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### 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 0:00
• Objectives 0:07
• 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

### 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

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

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/s2.0448

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 AT2.0565

For rotational, we have δθ = ωinitial T + 1/2αT2 and final velocity2 = initial velocity2 + 2 × acceleration × δX.0580

The rotational equivalent -- final angular velocity2 = initial angular velocity2 + 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/s2.0763

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/s2 gives me a linear acceleration of 0.075 m/s2.0795

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/s2 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/s2 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/s2, 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