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### Course: AP®︎/College Physics 1 > Unit 4

Lesson 1: Uniform circular motion introduction- Angular motion variables
- Distance or arc length from angular displacement
- Angular velocity and speed
- Connecting period and frequency to angular velocity
- Radius comparison from velocity and angular velocity: Worked example
- Linear velocity comparison from radius and angular velocity: Worked example
- Change in period and frequency from change in angular velocity: Worked examples
- Circular motion basics: Angular velocity, period, and frequency
- Uniform circular motion and centripetal acceleration review

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# Angular velocity and speed

In uniform circular motion, angular velocity (𝒘) is a vector quantity and is equal to the angular displacement (Δ𝚹, a vector quantity) divided by the change in time (Δ𝐭). Speed is equal to the arc length traveled (S) divided by the change in time (Δ𝐭), which is also equal to |𝒘|R. And arc length (S) is equal to the absolute value of the angular displacement (|Δ𝚹|) times the radius (R).

## Want to join the conversation?

- Are speed and linear velocity the same?(7 votes)
- Kind of. Speed is a scalar quantity and velocity is a vector quantity so it has direction and magnitude.(1 vote)

- Can't we also use the Speed=Distance/Time formula where the distance is 1/4th the circumference of the circle to find out the speed?(6 votes)
- Absolutely! The arclength formula is actually derived from multiplying the portion of the circle considered (in this case, 1/2pi rad out of 2pi rad, or 90' out of 360') by 2pi(radius). From this formula (2piR times portion of circle), you get a simplified formula for arclength: S = r|delta theta| For a detailed explanation, see arclength from angular displacement video.

Anyway, finding speed in this video, you can use that arclength formula divided by time to find distance travelled over time -- speed.

Note that r(delta theta)/t equals r multiplied by (delta theta)/t, which is the same thing as angular velocity (w). So we conclude that v = wr

To state clearly the answer to your question, you absolutely do use distance/time by the circumference of the circle in the way arclength is calculated! Sal just draws another relationship between angular velocity and speed.(2 votes)

- So at6:40, Sal expresses the speed in meters/second, when he expresses the angular velocity in radians/second. The only thing that was different about the speed and angular velocity was that the magnitude of the angular velocity was multiplied by a scalar with the "meters" unit. Therefore, following algebraic logic, the unit for speed should be meters*radians/second.

Where did the "radian" part go? Does it not count as an actual unit? And, if it doesn't, couldn't you replace the "radian" part with 1 so that you would have 1/second=1 Hertz? Hertz would be a much easier unit to use in cases like these.

Please tell me whether or not I am just confusing myself for no reason. Thanks!(6 votes)- pi radian = R pi meter. Both radian and meter are units for length. Radian is only a more convinient way of describing circular distance.

In this case, pi radian=7pi. By multiplying pi/2 by 7 (R=7), he automatically converts the unit from radian to meter.(2 votes)

- Since this is the average angular velocity how do we calculate the instantaneous angular velocity?(3 votes)
- if the object has a constant angular acceleration, the instantaneous angular velocity at any time "t" is just the initial angular velocity plus(angular acceleration*t). sorry or bad writing though(3 votes)

- what i understand is that for calculating Angular velocity we take the difference of angles and divide it over the time.but what if the ball moves back and forth.in this case speed would be different from Angular velocity.what i'm missing?(2 votes)
- Remember that velocity is defined as the change of position over the change of time.

To find angular velocity for a ball moving back and forth, you will have to find the beginning angle and the final angle at where the ball stops moving. Then find the difference between the two angles and divide by time.

To find the speed for a ball moving back and forth, you will have to find the total distance the ball moved. And don't forget to avoid a common misconception. For example, if the ball moved 90 degrees and then -90 degrees, the total distance is NOT ZERO. The displacement equals zero, but not the total distance.(2 votes)

- I have a question about how to solve a problem like this:

An object with an initial velocity of 0.12 rad/s (0.12 radians per second) accelerates at 0.11 rad/s^2 (0.11 radians per second squared) over a distance of 0.25 radians. What is the final angular velocity (ω) of the object in r/s (radians per second)?

I know that to solve for angular velocity, you can use this equation:

ω = ωₒ + α * t

where ωₒ = initial velocity, α = angular acceleration, and t = time.

However, I was not able to substitute everything needed to solve for ω:

ω = 0.12 rad/s + (0.11 rad/s^2 * t)

I wasn't given a time. Originally I figured, "Oh, well no big deal. I can probably just do something with the numbers to solve for time, or maybe look through all my equations and then solve one for t..."

I was wrong (at least to my knowledge).

I had no equations where I could solve for t, because I had multiple unknowns...

Eventually I started using intuition rather than solid numbers & equations, and I tried thinking about graphs. I figured that if the initial velocity was 0.12 rad/s, t at that point would equal 0, and that before the origin it was a constant line (with a y coordinate of 0.12). Once it got past 0, however, there would be an (angular) acceleration of 0.11 rad/s^2, so it would be a parabola shape.

It helped me to think of it this way (ignore the part where it's a really, really fast car and this car would have destroyed itself quite quickly):

You have a car going at a constant velocity of 12 km/s (ωₒ), and once you pass a white line across the ground, t is now > 0 and you start accelerating 11 km/s (α) until you have driven exactly 25 km (x) past the white line. I figured that if you started at 12 km/s (ωₒ) and started accelerating for 25 km, at roughly the same speed, it's kind of like doubling it, right? You were adding 11 km/s every second, and so after 1 second, you were going 23 km/s, and so t would have had to have been just over one second to reach the 25 km mark. Of course, I didn't know*exactly*how much time had elapsed; I just knew it was a bit more than a second.

So I just started entering numbers into the computer, starting at 0.24, going up one hundredth each time. I had only gotten to 0.25 when it said I was correct.

"Aha! I...*guessed*the right answer, how the heck are you supposed to calculate t**without**guessing?"

I put 0.25 in my original angular velocity equation:

0.25 rad/s = 0.12 rad/s + (0.11 rad/s^2 * t)

and then solved for t this way:

t ≅1.2 seconds

I'm glad I finished the problem after agonizing over it for hours, but I am still not satisfied with the issue: How are you supposed to calculate t when given initial velocity, angular acceleration, and distance? Maybe I am just missing an equation from my equations sheet, but this was (and is) a very frustrating problem... help is warmly welcomed.(2 votes)- i suggest you to use the eqn

(final angular velocity)^2=(initial angular velocity)^2+2(angular acceleration)(angular displacement)(2 votes)

- What is the difference between Δ𝚹 and |Δ𝚹| ?(1 vote)
- |Δ𝚹| means no matter what the sign of Δ𝚹 is, the amount is positive, as indicated by | |.(2 votes)

- Let's say a ball with a radius of .11 m is rolling at 5m/s. To find angular velociy, you multiply by the radius?? Why? What are the new units?(1 vote)
- well, speed=abs(angular velocity)*r. and if you want to find the angular velocity. you would just divide the speed by the radius.the direction depends on the fact of being clockwise or not. if its clockwise, its negative and if its counter clockwise its positive.and the units are radians/sec.and its because that's how we derived the formula.(1 vote)

- How do we find constant velocity if we do not have the radius, but we have change in time, acceleration, and angular displacement?(1 vote)
- i suggest you watch the "rotational kinematic formula" video in the APphysics 1( torque and angular momentum)play list(1 vote)

- how to find angular velocity about a point on the circumference? how is it different from angular velocity about center of rotation?(1 vote)
- I think you're talking about the ball's velocity around the circumference, right?

If yes, then it's the same thing as finding the speed in the "Angular velocity and speed" video and putting a "+" or "-" to indicate direction.(1 vote)

## Video transcript

- [Instructor] What we're
going to do in this video is look at the tangible example where we calculate angular velocity, but then we're going to
see if we can connect that to the notion of speed. So let's start with this
example where once again we have some type of a ball tethered to some type of center of
rotation right over here. Let's say this is connected with a string. And so if you were to
move the ball around, it would move along this blue
circle in either direction. Let's just say for the sake of argument, the length of the string is seven meters. We know that at time is
equal to three seconds, our angle is equal to, theta is equal to pi over two radians, which we've seen in previous videos. We would measure from the
positive x-axis, just like that. And let's say that at T
is equal to six seconds, T is equal to six seconds,
theta is equal to pi radians. And so after three seconds, the
ball is now right over here. And so if we wanted to actually
visualize how that happens, let me see if I can rotate
this ball in three seconds. So it would look like this. One Mississippi, two
Mississippi, three Mississippi. Let's do that again. It would be... One Mississippi, two
Mississippi, three Mississippi. So now that we can visualize or conceptualize what's going on, see if you can pause this
video and calculate two things. So the first thing that I
want you to calculate is what is the angular velocity of the ball? And actually, it would be the ball and every point on that string. What is that angular velocity
which we denote with omega? And then I want you to figure out what is the speed of the ball? So what is the speed? See if you can figure
out both of those things, and for extra points,
see if you can figure out a relationship between the two. Alright, well let's go
angular velocity first. I'm assuming you've had a go at it. Angular velocity, you might remember is just going to be equal
to our angular displacement, which we could say is delta theta, and it is a vector quantity. And we are going to divide
that by our change in time. So delta T. And so what is this going to be? Well this is going to be
our angular displacement. Our final angle is pi. Pi radians minus our initial
angle, pi over two radians. And then all of that's going
to be over our change in time, which is six seconds,
which is our final time minus our initial time,
minus three seconds. And so we are going to
get in the numerator, we have been rotated in
the positive direction, Pi over two radians. Because it's positive, we
know it's counterclockwise. And that happened over three seconds. And so we could rewrite this as this is going to be equal to pi over six. And let's remind
ourselves about the units. Our change in angle, that's
going to be in radians, and then that is going to be per second. So we're going pi over
six radians per second, and if you do that over three seconds, well then you're going to
go pi over two radians. Now with that out of the way, let's see if we can calculate speed. If you haven't done so
already, pause this video and see if you can calculate it. Well speed is going to
be equal to the distance the ball travels, and we've
touched on that in other videos. I encourage you to watch
those if you haven't already. The distance that we travel
we could denote with S. S is sometimes used to denote arc length, or the distance traveled right over here. So the speed is going to be our arc length divided by our change in time. Divided by our change in time. But what is our arc length going to be? Well we saw in a previous video where we related angular
displacement to arc length, or distance, that our arc
length is nothing more than the absolute value of
our angular displacement, of our angular displacement,
times the radius. Times the radius. And in this case, our radius
would be seven meters. So if we substitute all of this up here, what are we going to get? We are going to get that our speed, I'm writing it out because
I don't want to overuse, well I am overusing S, but I don't want people to get confused. Our speed is going to be equal
to the distance we travel, which we just wrote is the magnitude of our angular displacement. And this is all fancy notation, but when you actually apply it,
it's pretty straightforward. Times the radius of the circle. I guess you can say we
are traveling along. Let me write that in a different color. So times the radius. All of that over our change in time. All of that over our change in time. Well we could put in the
numbers right over here. We know that this is
going to be pi over two. You take the absolute value of that. It's still going to be pi over two. We know that our radius in this case is the length of that string. It is seven meters. And we know that our change in time here, we know that this over here
is going to be three seconds, and we can calculate everything. But what's even more
interesting is to recognize that what is this right over here? What is the absolute value
of our angular displacement over change in time? Well, this is just the absolute value of our angular velocity. So we could say that speed is equal to the absolute
value of our angular velocity. Absolute value of our angular
velocity, times our radius. Times our radius. And now so this is super useful. Our speed in this case,
is going to be pi over six radians per second. So pi over, pi over six. Times the radius. Times seven meters. Times seven meters. And so what do we get? We are going to get seven
pi over six meters per, meters per second, which will
be our units for speed here. And the reason why we're
doing the absolute value is 'cause remember, speed
is a scalar quantity. We're not specifying the direction. In fact, the whole time we're traveling, our direction is constantly changing. So there you have it. There's multiple ways to approach
these types of questions, but the big takeaway here is one, how we calculated angular velocity, and then how we can relate
angular velocity to speed. And what's nice is there's a
nice, simple formula for it. And all of this just came out of something that relates to what we
learned in seventh grade around the circumference of the circle, which we touched on in the video relating angular
displacement to arc length, or distance traveled.