Bam, Mr. Tarrou. We are going to look at some application problems in this video. Uh, those
applications being linear speed, angular speed, and arc length. Before it started though,
I am going to do four examples in this video, let me explain how radian measures make some
equations easier work with. Now arc length in geometry when you’re studying... Or using
degrees is the entire circumference of the circle 2 pi r times what percent of the central
angle inside the circle you’re dealing with, or how much rotation are you dealing with
over 360. So 90° would be one fourth of the circle. So the arc length would be one fourth
of the entire circumference. If I take out 360° and replace it with two pi, the radian
equivalent, look what happens the two pi’s cancel out and you get a much simpler arc
length formula of arc length is r time theta. Arc length S equals R times theta. Again with
two pi replacing 360° most of your equation appears to cancel away giving you a simpler
version. Instead of a bunch of multiplication and divisions you just have to numbers being
multiplied together to find arc length. So this is just one example of how working with
radians can make life a little bit easier compared to working with degrees. Linear speed
and Angular speed. Let’s take my arm like a spoke or propeller blade. You can also be
thinking about watching the Rose Bowl Parade where there’s a big long line of people
in the band. I am going to rotate 90° and point my arm right to the camera. Ready, 1,2,
3 so in three seconds my entire body made the same 90 degree movement. Every point in
my body and along my arm rotated 90° or pi over 2 radians in three seconds. However the
linear speed is much different. This point right here on my chest has a linear velocity
of basically zero, if I can put it in the center of rotation. However my fingertips
are moving quite fast to get my hand to travel the distance traveled by rotating 90 degrees
or pi over two radians in three seconds. Think of it as.. I always keep my keys in lanyard
if, I were to spin them my keys at the end of the lanyard going very very quickly would
start to whistle if I could spin them fast enough. The linear speed will pick up as you
pull away from the center rotation. So we have this standard formula in our textbook
of V is equal to S times T. Now S stands for arc length, so depending on how the numbers
are given to you in a text book you might see this as r theta divided by T or you might
want to think of it that way depending on the variables are given to you. However in
another form of linear speed that my new textbook doesn’t emphasize very much points out that
theta over t is angle velocity. Another form of this equation is R, the radius, times W,
the angular speed. This really emphasizes that the linear velocity is going to pick
up as the radius increases away from the center rotation. However the angle speed is not affected
by the radius. You don’t even see that variable r in the equation. Again, the radius length
is nowhere in that equation. So let’s start with an easy question. I want the Arc length
of a circle with a radius equal to 10 and a central angle measure of 225°. That diagram
is going to be a circle with a radius of 10 and you have marked of a rotation of 225°.
How much is this arc length here? Well arc length needs to be done, at least in its simplest
form in radians. So, s is equal to ten times... Well, I have got 225 degrees and if I know
my unit circle I know that is five pi over four. But if you are not sure about that,
we can take this and multiply it by pi over 180 and just get that conversion done. So
this is going to come S is equal to 10 times 5 pi over 4. Which is going to equal, and
I’m going to cheat and look over there at my decimal answer, 50 time pi over four ( know
that reduces to 25 pi over 2 ) but as a decimal that is 39.3 units. Whatever units I have
in my problem. I just had 10 units could be 10 feet and miles 10 inches whatever. 39.3
units would be the length of the arc. Okay so that’s how you find arc length, it is
just S equals R times theta. Linear speed, let’s talk about the tip of the minute hand.
That is 8 cm in length you know the long hand o’clock the minute hand, the one that goes
run a little bit quicker than a shorter minute hand. What would be.. Let’s do both. What
would be the angle speed of the minute hand? How long does it take for the minute hand
to go all the way around the clock? Well from 12 1,2,3,4,5,6,7,8,9,0,11, and back to 12
that full rotation of 2pi , takes a total of, well it’s the minute hand, so it goes
all the way around every hour or sixty 60 minutes. So the angular speed of the minute
hand, we’re to do this by minutes and not by hour, is pi over 30 radians per minute.
Yes okay spacing out there little bit. Now what’s the linear speed of the minute hand?
That minutes and it takes an hour to go around the clock. Linear speed can be found by taking
the arc length and dividing by time, r times theta divided by time, or if you’ve already
found the angular speed r times w. So the minute hand has a radius of eight and every
minute that minute hand is moving pi over 30 radians. So eight, the radius, times W
the angular velocity. And a lot of times with linear velocity I do prefer to have the angular
velocity first. That is going to be eight pie over 30 or, let’s see divided by two
or for pie over 15. Now this is a linear velocity it’s how far is it is moved. So, what is
going to be four pi times 15 centimeters per minutes, not radians per minute but centimeters
per minute. So rotation or distance, rotation is what angular velocity is about and a distance
over time is what’s linear speed is about. Alright, so that is angle leader velocity
and linear velocity for the tip of the minutes hand on a clock. Okay, let’s say that we
are talking about the spokes of the wheel like a bicycle wheel. Or maybe the car with
spoke wheels and we have a wheel with a diameter of 26 inches spinning at 200 revolutions per
minute or RPM. So I’m going to the same thing where we find the angle velocity first
and then we find linear velocity. So we have this wheel and it doesn’t matter how big
or small it is the wheel is turning 200 revolutions per minute. Well angler velocity is not about
revolutions per minute, is about an angle measure and usually we’re talking about
radians over a given per period of time. So each of those full revolutions, the 200 revolutions,
each one of those revolutions in radians is going to be two pie or 360°. I am doing using
radians since we’re just learning those and we are very comfortable with them. So
200 times to pi is 400 pi radians... Well I forgot about the time period. It’s 200
revolutions per minute so if I keep it in terms of the minute we’ll have 400 pi radians
per minute. Or, if I like I am going to break this down into seconds and say that it is
this many radians per second and there is 60 seconds in a minute. So... break that down
and.... This brings up another topic and you can… we can cancel out the zeros and reduce
this answer before I talk too much and make a mistake. Now you can set up your angular
velocity it is going to be about the time period, what the time period that is wanted
in the problem. Do you want it angle speed per hour, per minute, per second, that is
going to change your answer. So make sure that you....now this not completely off the
top of my head but I sort of just slapped this down here and did not put anything in
the problem about what time period I wanted to set these up with. So we are going to do
per second. So we have an angle speed that is in seconds based on the fact that we are
rotating 200 times, each 200 times being 2 pi radians and it is given per minute. But
we broke the time down to seconds, so instead of dividing those by 1 for a minute, we divided
by 60 for 60 seconds. Ok, so we have 20 pi over 3 radians per second. Lets find out how
far a tip of a spoke on that wheel is going to travel over that one second period. Well,
linear speed if we already have angular speed is r, the radius. If the diameter is twenty-six
inches the radius is going to be thirteen inches. So we have r w and we have w already
worked out. So it is the radius, half the diameter, times the angular speed of 20 pi
over 3 radians per second. So we are going to have a linear speed, how many inches that
spoke on the outside of the wheel travels in a one second period. So that is going to
be...uhmm... Did I work that out in decimal form? Yep! So thirteen times twenty divided
by three and then multiplied by pi comes out to be 272.3 inches per second. Which is a
pretty good distance over a one second period, but then 200 revolutions in a minute is pretty
quick. Angular and Linear Speed of a wheel. And again remember angular speed, here is
pi over two radians or ninety degrees in a given period of time. My whole body is turning
at that same rate, but my actual finger tip is moving much farther than my chest. The
tip of my fingers are going to have a higher linear speed the farther away from the center
of rotation my finger tips get. Ok... One more example if I have time which I don't
so I won't. I only have two minutes left. So that is a couple of examples of setting
up Linear Speed, Angular Speed, and Arc Length. Thank you very much. BAM!!! Go Do Your Homework.