For more information, please see full course syllabus of AP Physics 1 & 2

For more information, please see full course syllabus of AP Physics 1 & 2

### Rotational Kinematics

- Much like linear motion can be described by displacement, velocity, and acceleration, rotational motion can be described by angular (or rotational) displacement, angular velocity, and angular acceleration.
- One complete revolution of a circular path describes 360 degrees, or 2 Pi radians, where a radian measures the distance around an arc equal to the length of the arc's radius.
- Angular velocity describes the rate of change of an object's angular displacement. The right-hand rule describes the direction of the angular velocity vector, where counter-clockwise rotations correspond to positive angular velocities.
- Angular acceleration describes the rate of change of an object's angular velocity. The right-hand rule also describes the direction of the angular acceleration vector.
- Rotational analogs to the kinematic equations can be used to solve problems involving rotational motion.

### Rotational Kinematics

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Objectives
- Radians and Degrees
- Example 1: Degrees to Radians
- Example 2: Radians to Degrees
- Linear vs. Angular Displacement
- Linear vs. Angular Velocity
- Direction of Angular Velocity
- Converting Linear to Angular Velocity
- Example 3: Angular Velocity Example
- Linear vs. Angular Acceleration
- Example 4: Angular Acceleration
- Kinematic Variable Parallels
- Kinematic Variable Translations
- Kinematic Equation Parallels
- Example 5: Medieval Flail
- Example 6: CD Player
- Example 7: Carousel
- Example 8: Circular Saw

- Intro 0:00
- Objectives 0:07
- Radians and Degrees 0:26
- In Degrees, Once Around a Circle is 360 Degrees
- In Radians, Once Around a Circle is 2π
- Example 1: Degrees to Radians 0:57
- Example 2: Radians to Degrees 1:31
- Linear vs. Angular Displacement 2:00
- Linear Position
- Angular Position
- Linear vs. Angular Velocity 2:35
- Linear Speed
- Angular Speed
- Direction of Angular Velocity 3:05
- Converting Linear to Angular Velocity 4:22
- Example 3: Angular Velocity Example 4:41
- Linear vs. Angular Acceleration 5:36
- Example 4: Angular Acceleration 6:15
- Kinematic Variable Parallels 7:47
- Displacement
- Velocity
- Acceleration
- Time
- Kinematic Variable Translations 8:30
- Displacement
- Velocity
- Acceleration
- Time
- Kinematic Equation Parallels 9:09
- Kinematic Equations
- Delta
- Final Velocity Squared and Angular Velocity Squared
- Example 5: Medieval Flail 10:24
- Example 6: CD Player 10:57
- Example 7: Carousel 12:13
- Example 8: Circular Saw 13:35

### AP Physics 1 & 2 Exam Online Course

### Transcription: Rotational Kinematics

*Hi everyone and welcome back to Educator.com.*0000

*I am Dan Fullerton and today we are talking about rotational kinematics.*0003

*Our objectives are going to be to understand the analogy between translational and rotational kinematics, to use the right-hand rule to associate angular velocity with a rotating object, and to apply equations of translational and rotational motion to solve a variety of problems.*0008

*Let us start by talking about radians and degrees.*0027

*In degrees, one time around a circle is 360 degrees.*0029

*In radians though, once around a circle is 2π, where a radian measures the distance around an arc equal to the length of an arc's radius.*0034

*So distance around a circle -- oftentimes written δS -- is the circumference, which is 2π radians or it would be 360 degrees if you are looking at an angular measurement.*0043

*Let us convert 90 degrees to radians.*0058

*If we start off with 90 degrees -- if we want to convert that to radians, we are going to multiply this by -- well we want degrees to go away, so 360 degrees = 2π radians.*0060

*So the degrees will cancel out and I will be left with 90/360, that is 1/4 and that is going to be π/2 radians or 1.57 radians.*0075

*Let us convert 6 radians to degrees, going the other way.*0091

*We have 6 radians, and we are going to multiply that -- we want radians to go away -- I know there are 2π radians in a complete circle and 360 degrees in a complete circle.*0096

*So we will be left with 6/2π × 360 or 344 degrees.*0109

*As we do this, let us talk for a few minutes about linear versus angular displacement.*0120

*Linear position displacement, is given by δR δS.*0125

*If we talk about angular position or displacement though, we can talk about how much this angle changes. *0129

*That is given by δ θ, and there is a conversion between these.*0135

*The linear distance is equal to R × θ, or δS = R × δθ.*0139

*Multiply the angular displacement, δθ by the radius to get a linear displacement.*0145

*We can also look at this for linear versus angular velocity.*0155

*Linear speed or velocity is given by the symbol V.*0159

*Angular speed or velocity is given by ω, kind of a curly W.*0162

*Now whereas velocity was the change in displacement over time, angular velocity is the change in angular displacement over time.*0168

*Dθ, DT, or δθ with respect to T.*0176

*If we want to take a look at the direction of angular velocity, we use the right-hand rule.*0186

*And the way we do that is you wrap the fingers of your right hand in the direction of the angular velocity -- your thumb will point in the direction of that vector.*0191

*Having a typical vector is not going to work because angular velocity, the direction linearly, is constantly changing, so you have to define it with something perpendicular.*0200

*Wrap the fingers of your right hand around the circle -- your thumb will point in the direction of the angular velocity vector.*0210

*In this case, as we have here on the screen, angular velocity is around this way, so as I wrap the fingers of my right hand around that direction, my thumb points out toward me.*0217

*I show that by showing a dot coming toward me, almost as if there is an arrow being pointed toward me -- that is what I would see.*0228

*So that is out of the plane of the board.*0235

*If it was into the plane of the board, the way I would draw it would be an x like I am looking at the fletchings of an arrow as it is moving away from me, so that would be into the plane.*0239

*In this case though, angular velocity points out of the plane.*0249

*Angular velocity is the cause of counterclockwise rotations, typically referred to as positive, and those that cause clockwise, negative.*0251

*How do we convert linear to angular velocity?*0263

*Well, linear velocity is just equal to angular velocity times the radius or angular velocity equals linear velocity divided by the radius.*0268

*Let us take an example.*0282

*Let us find the magnitude of Earth's angular velocity in terms of radians per second (rad/s).*0283

*Angular velocity is going to be a change in angular displacement divided by the time.*0290

*The Earth goes once around on its axis or 2π radians every 24 hours, once a day.*0296

*And let us multiply that to get radians per second -- let us convert hours into seconds.*0305

*One hour is 3,600 s, so my hours make a ratio of 1 and I am left with ω = 2π/24/3,600 or 7.27 × 10 ^{-5} rad/s.*0312

*As we look at linear versus angular acceleration, linear acceleration is given by A, and angular acceleration is given by the symbol α and it too is a vector.*0336

*Just like linear acceleration is change in velocity over time, angular acceleration is change in ω over time.*0346

*The rate of change of the angular velocity with respect to time, or we can write that as δ ω/δT.*0353

*The conversions between them are pretty straightforward as well -- A = Rα or α = A/R.*0362

*Another example -- angular acceleration.*0376

*A clown rides a unicycle. If the unicycle wheel begins at rest and accelerates uniformly in a counterclockwise direction to an angular velocity of 15 rpms in a time of 6 s, find the angular acceleration of the unicycle wheel.*0378

*Let us start by converting this 15 rpms to radiants per second. We have 15 rpms or revolutions per minute. *0393

*We need minutes to go away, so I will put minutes on the top and I want seconds here, so I know 1 minute is 60 seconds and now I have revolutions per second (rps).*0403

*So, I also need to multiply to make the revolutions go away.*0413

*One revolution is 2πradians.*0417

*Unit conversions then -- minutes make a ratio of 1, revolutions make a ratio of 1 and I am left with 15 × 2π/60, or 1.57 rad/s.*0422

*Similarly, the angular acceleration -- now I can find as change in angular velocity divided by time.*0440

*That is going to be final angular velocity minus initial angular velocity over time or 1.57 - 0/6s, which is 0.26 rad/s ^{2}.*0448

*So let us put this all together to talk about kinematic variable parallels.*0467

*We talked about displacement in the translational or linear world.*0472

*Displacement -- we are writing as δS or D, or δX or we would even have it as R.*0478

*In the angular world, it is δθ.*0485

*Velocity is V. Angular velocity is ω, acceleration is A, angular acceleration is α, and time is the same translationally and in the angular world.*0490

*And there are more parallels we can draw, such as kinematic variables -- we can convert them.*0509

*Displacement is S = Rθ or if we want the angular version we have θ = S/R. *0514

*Velocity or V = Rω, angular velocity is ω = V/R, acceleration linear is A = Rα, angular is α = A/R, and same as before, time equals time.*0522

*So it is very easy to translate back and forth to these variables.*0543

*We even have parallels with the kinematic equations -- translational kinematic equations, V final = V initial + AT.*0548

*In the rotational world, we have kinematic equations too -- we just replace the variables with their angular equivalents.*0559

*So ω = ωinitial + αt and for translational, δX = V initial T + 1/2 AT ^{2}.*0565

*For rotational, we have δθ = ωinitial T + 1/2αT ^{2} and final velocity^{2} = initial velocity^{2} + 2 × acceleration × δX.*0580

*The rotational equivalent -- final angular velocity ^{2} = initial angular velocity^{2} + 2 α δθ.*0600

*So really there is not a whole lot new to learn here. It is just using different variables to cover the rotational kinematics as opposed to just the linear kinematics.*0614

*Let us take an example of a medieval flail. *0625

*A knights swings a flail of radius 1 m in 2 complete revolutions. What is the translational displacement of the flail?*0627

*Well S = R θ, so R is going to be 1 m, θ is 4π radians, twice around the circle.*0636

*So that is just going to be 4π × 1 or 12.6 m.*0647

*Or let us look at a CD player.*0657

*A compact disc player is designed to vary the disc's rotational velocity so that the point being read by the laser moves at a constant linear velocity of 1.25 m/s.*0659

*What is the CD's rotational velocity in revolutions per second when the laser is reading information on an inner portion of the disc when the radius is 0.03 m?*0669

*Angular velocity is linear velocity divided by radius, so that is going to be 1.25 m/s over the radius of 0.03 m which is 41.7 rad/s.*0680

*We want that in revolutions per second so let us convert it.*0699

*41.7 rad/s times -- there are 2π radians in one revolution, so radians make a ratio of 1, 41.7 × 1/2π -- I get 6.63 rps.*0701

*We can even look at a carousel.*0733

*A carousel accelerates from rest to an angular velocity of 0.3 rps in 10 s. What is its angular acceleration?*0735

*Well, just like we did in kinematics, we can make a table -- ω initial = 0, ω final is 0.3 rad/s -- δθ, α and we know t is 10 s.*0743

*What is its angular acceleration? We can use our kinematic equations.*0759

*α = ω - ω initial/t or 0.3 rad/s - its initial 0/10 s for an angular acceleration of 0.03 rad/s ^{2}.*0763

*What is the linear acceleration for a point at the outer edge of the carousel 2.5 m from the axis of rotation?*0783

*Well to do that we just need to find the linear acceleration from the angular acceleration.*0790

*A = Rα, where R = 2.5 m and our α -- we just determined, 0.03 rad/s ^{2} gives me a linear acceleration of 0.075 m/s^{2}.*0795

*Or a circular saw example -- a carpenter cuts a piece of wood with a high powered circular saw.*0814

*The saw blade accelerates from rest with an angular acceleration of 14 rad/s ^{2} to a maximum speed of 15,000 rpms.*0821

*What is the maximum speed of the saw in rad/s?*0830

*Well 15,000 rpms or revolutions per minute -- let us convert those minutes to seconds -- 1 min is 60 seconds, and instead of revolutions, we need this in radians.*0835

*So we have 2π radians per revolution -- revolutions make a ratio of 1, minutes make a ratio of 1 and I come up with 1,570 rad/s.*0852

*How long does it take the saw to reach its maximum speed? Well, that is a kinematics problem.*0871

*ω initial = 0 -- final, its maximum speed is 1,570 ras/s, δθ, α -- which we said was 14 rad/s ^{2} and time.*0878

*If we are looking for how long -- we are looking for time -- I will use the formula ω = ω0 + αT and rearrange this for the time.*0896

*Time = ω - ω0/α or 1,570 - 0/14 rad/s ^{2}, which gives me a time of 112s.*0907

*Hopefully that gets you started with rotational kinematics.*0928

*It is really similar to what we did with linear kinematics, it is just we have some slightly different variables dealing with objects going around the circle.*0932

*Once you know the change -- the parallels with the variables -- and you know the equations already, it is just a matter of being careful with your variables.*0939

*Thanks so much for your time and thanks for watching Educator.com.*0949

*Make it a great day.*0952

2 answers

Last reply by: Peter Ke

Thu Jul 7, 2016 4:23 PM

Post by Peter Ke on June 26 at 07:10:34 PM

Hello, I understand all the equation, but I just don't understand what the equation mean. Like what is the difference between:

linear position/displacement vs angular position/displacement

linear velocity vs angular velocity &

linear acceleration vs angular acceleration.

Please explain.

1 answer

Last reply by: Professor Dan Fullerton

Wed Mar 4, 2015 7:00 AM

Post by Derek Boutin on March 3, 2015

How are you able to relate linear velocity and angular velocity? What is the process?

1 answer

Last reply by: Professor Dan Fullerton

Sun Aug 17, 2014 9:31 AM

Post by SD Ryo on August 16, 2014

Why is example 5 not (1) x (4 pi 57.3)? (since rad = 57.3)

1 answer

Last reply by: Professor Dan Fullerton

Wed Jun 5, 2013 6:04 AM

Post by Ty Forgey on June 4, 2013

In example 4, you first convert to radians. Would it have been okay to answer in terms of degrees or is it convention to answer in radians?