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Lecture Comments (6)

1 answer

Last reply by: Professor Selhorst-Jones
Fri Mar 25, 2016 6:18 PM

Post by Peter Ke on March 8, 2016

For example 3, I really don't understand the diagram you draw.
Why the arrow for mg and T is about half the centripetal force at the top?

3 answers

Last reply by: Professor Selhorst-Jones
Mon Jun 23, 2014 9:47 AM

Post by Mitrica Dragos on June 20, 2014

Where does the centrifugal force come from ? How we actually use Newton 3rd low to explain the centrifugal force. At example 3, when the rock was at the side, we have just the tension witch equals the centrifugal force.

Force & Uniform Circular Motion

  • For something to stay in a circle at a uniform speed, it must have an acceleration constantly pointing in to the center of the circle. The acceleration's magnitude is |a | = [(| v |2)/r]. This is called centripetal acceleration.
  • To have an acceleration, there must be a force. We know F = ma, so we can combine that with our centripetal acceleration formula to get a formula for centripetal force:
    F = m


  • The centripetal force on the object always points from the object to the center of the circle.
  • Centripetal force must be supplied by something. It must come from something else: a string, the rails of a roller coaster, etc.
  • Centripetal force is not a force in and of itself. It is a relationship that must be fulfilled by the net force on an object if the object is to remain in a circle.

Force & Uniform Circular Motion

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
  • Centripetal Force 0:46
    • Equations for Centripetal Force
    • Centripetal Force in Action
  • Where Does Centripetal Force Come From? 2:39
    • Where Does Centripetal Force Come From?
  • Centrifugal Force 4:05
    • Centrifugal Force Part 1
    • Centrifugal Force Part 2
  • Example 1: Part A - Centripetal Force On the Car 8:12
  • Example 1: Part B - Maximum Speed the Car Can Take the Turn At Without Slipping 8:56
  • Example 2: A Bucket Full of Water is Spun Around in a Vertical Circle 15:13
  • Example 3: A Rock is Spun Around in a Vertical Circle 21:36

Transcription: Force & Uniform Circular Motion

Hi, welcome back to, today we are going to be talking about force and uniform circular motion.0000

Last time we talked about objects moving in a circle, we talked about uniform circular motion, we realized that for something to be able to continue moving in a circle, if an object is going here, here, and keep up the uniform speed, it has got to have an acceleration pointing to the middle of the circle at all times.0007

We realized that we needed this centripetal acceleration, an acceleration pointing in, centripetal - towards the centre.0024

From Newton's second law, we now know that for any acceleration to exist, we have to have a force creating that acceleration.0032

This implies that there has to exist some sort of centripetal force.0041

What is that centripetal force?0045

How much is it?0048

From previous work, we know that, force = ma (Newton's second law), and we learned that centripetal acceleration's formula was, acceleration = (speed)2 / radius . (radius of the circle)0049

Remember that the acceleration always points to the centre, it is the centripetal acceleration, to the centre of the circle, wherever your object is on the circle.0065

Putting these together, we get that the magnitude of the centripetal force has to be equal to, mv2 / r ,and it always points towards the centre of the circle.0074

Here is some examples of centripetal force in action:0088

When do we see it in real life!0090

Anytime we see rotational movement, centripetal force has to be in place.0092

If a rock is spinning on a string, what is keeping it in that string?0096

If we have got a rock, and it is spinning, and the thing that is keeping it, that force, is the tension in the rope, if that it is horizontal, if it is vertical, it is going to be combination of tension and gravity, sometimes working together, sometimes working against each other.0100

If a car is rounding a corner, then the force of friction on the tyres, is going to be the friction of the tyres, has to be pulling in to the centre of the circle, throughout the corner.0113

A roller coaster doing a loop, what keeps you at the top?0124

It keeps on the top, because the normal force of the car, at the speed that you are going through, during that top of the circle, and that radius, that is what is going to hold you at the top.0128

An airplane banking in the air, turning, banking like this, and pulling up, if it is turning in the air, similar to the car rounding the car, but now it is the force of pressure, the air pressure, and the fact that the winds are making a certain shape, allowing it to force itself around.0136

It is able to use air pressure and friction, through that it is able to turn it through the air.0153

Where does centripetal force come from?0160

Remember, in all the examples that we talked about, centripetal force was not just inherently present.0162

To have that circle, we needed centripetal force.0167

To have that rock go in a circle, we needed that rope to provide tension.0171

To have that car turn in a circle, we needed those tyres to have friction.0175

To have that roller coaster stay atop, we need the normal force keep it pushing in.0180

Centripetal force is created by other things, centripetal force does not get added in, we just know that the sum of our forces has to equal the centripetal force we are dealing with.0186

So all those previous examples, there is always some force keeping the object moving in a circle, there is always some force satisfying that centripetal force relationship.0197

Centripetal force will not show up, we will not see it in our free body diagrams directly, we see it as the sum of the forces will have to be this centripetal force, otherwise you cannot have a circle.0204

You have a circle, it means that you need a constant centripetal acceleration, to have a constant centripetal acceleration, you need to have a constant centripetal force.0214

But, that does not guarantee that you are going to have that.0221

It is just the qualification.0224

If you know something is in a circle, then you know it is a centripetal force.0225

If you have a centripetal force, then you know it is going to be a circle.0228

But it does not create the centripetal force, it has to be given by something else.0231

So, it is a relationship that must be upheld by the forces acting on the object, to maintain the circle.0236

We do not get centripetal force, it is created form what we already have.0241

Centrifugal force:0246

You probably have heard of the idea of centrifugal force, at some point.0248

Centripetal points into the middle of the circle, so centripetal points in, whereas centrifugal points out.0251

So, centrifugal, and centripetal, like that.0263

Centripetal points in, centrifugal points out, centripetal meaning centre seeking, centrifugal meaning centre fleeing.0271

You may have heard before that centrifugal force is not real, that it is a fictitious force.0278

Sort of!, the reason why people call that a fictitious force is, say you are in a circle, and you might have seen these at a fair.0285

You are in a circle, and they pt you up against the wall, and you are spinning really fast, you get pushed up against it, because of centrifugal force, sort of.0292

What is really going on, is the combination of friction and the centripetal force, is keeping you, the fact that your body wants to move in a certain path, and then the wall is there, and it is spinning, all these things combined to give you a centripetal force created by other things, centripetal force pulling into the centre, so you get pushed against the wall.0303

But, when we experiences it, it feels from our point of view that our back is being pushed by a constant pressure, that we have got this pressure of the wall pushing on us, constant, and just that it is down.0322

Because from our point of view, we are not moving, the circle is spinning, but it is like we are still, and the world around us is spinning.0335

And that is to do with the reference frames.0341

So this is an example of a non-inertial reference frame.0343

A reference frame where laws of inertia, laws of Newton's laws, they do not hold like they used to.0346

So, it is important to pay attention, we have to step outside and we have to observe the circle moving from outside, otherwise things get to start really funky.0351

That said, centrifugal force is not necessarily false, it is just, this is not quite the right way to think about it.0359

It is not that the force is constant, and just pushing down, and created out of nowhere, it is created out of a variety of other things, it is created out of the centripetal force, it is the reaction force, it is Newton's third law in action.0365

Remember Newton's third law?0377

Equal and opposite force stuff?!0378

That is exactly where this is coming from.0380

If we have a centripetal force keeping an object in a circle, say we have got that circle, and we know that there is a centripetal force, of some amount pulling in.0381

Based on Newton's third law, there is also a force of this much, pulling the other way.0390

Which one do we experience in our bodies?0397

We experience centrifugal force.0401

So, why does your body feel centrifugal force, and not centripetal?0404

Human body is built to feel pressure, the way our skin cells, the way our nerve cells combines together to work, is one, something pushes on you, you feel the push, you do not feel the other force involved.0410

When you push on a book, you do not feel the book pulling away from you, you do not feel the push on the book, you feel the push on your hand, you feel the reaction force.0423

When you turn a corner in a car, you feel the car, the side of the car push on you, even though you are really being pulled to stay in the circle.0432

There is two forces at hand always, but you only feel one of them, because of the way your body is designed to sense things.0440

So, that is just why we feel it.0446

But the centrifugal force is there, centrifugal force is just the reaction to the centripetal force, it is the reaction force.0449

It is Newton's third law in action.0455

So, centrifugal force is not false.0457

It is a real thing, but the way that it gets explained and talked about, sometimes, that is not always true.0461

Sometimes, people talk about it as if it is its own thing, but really it is because it is a matched pair, the centripetal force.0469

We talked about centripetal force as really a force in itself, so it can get a little bit confusing, but we understand that centripetal force is created by other forces, and centrifugal force is the equal and opposite half of that, that is the other side of the coin, iti allows us to see what the other part is, it allows us to see what our bodies feel.0473

So we have got plenty of understanding to start doing some problems.0493

We got a car of mass, m = 1500 kg, and it makes a quarter circle turn of radius = 200 m, while maintaining a constant speed of 20 m/s.0496

What is the centripetal force on the car?0509

We just pump it all into our equation, F = mv2 / r = 1500 × (20)2 / 200 = 3000 N, to maintain a constant speed, that car has to experience 3000 N of force pulling it in.0512

Where does that force come from?0537

That force has to come from friction of the tyre on the road.0540

The only interaction that the car can make with the world around it, is through the road, through friction, through what is touching it through the wheels.0543

So, the friction of the wheel son the road is what creates it.0551

Given that the coefficient of friction between the tyres and the road is, μs = 0.8, because remember, we talked about in our section on friction, μs is what we use for tyres, because they are in constant contact.0555

The contact patch is effectively still on the ground.0568

So we use μs.0572

What is the maximum speed the car can take the turn out without slipping?0574

This will take a little bit more thinking.0577

Here is all our important stuff, we got the radius, the coefficient of friction (static), what speed will it slip at?0580

In this case, we are not going to need mass, we will see in a few seconds.0588

So, let us look at it in a top-down perspective.0592

Here is our car driving along, and the car needs to have a curved path.0596

That means, for the whole time, it needs to have static friction pulling in, like this, at some certain amount.0600

The static friction, otherwise the car will just slip, because if all of a sudden it were not able to do that, its wheels would not be turned at some angle, for the turn.0608

The wheels, here is one wheel of the car, if instead it were to just escape, it would not be rolling anymore, the tyre would slide along the ground, it will start going to a skid, a bad thing of course.0620

But it does not want to go to skid, it wants to stay at rest, friction wants to keep two materials together, wants to keep them from sliding.0634

It does not want things, it is force, but friction is going to keep it together as opposed to letting it slide.0648

As long as we do not exceed the maximum static friction, will be able to keep that turn.0651

How much is it?0657

The maximum static friction is going to tell us what the maximum centripetal force is.0658

The maximum static friction is the most centripetal force we can get.0667

The most centripetal force, assuming that our mass and radius stays the same, which they do, is going to be, the maximum velocity.0673

What we have to do, is we have to solve for what is the maximum static friction going to say about that maximum centripetal force.0682

Maximum static friction, μs × FN , this case, car is flat on the ground, so, FN = mg , since the car is flat on the road, when it turns it is going to still experience its full normal force.0691

So, the maximum static friction has to be equal to the maximum centripetal force.0713

What is the maximum centripetal force?0718

It is going to be, m × v2 / r .0720

We get, μs × mg = mv2 / r .0727

Cancel those m's, and now we get, μs × g × r = v2 .0735

Taking square root, we get, v = sqrt( μs × g × r) = the maximum velocity.0747

So, what is that maximum velocity?0755

Just when it is on that razor's edge of slipping, what is it?0757

Punching in numbers, v = 39.6 m/s = the maximum velocity that it can take without starting to slip.0760

At the same time, that is also where it starts to slip, because it is hard to stay precisely on 39.6, you might accidentally go to say, 39.6000000001 m/s, you cannot stay precisely.0785

So, that is the moment of slipping, just when you pass 39.6, that is when the car suddenly lose traction, and that is really bad, because we have been relying on the fact that, if we had static friction, once we exceeds static friction, (as we know from our talk about friction), you flip into using kinetic friction.0796

Kinetic friction for a tyre is considerably less than static friction, that means you had a certain amount of control, and then you have even less control, once you start to slide.0815

So, if things start to slide, you are on a slippery slope, things are going to get worse than worse.0825

Also, something to keep in mind, we are assuming that μs = 0.8, that is a reasonable amount for a car tyre on a dry, clean road.0830

What happens if all of a sudden, you hit a patch of wet water, or it has been raining recently, or there is an oil slick on the ground?0839

Your μs could drop to something really low.0847

If it was just wet, it is perfectly reasonable for it to be 0.4, and if you hit like, a patch of grease, or hydroplane (causing almost no friction), there is a large pool of water and you go at high speed, suddenly your friction force is going to drop way down all of a sudden.0850

That means you have got that much less control.0868

Yu $mu;s drops to 0.4, all of a sudden, your top speed, you can take that corner at without starting to slide out, without skidding and fish-tailing, is much lower.0870

This is an important reason, this is why you have to drive carefully when it has been raining, is because you could rely on a dry safe road and go fast, when it is dry and safe.0880

But if it is wet, all of a sudden, the top speed that you can take a turn at drops massively just because of the laws of Physics.0890

There is no way around it, it is more dangerous, because the maximum forces that you can attain with your car becomes much less once it is wet.0897

There is less friction to go around, so less forces to around, which means that speeds that you can go at, have to go down.0905

Example 2: We have a bucket full of water, and it is being spun around in a vertical circle on a rope that is 1 m long.0914

How fast does the bucket need to spin to keep that water from sloshing out?0921

We have almost all certainly seen this demonstration.0927

You have some bucket on a rope, and you spin it really fast, and the water inside the bucket will not fall out.0929

If you spin it fast enough, it gets held in.0935

One way to talk about it being held in, is the centrifugal force.0938

We can say that it is the centrifugal force that is holding it in.0941

But we can also think about this problem, we do not even actually need force to solve it, it just helps us to have force to think about it.0943

Let us say that the total centripetal force needs to be this long a vector, pointing in the top.0951

We are going to need to make the actual distance be our, the actual distance is going to be this length here.0961

The distance is the length, would be the magnitude for the vector.0971

Say you need this much centripetal force, to stay in that circle.0975

If you need this much centripetal force and gravity is going to pull down by this much, then we need, so here is 'mg', 'mg' is definitely guaranteed, as you are going to the top, there is going to be some weight pulling you in.0983

You are going to have it, because you are on Earth, you are in a vertical circle, so throughout the circle, you are going to have some 'mg' pulling you.0998

But if you are at the top, then we need more.1006

This is longer, so we need extra bit and we get that from tension.1010

The tension in the rope is what is going to keep it, extra, so that is going to make out for that extra amount of centripetal force that we need.1015

So we are going to have some tension in the rope.1023

What happens all of a sudden if we need less centripetal force?1024

If we are going at a lower speed, and we only need this much centripetal force.1030

Say this is the amount of centripetal force we need.1035

If we need this much centripetal force, then we do not need any tension at all.1038

There is no tension whatsoever.1043

But, we still have all this gravity.1046

So this gravity suddenly, we have got this much 'mg' and this much 'mg', so 'mg' for the centripetal force, and then 'mg' left over.1050

We cannot get rid of it, if it is not being, you know, used to keep it in the circle, it is still going to have an effect on it, so all of a sudden, this bucket is going to fall out.1066

You do not take the top of it at a fast enough speed, and your spin decays, your circle decays, and you fall out of your orbit, you fall out of going around that centre point, and the bucket is going to fall out, if we do not have enough.1078

What we need is, we need to have enough centripetal force, so that all of 'mg' gets used up.1092

We need to have the centripetal force be greater than or equal to the weight.1100

Because if it is less than 'mg', then that means we have some 'mg' left over, and we do not, we still have to use 'mg', it is the Physics, it does not get like, "well, we have this remainder, we do not need it, we will not use it", No!, weight is still going to have an effect, it is still going to pull the bucket down, so it is going to pull it out of a circle.1108

We have to need all of the weight, we can also go over the weight, and then we will make up the extra with the tension in the rope.1123

But for it to stay in it, we are going to have to have the centripetal force be greater than or equal to 'mg'.1133

Smallest centripetal force that we are allowed to use will give us the smallest speed.1138

The lowest, how fast the bucket has to go at a minimum, to keep the water from sloshing out.1144

Basic idea that we come up from all of this, is we got the fact that the centripetal force has to be greater than or equal to mg, or it sloshes out.1150

Otherwise, we will have left over gravity, and that left over force of gravity will still have an effect, it will pull out the water, it will pull out the bucket from the circle.1168

So, to keep the water in the bucket, to keep the bucket in the circle, centripetal force has to be greater than or equal to mg.1177

We want to find out what the smallest speed we can go is, then we consider one centripetal force equals mg.1188

Centripetal force is, mv2 / r = mg1191

One thing to see is, we actually did not have to do this, by using forces.1205

As we can see now, we can easily cancel out those m's, because really what we are saying is, we need the centripetal acceleration, to be greater than the acceleration due to gravity.1209

Centripetal acceleration is less than the acceleration of gravity, gravity is still going to accelerate you, so it is going to make up the extra, no matter what, so you have to use all that centripetal acceleration.1219

Same basic idea, we could have approached it by just talking about acceleration, but little bit easier to see it as forces, because we have a better understanding of how force works.1229

Just intuitively, as humans, we are more used to working in forces than just the acceleration.1238

At this point, if we want to solve for this, we know the radius is, we got v, take the square root, is just, rg.1243

Plug in numbers, sqrt(rg) = sqrt(1 m × 9.8)1251

So we have got that speed has to go at the minimum speed it has to go at, is 3.13 m/s, or greater.1260

If we go at that speed or more, for a 1 m radius, it will be able to keep it in the bucket.1272

Keep in mind, if the radius changes, the minimum speed that you have to go at, keep the water in the bucket will change.1281

But for this case, the 1m long rope, we have to have 3.13 m/s or greater to be able to keep the water from sloshing out.1288

Last example: Say we have got a rock of mass 1 kg, attached to a string of length 0.5 m.1297

So, r = 0.5 m, and we have got the rock up here, 1 kg.1304

And the string snaps at a tension = 75 N, is our snap, that is when the string all of a sudden will snap.1312

So, we very slowly increase the speed of the rock traveling in a vertical circle, until the string snaps.1321

Starts off, going through slowly, then faster and faster, then snap!, all of a sudden it snaps.1329

Let us think about, at what point on the circle will the string snap.1335

Where does it snap?1339

Let us do a quick free body diagram.1341

At the top, we have got mg, and say for ease that (just to understand it graphically), we have to have a full length that is equal to the length of the radius.1343

In this case, we put these two vectors together, we get some tension pulling into the centre, some gravity pulling in to the centre, when it is at the top.1358

What is at the side, we have got gravity not really having an effect, because it is going perpendicular, so it is not doing much to us right now.1367

But we still got that tension, so now the tension has to go all the way in.1375

Clearly, when you are on the side, you are going to have more tension, then when you are at the top.1380

What happens when you are at the bottom?1384

Now, we have got mg pulling down, but we also got to have a tension, that is able to make up for mg, so we can still maintain the same centripetal force.1386

To maintain the same centripetal force at the top, gravity is working with us, it means we need less tension in our string.1397

When we are on the side, gravity does not have any effect, it means that we need just the same amount of tension that we needed as our centripetal force.1405

When we are at the very bottom, gravity is going to work the most against us, and so the tension is going to be the maximum.1412

So, string snaps at bottom.1419

String snaps at the bottom of the circle, because that is when gravity, and tension are going to be butting heads.1424

The tension has to be the biggest, because that is when gravity is working against us.1431

What speed we will have to go?1435

Now we have that centripetal force, = the net force.1437

The sum of the things, because centripetal force is only created from other things, centripetal force = net of the forces.1449

What forces are acting on the rock?1455

We have got gravity pointing down, and we have got the tension pointing up.1458

In this case, we will make up positive, so, tension - mg .1464

That is what the net force is.1469

Centripetal force = mv2 / r1471

We know pretty much what all these numbers are, except, what is tension?1482

We are looking at the instant of snapping, because we very slowly sped up to this, if we sped up to this suddenly, we might have accidentally put on a 75 N tension at a different spot.1485

That is what we had to speed up slowly, so we can be sure that it would snap at the bottom.1494

We have got that tension = 75, sub things in.1498

So, at the instant of snapping, the tension 75, we know what mass is, we know what gravity is, we know what the radius is, now we just have to figure out what v2 is.1503

So, v2m / r = 75 - (1 × 9.8) , move things around, we get, v2 = 0.5 × (75-1.98) / 1 , v2 = 32.6, take square root of both sides, we get that the speed of snapping is going to be, v = 5.71 m/s.1514

Once we get faster than 5.71 m/s, it will snap at the bottom of our circle.1585

As soon as we get to 5.71 m/s, and the rock passes through the bottom, that is the, just enough tension to create 75 N pull in the string, and snap it.1590

Hope everything made sense, hope you learned a lot.1600