default,,0000,0000,0000,,Times seven meters.
Dialogue: 0,0:06:34.47,0:06:36.30,Default,,0000,0000,0000,,And so what do we get?
Dialogue: 0,0:06:36.30,0:06:40.39,Default,,0000,0000,0000,,We are going to get seven\Npi over six meters per,
Dialogue: 0,0:06:43.76,0:06:47.93,Default,,0000,0000,0000,,meters per second, which will\Nbe our units for speed here.
Dialogue: 0,0:06:48.83,0:06:50.94,Default,,0000,0000,0000,,And the reason why we're\Ndoing the absolute value
Dialogue: 0,0:06:50.94,0:06:53.61,Default,,0000,0000,0000,,is 'cause remember, speed\Nis a scalar quantity.
Dialogue: 0,0:06:53.61,0:06:55.49,Default,,0000,0000,0000,,We're not specifying the direction.
Dialogue: 0,0:06:55.49,0:06:56.85,Default,,0000,0000,0000,,In fact, the whole time we're traveling,
Dialogue: 0,0:06:56.85,0:06:59.22,Default,,0000,0000,0000,,our direction is constantly changing.
Dialogue: 0,0:06:59.22,0:07:00.41,Default,,0000,0000,0000,,So there you have it.
Dialogue: 0,0:07:00.41,0:07:03.49,Default,,0000,0000,0000,,There's multiple ways to approach\Nthese types of questions,
Dialogue: 0,0:07:03.49,0:07:05.48,Default,,0000,0000,0000,,but the big takeaway here is one,
Dialogue: 0,0:07:05.48,0:07:07.68,Default,,0000,0000,0000,,how we calculated angular velocity,
Dialogue: 0,0:07:07.68,0:07:12.59,Default,,0000,0000,0000,,and then how we can relate\Nangular velocity to speed.
Dialogue: 0,0:07:12.59,0:07:15.85,Default,,0000,0000,0000,,And what's nice is there's a\Nnice, simple formula for it.
Dialogue: 0,0:07:15.85,0:07:18.00,Default,,0000,0000,0000,,And all of this just came out of something
Dialogue: 0,0:07:18.00,0:07:20.08,Default,,0000,0000,0000,,that relates to what we\Nlearned in seventh grade
Dialogue: 0,0:07:20.08,0:07:21.81,Default,,0000,0000,0000,,around the circumference of the circle,
Dialogue: 0,0:07:21.81,0:07:23.84,Default,,0000,0000,0000,,which we touched on in the video
Dialogue: 0,0:07:23.84,0:07:27.68,Default,,0000,0000,0000,,relating angular\Ndisplacement to arc length,
Dialogue: 0,0:07:27.68,0:07:29.42,Default,,0000,0000,0000,,or distance traveled.