AP Physics C: Mechanics Circular & Relative Motion

Hello, everyone, and welcome back to www.educator.com.0000

I am Dan Fullerton and in this lesson we are going to start to explore circular and relative motion.0004

Our objectives include understanding and applying relationships between translational and rotational kinematics which we are going to continue in future lessons.0009

Use the right hand rule to determine the direction of the angular velocity vector.0017

Describe the meaning of the phrase motion is relative and calculating the velocity of an object relative to various reference frames.0023

Let us start by talking about how we measure circular motion in degrees and once around the circle is 360°.0031

In radians however once around the circle is 2 π.0040

A radians measures the distance around an arc equal to the length of the arcs radius.0044

If there is a circle there is our radius it takes 2 π × this value to go once around the circle the circumference.0049

We call that distance around the circle δ s or C circumference = 2 π × the radius of that circle.0058

It is going to be useful to convert from radians to degrees and back again.0068

Especially, in this APC course, we are using both types of measurements depending on the problem.0072

If you are using your calculators, be very careful you got them in the right mode that when you are inputting angles0077

you know whether you are using degrees or radians in your calculator is also on the same page as you are.0082

First let us convert 90° to radians.0088

If we start with 90° and we want radians from 90° / 1 we will multiply that there are 2 π radians in one time around the circle or 360°.0091

Therefore I can say that we have π / 2 radians is equal to 90° or 1.57 radians if you to put it in decimal form.0105

Interesting to know radians is not an actual unit of measure.0119

It is helpful for us to use mentally but radians are officially unit less.0123

Let us convert 6 radians to degrees going the other way.0129

We have 6 radians and we want to convert to degrees.0132

I know there are 2 π radians and 360° therefore our radians will make a ratio of 1 and 6 × 360 degrees / 2 π is about 344°.0137

Nice and straightforward.0156

How do we use these?0158

Let start talking about displacements.0160

Linear position their displacement we have talked about is being given by δ r or we are also going to see it now especially as we talk about things in circles as δ s.0163

Angular position or displacement is given by δ θ.0172

How many degrees you have is you are going around the circle.0176

S = r × θ if you know your radius and the angle through which you pass you can find how far you have traveled0180

this distance along the circumference s = r θ or δ s = r δ θ.0191

We can also take a look at that in terms of velocity.0198

Linear speed and velocity we have been talking about is being given by V.0201

Angular speed or velocity is given by ω that is the curvy w symbol.0206

If we look V our average velocity is dr dt or ds dt the time rate of change of position.0214

Angular velocity is the time rate of change of θ your angular displacement.0222

Know that any point here your velocity vector is constantly changing the pointing in this direction.0228

Ω is around the circle your angular velocity as you go around there.0234

Now the direction of angular velocity is a little bit tricky because if you look here depending0240

on where you are in the circle your linear velocity is constantly changing.0244

Trying to describe a constant velocity for something around the circle is very tough in the angular world.0250

When we talk about angular velocity to define its direction we are going to use the right hand rule.0256

For something that is moving counterclockwise around the circle take the fingers of your right hand and wrap them in that counterclockwise direction.0262

Your thumb points in what we call the direction of the angular velocity vector.0271

To illustrate that I got a diagram on our next slide here.0275

If something is moving in this direction wrap your hands around it and your thumb points in the direction of the angular velocity vector.0279

It is very interesting to note that the angular velocity vector does not tell you the direction the object is moving at that specific point in time.0286

It is of very nebulous concept the first couple of times you see that.0294

We will also see this with angular acceleration and some other things angular momentum coming up as well.0298

How do we convert linear to angular velocity?0307

Linear velocity was ds dt or dr dt and using those interchangeably at this point.0312

S = r θ how we found our linear displacement along the curve from an angular displacement.0319

We can write this then it as velocity =dr θ dt.0328

But as we go around a circle the radius is a constant that is what makes a circle.0336

We can pull that out of the derivation.0341

We have V =r d θ dt but we just to find d θ dt as ω.0344

ω is d θ dt and we can write that our velocity vector is equal to r × ω vector V= rω.0354

We have a conversion method between the 2.0367

Let us take an example.0374

If we look at Earth's angular velocity find the magnitude of Earth's angular velocity and radians per second.0375

Ω angular velocity is angular displacement over some time interval.0384

Once around is 2 π radians and how long does it take the Earth once all the way around.0392

That is a day or 24 hours so that is going to be π /12 radians / hr.0400

radians / hr is kind of goofy metric.0411

Let us make that into radians / s to more standard unit.0414

I know that 1hr is 3600s.0417

My hours make a ratio of 1 and I come up with about 7.27 × 10⁻⁵ radians /s.0423

We will look at angular displacement angular velocity what come next? angular acceleration.0442

Linear acceleration we said was given by the A vector.0449

Angular acceleration is given by the α vector.0452

Angular acceleration was a derivative of velocity with respect to time.0457

Angular acceleration is the derivative of angular velocity with respect to time.0462

How fast something is accelerating as it goes around that circle?0468

By accelerating, we are not talking about the centripetal acceleration towards the center of a circle0473

that allows it to keep moving a circle or talk about whether it speeding up or slowing down in its path around the circle.0477

Looking at centripetal acceleration in much more detail and putting this together.0484

We can express the position vector in terms of unit vectors.0490

If we say that the position vector r(t) = function x is sum function of time in the i hat direction0494

+ y position as function of time in the j hat direction to give us r vector.0505

When we do this is we are going around the circle we should be fairly easy to see.0514

If we broke this up into components our x position is going to be r cos θ.0521

R y is going to be r sin θ.0527

x(t) is given by r cos θ or θ is going to be a function of time.0531

Ry function, y is a function (t) is r sin θ.0540

We could then write that r(t) Position vector is our cos θ in the x direction × I hat + r sin θ in the y direction or × j hat.0546

We also know θ = ωt angular displacement is angular velocity × time.0565

It is like in the linear world linear displacement is linear velocity × time.0576

R(t) = r cos ωt i hat + r sin ωt j hat.0584

We have gone that far let us take a look and say what if we want velocity though?0602

Velocity was dr dt which is going to be the derivative with respect to time of all the stuff we have up here.0607

R cos ωt i hat + r sin ωt j hat.0619

Let us take the derivative there then that is going to be equal to the derivative of r cos ω t0635

is going to be - ωr sin ωt i hat + derivative of r sin ωt is going to be ωr cos ωt j hat.0642

There is our velocity function.0666

Just write rv in here so we do not forget.0668

There is velocity, let us go to acceleration.0673

Let us keep going.0675

We know the acceleration is dv dt so that is going to be the derivative with respect to time of r - ωr sin ωt i hat + ωr cos ωt j hat.0677

Acceleration, as we take that derivative am going to have -ω² r cos ωt I hat - ω² r sin ωt j hat.0703

We can factor an ω² out of that so I could write that as acceleration is - ω² will have r cos ωt i hat + r sin ωt j hat.0730

That looks mighty familiar this term right here because we defined that way up here.0756

If we look up above we said that r cos ωt I hat + r sin ωt j hat that was our r vector.0763

We could then say that acceleration is - ω² r.0779

Why that negative sign?0786

This is because the acceleration points in the opposite direction of the radius.0789

If our radius vector r is going out to that position this implies that acceleration is in the opposite direction towards the center of the circle.0793

That is centripetal acceleration.0802

We can make it look a little more familiar if you like.0805

If you want to look at the magnitude of that centripetal acceleration that is ω² r still not familiar0808

but what if they put in our substitutions V = ωr or ω = V / r.0818

We can say that the magnitude of a = V² / r² × r or V² / r.0825

Centripetal acceleration V² / r it all works out.0835

Let us do an amusement park example.0841

Riders on a merry go round move in a circle of radius 4m executing 4 revolutions every minute or 1 revolution every 15s.0845

Find the angular velocity and the centripetal acceleration of a rider on the merry go round.0854

Let us start with the angular velocity that will be angular displacement divided by the time interval0860

which is going to be 4 × around in each time around is 2 π radians that takes 60s for all of that which will come out to be 0.419 radians / s.0867

If we want centripetal acceleration that is going to be ω² r or 0.419² × r radius 4m or 0.702 m/s².0882

Another example, let us look at the moon.0904

The Moon revolves around the earth every 27.3 days and the radius of the orbit is 382 mega meters.0906

What is the magnitude and direction of the acceleration of the moon relative to earth?0914

Couple ways we can do this let us start with the old fashioned way.0920

We know that the speed is distance /time.0924

It travels one time around 2π r one circumference in the period of t which is going to be 2π × r radius 382,000,000 m0928

in our time period is going to be 27.3 days but we want that in seconds.0944

Let us do the conversion.0949

27.3 days × we know there are 24 hr in one day and we know that there are 3600s in 1hr so why do all that and I come up with about 1018 m/s.0950

The centripetal acceleration ac =V² / r which is going to be r 1018 m / s² divided by our radius again 382,000,000m or about 0.00271 m/s².0972

That is one way to do it.0996

Let us take what we just learned during the different way.0998

Our angular velocity is δ θ divided by the time interval below 2π radians in again 27.3 days × 24 hours in the day × 3600 s/hr.1001

That comes up with about 2.66 × 10 -6 radians / s.1024

If we want the centripetal acceleration that is going to be ω² r which is 2.66 × 10⁻⁶² × r radius 382,000,000m.1032

I get the same thing 0.00271 m/s².1050

Let us move on and talk about reference frames for couple moments.1063

A reference frame describes the motion of an observer watching something else moving.1067

Our most common reference frame is earth that is the one we deal with every day.1071

We typically measure the motion of things compare to the earth because that is what we typically see is its surroundings.1075

Now the laws of physics, we study in this course assume we are in an inertial not accelerating reference frame.1081

That is not quite true.1087

As we are spinning on the earth we are constantly accelerating1089

and have other motion from the universe but it is negligible compared to what we are typically dealing with.1092

Let us assume we are going to call the earth an inertial reference frame.1098

There is no way to distinguish between motion at a rest and motion at the constant velocity in an inertial reference frame.1102

What that means is, let us think about what is going on if you are in an airplane as an example.1109

Let us take the airplane and put a wing in here.1114

Here we are nice and happy in our airplane.1119

Assuming this is an extremely smooth airplane, no turbulence, no bumps, whatsoever.1123

You cannot tell the difference whether you are sitting on the runway or whether you are in the air moving 500 miles/hr if all the window shades are down.1130

As long as you are moving it that constant speed at the inertial reference frame there is no way to distinguish the difference between the 2.1140

That is what we are talking about when we are talking about an inertial reference frame.1147

How can you tell if you cannot look out the window?1151

There is no experiment you can do to tell the difference between those 2.1153

Once we have that settled what we have talked about that reference frames we will talk about motion being relative.1161

For example imagine you are sitting in a lawn chair watching a train travel past you.1166

It is going to the right at 15m/s.1170

From your reference frame if you saw a cup of water through the train’s window it would look to you at your lawn chair like it is moving at 50 m/s.1175

If you are sitting on the train right in front of a cup of water though the couple of water would appear to be at rest not moving at all.1184

It all depends on your frame of reference.1192

Now imagine, instead you are on the trains staring out of the window you are watching some students sitting on a lawn chair for some unknown reason.1197

From your reference frame, the cup of water on the train remains still but you see that student as though they are moving to the left that 50 m/s.1204

Again, it all depends on your frame of reference.1213

What is going on as far as motion and how you describe that?1216

When we want to calculate velocities between 2 things, relative velocities, let us consider 2 objects a and b.1221

Taking the velocity of a with respect to some reference framed b helps us understand exactly what we are talking about.1228

It is pretty straightforward.1235

For example you might want to know the speed of a car with respect to the ground.1237

Or if you are walking on the train you might want to know the speed of a person with respect to the train.1241

It does not matter that the train is speeding along at 40 m/s.1246

What you are really concerned about is how fast that person is moving with respect to the rest of the train.1250

Let us do some examples here as we talk about strategies.1256

When we are calculating relative velocities here is a little trick I use to help keep things straight.1260

What we are going to do is say that the velocity with some object a with respect to c is defined as the velocity of a1265

with respect to some intermediate object b + the velocity of b with respect to some objects c.1275

As long as you change things like this a to b, b to c, c to d.1281

As long as you have them all daisy chained you can say that the total is the velocity of the first letter with respect to the last letter.1288

In the example we were doing for example.1296

The example we are doing for example a redundant again.1299

The example we were doing, we will call velocity of b with respect to c, the velocity of the train with respect to the ground.1302

We will call object a, that is a is our cup, b is our train, and c is the ground.1314

Velocity of a with respect to b then would be the velocity of the cup with respect to the train.1324

The velocity of a with respect to c then would be the velocity of the cup with respect to the ground.1335

If you want to know the velocity of the cup with respect to the ground you take the velocity of the cup1344

with respect to the train and add it to the velocity of the train with respect to the ground.1348

And the math will all work out.1351

As you look at that pattern here we could have any number of these that we daisy chain1354

for example the velocity of some a with respect to e would be the velocity of a with respect to b + the velocity of some b1359

with respect to c + the velocity of c with respect to d + the velocity of d with respect to e.1365

As we do all these all we have to do is take a look and go with we got a b and b they match.1375

C and c they match.1381

D and d they match.1382

Our total is going to be the first with respect to the last.1384

Velocity of a with respect to e.1389

It will be a little easier to show you how that is done with an example or 2 that is by just talking about it.1392

Let us do a couple.1398

Let us say we have a train we will call that a traveling at 60 m/s to the east with respect to the ground.1400

We will call the ground c.1408

A businessman b on the train runs it 5 m/s to the west with respect to the train.1410

Find the velocity of the man b with respect to the ground c.1415

We are looking for the velocity of b with respect to c.1422

One way I could do that is I can find the velocity of b with respect to a + the velocity of a with respect to c.1427

That would work because we have the a's in the middle so we end up with b and c for a total.1436

Based on what we are given I think we will be ok there.1441

The velocity of b with respect to a would be the velocity of the man with respect to the train.1445

It says the man on the train runs 5 m/s to the west with respect to the train.1449

Velocity of b with respect to a is 5 to the west.1455

Let us call it to the east positive.1458

That will be -5 m/s because the man is running to the west with respect to the train + V ac the velocity of the train1461

with respect to the ground that is 60 m/s to the east.1470

That will be + 60 m/s I will end up with 55 m/s east.1474

The velocity of the businessman with respect to the ground is 55 m/s east.1481

You probably could have done that in your head on a simple problem like this but1487

as we get more involved having a way to keep track like this can be very valuable.1491

Let us take a look at an airspeed example.1497

An airplane let us call that P flies at 250 m/s to the east with respect to the air.1500

The air we will give that an a is moving at 15 m/s to the east with respect to the ground.1505

We will call to the east our positive direction again.1512

Find the velocity of the plane with respect to the ground.1516

We are looking for the velocity of the plane with respect to the ground.1521

We are given the velocity of the plane with respect to the air that is 250 m/s and we are given1525

the velocity of the air with respect to the ground which is 15 m/s.1533

If we want the velocity of the plane with respect to the ground that will be the velocity of the plane1541

with respect to the air + the velocity of air with respect to the ground.1545

Those matches in the middle are left with P and g that should work.1550

So that is going to be 250 m/s + 15 m/s or the velocity of the plane with respect to the ground is 265 m/s.1555

This works in multiple dimensions as well.1570

An airplane P flies a 250 m/s to the east with respect to the air.1574

The air is moving at 35 m/s to the north with respect to the ground.1581

Air with respect to the ground is 35 m/s north.1586

Find the velocity of the plane with respect to the grounds.1593

We want plane with respect to the ground.1596

The velocity of the plane with respect to the ground is going to be the velocity of the plane1600

with respect to the air + the velocity of the air with respect to the ground.1604

It will be Pg Pg that should work.1610

What I am going to do is draw this out.1613

The velocity of the plane with respect to the air that is 250 m/s to the east.1615

The velocity with of air with respect to the ground is 35 m/s to the north and we are going to add that1623

so I will line them up to tail to remember as we talked about vectors.1628

There is our velocity of air with respect to the ground which is 35 m/s.1632

If I want the sum of those the velocity of the plane with respect to the ground I will draw a line1640

from the starting point of my first factor to the ending point of my last once they are lined up to tail.1647

To add these now in vector form let us find a magnitude and find the speed here first.1651

We can do that using the Pythagorean Theorem.1657

The magnitude of the velocity of the plane with respect to the ground is going to be √250² + 35² or about 252 m/s.1659

If we wanted to know this angle θ we can find that as well.1676

Θ equals the inverse tan.1681

We know the opposite and the adjacent side.1683

Inverse tan of opposite / adjacent which will be the inverse tan of 35 m/s / 250 m/s or about 7.97 degrees.1685

My answer the velocity of the plane with respect to the ground is 252 m/s and angle of roughly 8° NE.1700

One last 2D problem here with relative velocities.1711

An oil tanker let us call it T travels east at 3 m/s with respect to the ground while a tugboat moves north at 4 m/s with respect to the tanker.1717

Our tug boat we will call b with respect to our tanker T.1728

What is the velocity of our tug boat with respect to the ground?1735

We want the velocity of our tug boat with respect to the ground and that is going to be the velocity of our tugboat1740

with respect to the tanker + the velocity of the taker with respect to the ground.1746

All right drawing these out, we know that we have as I look at the problem tanker travels east that 3 m/s with respect to the ground.1753

The tanker with respect to the ground VT g is 3 m/s.1761

We also have the tugboat pushes it north or moves forward moves north at 4 m/s with respect to the tanker.1771

The velocity of our boat with respect to that tanker is going to be 4 m/s.1778

We can add these in any way we want I am just going to drop this way so tip to tail again.1784

Velocity of tugboat with respect to the tanker is 4 m/s.1788

Find the velocity of the tugboat with respect to the ground.1793

To do that I will add these up starting at the starting point of my first going to the ending point of my last.1796

And I can do that when my head that is a 345 right triangle.1803

The velocity of our tugboat with respect to the ground must be 5 m/s.1807

And we can use trigonometry if they wanted to if we needed to know that angle exactly.1812

Hopefully that gets you a good start on circular and relative motion.1817

Thank you so much for watching www.educator.com and make it a great day everybody.1821