Kamehameha Schools 2010-2011
Circular Motion
  • Intro
  • Kinematics
  • Centripetal
  • Dynamics
  • Statics
  • Exams
  • Homework
  • Labs & Activities
The wheel is at the center of our "modern" society and everything we have revolves around it.

In our mechanical world, it is probably the most important mechanical invention of all time. Nearly every machine built since the beginning of the Industrial Revolution involves the principles of the wheel. 

It’s hard to imagine any mechanized system that would be possible without the wheel or the ideas of circular motion that we learn by watching the wheel.  From tiny watch gears to automobiles, jet engines and computer disk drives, the principle is the same.

The Wheel in Civilization

The wheel is so important that it is hard for us to imagine advanced civilations that did not use it.  But it seems there were many.  Either they were more advanced, or the wheel leads to places they did not wish to travel.  Wheels may be a step backward.

Why? Well, it seems that the wheel just does not exist in the natural (animal) world on earth!  Physics is a natural science and seeks to explain natural motion.

Later on we'll learn about circular motion of the planets and sub-atomic particles, but as far as mother nature here on earth, she either didn't like wheels or just didn't need them!

It appears that the creation of the wheel was done by man.  Why?  If the natural world doesn't need wheels, why do men?

The wheel appears to have been a latecomer to "civilized" cultures and does not appear to have made much of an impact at all on cultures that the western world calls "uncivilized." 

Most "stone-age" peoples of the "new world" had little use for the wheel in any capacity, but it would probably not be accurate to assume that they were any less civilized than "old world" cultures.

Hawaiians among others had a very highly developed society and were able to travel long distances over the open seas without any recourse to the wheel.

Walls Constructed out of stones with tolerances greater than modern technology can achieve.

The Inca (or pre-Inca among others) are famous for having been able to construct buildings with massive stone blocks (many of which cannot be moved today) with tolerances that cannot be achieved with today's technologies.    

The ancient pyramids remain a mystery.  Since "modern" man cannot conceive of methods to construct them without a wheel of some type, most theories rely on rolling the stones over circular objects that functioned like wheels (dragging the blocks over trees, etc.).

Virtually all modern transportation systems are based on the wheel. All electrical power generation and distribution systems need a wheel, our information systems are still dominated by the wheel.  Future ones may not be.

So is the wheel a symbol of an advanced civilization, or is it a limiting idea?














































Wheel of Fortune: More than just a game show.

The Wheel of Fortune, or Rota Fortunae, is much older than Pat Sajak. In fact, the wheel, which the goddess Fortuna spins to determine the fate of those she looks upon, is an ancient concept of either Greek or Roman origin, depending on which academic you talk to. Roman scholar Cicero and the Greek poet Pindar both reference the Wheel of Fortune. In The Canterbury Tales, Geoffrey Chaucer uses the Wheel of Fortune to describe the tragic fall of several historical figures in his Monk’s Tale. And William Shakespeare alludes to it in a few of his plays. “Fortune, good night, smile once more; turn thy wheel!” says a disguised Earl of Kent in King Lear.

Read more: http://www.smithsonianmag.com/science-nature/A-Salute-to-the-Wheel.html#ixzz19GNBmDTX

“Breaking on the wheel” was a form of capital punishment in the Middle Ages.

Read more: http://www.smithsonianmag.com/science-nature/A-Salute-to-the-Wheel.html#ixzz19GNikZOb

The earliest wheels in North America were used for toys.

In the 1940s, archaeologists unearthed wheeled toys—ceramic dogs and other animals with wheels as legs—in pre-Colombian layers of sediment in Vera Cruz, Mexico. The indigenous peoples of North America, however, would not use wheels for transportation until the arrival of European settlers.

Read more: http://www.smithsonianmag.com/science-nature/A-Salute-to-the-Wheel.html#ixzz19GOC1L2S

The term “fifth wheel” comes from a part that was often used in carriages.

By definition, a fifth wheel is a wheel or a portion of a wheel with two parts rotating on each other that sits on the front axle of a carriage and adds extra support so it doesn’t tip. But it’s superfluous, really—which is why calling someone a “fifth wheel” is a way of calling them unnecessary, basically a tagalong.

Read more: http://www.smithsonianmag.com/science-nature/A-Salute-to-the-Wheel.html#ixzz19GOpgWvd


How the bicycle ruined enlightened conversation.

As reported in the New York Times, an 1896 column in the London Spectator mourned the impact of the bicycle on British society: “The phase of the wheel’s influence that strike …most forcibly is, to put it briefly, the abolition of dinner and the advent of lunch….If people can pedal away ten miles or so in the middle of the day to a lunch for which they need no dress, where the talk is haphazard, varied, light, and only too easy; and then glide back in the cool of the afternoon to dine quietly and get early to bed…conversation of the more serious type will tend to go out.”

Read more: http://www.smithsonianmag.com/science-nature/A-Salute-to-the-Wheel.html#ixzz19GP7fgVN

In movies and on TV, wheels appear to rotate in reverse.

Movie cameras typically operate at a speed of about 24 frames per second. So basically, if a spoke of a wheel is in a 12 o’clock position in one frame and then in the next frame, the spoke previously in the 9 o’clock position has moved to 12 o’clock, then the wheel appears stationary. But if in that frame another spoke is in the 11:30 position, then it appears to be revolving backwards. This optical illusion, called the wagon wheel effect, also can occur in the presence of a strobe light.

One man actually succeeded in reinventing the wheel.

John Keogh, a freelance patent lawyer in Australia, submitted a patent application for a “circular transportation facilitation device” in May 2001, shortly after a new patent system was introduced in Australia. He wanted to prove that the cheap, streamlined system, which allows inventors to draft a patent online without the help of a lawyer, was flawed. His “wheel” was issued a patent.

Read more: http://www.smithsonianmag.com/science-nature/A-Salute-to-the-Wheel.html#ixzz19GPfUiDr



To study circular motion, we will need to have a way to determine how many times a circle spins (or how many times you traveled around it), and how fast and in which direction it is spinning.

You have done that in previous classes by using angles measured in degrees.  Now we are going to measure angles a different way, we are going to use the radius of the circle itself to measure the angle that the circle turns. 

Summary - What we're gonna do!

Measure angles using the radius of curvature.  Angles measured this way are called radians.

Convert the kinematic equations into their angular forms.  All the  same equations still work, but we are going to be talking about how far or how fast something spins.

Convert from linear kinematics to angular kinematics using the three equations that relate linear and angular properties.

Do lots of angular kinematics!

Let's get started.

Circles, Angles, and Radians

So, if you walk around a circle, how far did you walk?

You'll probably answer something like "it depends on how big the circle is."

But does it?

If you really are trying to mess with the physics teacher, you'll say ZERO because you are thinking about linear displacement.  And you would be right, except for the fact that I asked about how far (a distance) and not for your linear displacement.

Since this section is about circular motion, I might also ask you what your angular displacement was.  How much angle did you walk?  That is definitely not zero, unless you walk back around again (the opposite way).

Angular displacement is a vector, it tells us how much we (or whatever) have rotated since the start of the problem.  We usually use radians to measure angles when we are talking displacement, so it's time to learn about radians.  Work through the next animation to learn what a radian is.

So, the idea is really pretty simple.  A radian is just the distance you travel along the curve (arc length) divided by the radius of the curve. 

Definition of a Radian
Arc length (s) divided by the radius (r).

One radian is the angle that a radius makes along the curve (arc).

One radian is the angle that a radius makes along the curve (arc).

So if I ask you how far (in radians) you traveled by walking around a circle, it does not depend on how big the circle is.  The angular displacement is the same for all circles.  To put it simply, you walked all the way around.  The answer is the same for both circles; one revolution, 360 degrees, or 2π radians.

On the other hand, if I ask about linear displacement, then you obviously travel farther around circles that have a bigger radius.

One more thing to note.  A radian is a length divided by a length.  The basic dimensions cancel which means that a radian has no basic dimensions.  It is just a number.

... a radian has no basic dimensions.  It is just a number.

Linear and Angular Variables

Now, since s is a length, we can use the formula for a radian to help us make the kinematic equations for circular motion. 

Before we do that, let's make some letters/symbols that represent angular displacement, angular velocity (how fast is it spinning), and angular acceleration (is it spinning faster or slower).

Symbols for Angular Displacement
Velocity and Acceleration


The three letters we'll use are theta-θ, omega-ω , and alpha-α.

Let's start with the first one.  The greek letter theta-θ is normally used to indicate an angle.  Our angles will usually be in radians, so on the right I have the formula for radian measure (s / r).  The letter s is a linear displacement.  I know what you're going to say.  I used d or x for linear displacement in the past, so why am I using s now? 

It's simple!  S is a curvy letter, and we are dealing with curves!!!  Got it?

Now that you know how s is related to θ, you can use the formulas for angular velocity (omega), and angular acceleration (alpha) to create the relationships between linear and angular displacement, velocity, and acceleration.  Here they are.

Formulas to convert from Linear Kinematics
to Circular Kinematics

Sooo then.  All you gotta do to convert from angly stuff to straight stuff is slap it with a radius. (Uh, I mean multiply it.)  That should make sense.  The radius is the straight thing that creates a curvy one.  Each of the properties on the left above is a linear one.  It is also called tangential because they point in a direction that is tangent to the circle.  So when you hear the words tangential velocity, tangential acceleration, and tangential displacement, you know that we are talking about vectors that are parallel (or anti-parallel) to the direction of the motion AT THAT INSTANT in time!

One more thing here.  These angular properties are vectors!  Remember those?  They have magnitude and direction.  The magnitude tells us how fast it's spinning, how much it did spin and if it's speeding up or slowing down.  The direction tell us which way it's spinning or which way it's changing.

We will still use arrows to represent vectors, but these arrows are going to seem a little bit weird at first because they are not in the plane of the thing that is spinning.  Instead, they point along the axis of rotation - kind of just out into space!  In order to know which way the vector is pointing, you will need to use the right hand rule.  (Watch the rotaional vector video in the sidebar.)

Angular Kinematic Equations

And now, for the grand unveiling of the new kinematic equations for circular motion. . .    We'll do this by just substituting the new letters into the old kinematics equations.

Linear Kinematics formulas on the Left
Angular Kinematics formulas on the Right

Wow, no big surprise there huh. All we did was change the letters to greek.  Hey, it's all greek to me.

None of the basic ideas have changed.  If you understood linear kinematics, you should be able to do this too.  Everything we said about averages and instantaneous values for linear kinematics is also true for circular kinematics.

And now we can do problems that ask us to convert back and forth from circular to linear motion.  Wow, good fun!







































So how do objects move in circles?  Is there a special kind of circular force or something?

And if it is moving in a circle, how do you keep it moving that way?

Summary - What we're gonna do!

Explain how object's that move in circles are "seeking the center" of the circle because of a center-seeking force.

Use the center-seeking force to solve lots and lots of circular motion problems.  These will include revisiting Netwon's laws and drawing free-body diagrams.

Let's get started.

Centripetal Acceleration

To put it as easy as I can, things don't move in circles unless there is acceleration going on.  Why?  Because velocity is a vector.  It has magnitude and direction.  I've said that a million times now, and I'll probably say it a million more. 

If you are moving in one direction, you'll keep moving that way unless you turn.  Pretty simple huh?  Well if you turn, you just accelerated, even if you didn't change your speed. 

Things that are moving in a circle (including all the molecules in a wheel that is spinning), are constantly turning which means that they must be constantly accelerating.  The question is which way? 

So the acceleration - centripetal acceleration - is toward the center, toward the point around which the things rotate.  It's like they are constantly trying to get to the center but never make it there!  Don't forget that. 

There is no such thing as centrifugal acceleration (or centrifugal force) that makes something go in a circle. 

There is no such thing as centrifugal acceleration (or centrifugal force) that makes something go in a circle.

The formula for centripetal acceleration can be written 2 ways.  One involves the linear velocity (tangent to the path) and the other involves the angular velocity.  The derivation of the formula involves some basic geometry and similar triangles and is in the sidebar if you're interested.

Formula(s) for Centripetal Acceleration

Remember, this is a vector.  The direction of the vector is constantly changing in circular motion but it always points to the center of the circle (or of the rotation).

Centripetal Forces

Acceleration is caused by force.  You have to force things to revolve in a circle, they don't seem to do it willingly.

The ball on the rope is held in circular motion by the tension in the rope.  It is this rope that is causing the circular motion.

Newton's second law says that net force is the product of mass times acceleration. 

Circular motion is caused by real forces that you can identify, like a rope tied to a rock that you swing in circles, or the force of gravity causing the moon to go around the earth.  These are real forces (tension and gravity) and they are causing circular motion.




What I am trying to say is that circular motion is a NET force caused by one or more real forces.  It is not a new magical force.  Since it is a NET force, we can write an equation for it by using Newton's second law.  Here it is.


Formula(s) for Centripetal Force

We call the net force that makes something move in a circle a centripetal force.  I'll say this again, centripetal force is a NET FORCE!  It is not a new magical circular force!

We call the net force that makes something move in a circle a centripetal force.  I'll say this again, centripetal force is a NET FORCE!  It is not a new magical circular force!

Since it is a centripetal force, its effect is to push or pull objects toward the center in order to cause circular motion.  Circular motion is not caused by an outward force that some people call centrifugal force.  In fact, there is no such thing as a centrifugal force that pushes you to the outside of a circle.

Things that appear to be forced to the outside of a circle are just moving in a straight line like they're supposed to.  When you feel pushed to the outside of a turn in your car or on a carnival ride, you are moving straight.  Everything else is experiencing centripetal acceleration toward the center which makes it appear like YOU are being pushed to the outside, but it just ain't so!

If you want to make something that is moving with a velocity v move in a circle, you will have to push or pull on it with a force that is equal to the square of that velocity divided by the radius of the circle you want to have.  If you don't, it will not move in circular motion.

Applications of Centripetal Force

There are a lot of classic problems that test your understanding of centripetal forces.  I have grouped them into a few basic categories below.


Centripetal force is not a force that belongs on your free body diagram.

Why?  Because it is the NET force.  It is the force that is the result of the real forces that should be on your free body diagram.  Watch this ki`i training show.

Once again, the centripetal force is the result of the real forces.  There is no magical "circular force."  Centrifugal force is an illusion.  You only think you are being forced to the outside of a circle because you think the circle is natural.  It is not.  It requires a force to go in a circle, not to leave it.

Centrifugal force is an illusion.  You only think you are being forced to the outside of a circle because you think the circle is natural.  It is not.  It requires a force to go in a circle, not to leave it.


The axes on your free body diagrams should be setup so that the NET CENTRIPETAL FORCE is in the same plane.  Just because there is a slope does not mean you should rotate the axes. You should only rotate the axes if the motion is not in the same plane as one of your coordinate axes.


These types of circles can cause the object to fall (inside loop) or fly off the  circle (outside loop) if there is not enough energy (inside loop) or if there is too much (outside loop).

The critical energy is the same for both types of loops, but  the result is  not.


Everything that moves in a circle has a net force that is pulling toward the center.

Since the NET force must account for the speed of the object and the radius of the circle, moving too fast will cause the object to break free of the force and travel outside the circle.  Not only that, moving in too small of a circle will also cause the object to break free of the force and travel outside the circle.

On the other hand, moving too slow will cause the circle to shrink or break down altogether.

The rope is creating the centripetal force.  If you try to make the ball go in a circle too small or too fast, the rope will not be able to hold onto it.

Here we have the boy swinging a ball on a rope again.  The rope is under tension.  It is pulling the ball toward the center of the circle.

The faster he swings the ball, the stronger the force must be to keep it in a circular path.

Eventually, if he swings it too fast, the rope will not be able to hold the ball and it will break allowing the ball to go sailing off.

As the rope rotates, each part of the rope pulls on the parts next to it causing it to rotate in circular motion.  Since all parts of the rope have the same angular speed, the parts that are closer to the ball experience a greater tension than the parts closer to the boy's hand. 

Demolition of a smokestack in Kentucky.

This chimney was blasted at the base.  It began to rotate toward the ground.

Each brick of the chimney was bonded to the one next to it.

The bricks farther away from the rotation point will have to bond with a greater force in order to keep the chimney rotating in one piece.

Eventually, the increasing angular speed will cause some of the bonds to break because they can no longer provide the force required to maintain circular motion.












Object's in motion will stay in motion . . .

Remember that?  Well, it's true of rotating objects too.  If they are spinning, they are going to keep spinning unless something stops them, and if they aren't spinning, they won't unless something makes them spin.

This is your old friend Newton.  He's back again to make all your dreams come true (or at least make them spin).

Summary - What we're gonna do!

Use Newton's Second Law for rotational motion.  If something changes it's angular velocity, there must be a reason, we'll call it torque whose direction can be found with the right hand rule.

Explain why somethings are easier to spin than others and how it's related to the shape and the way you're trying to get it to spin - moment of inertia.

Solve problems involving rotational kinetic energy and combinations of rotational and linear kinetic energy.

Work with angular momentum and changes in angular momentum that are caused by outside torque.

Solve problems involving centripetal force.

Newton's Second Law Again - Torque

The last section was about the force required to get something to move in a circle.  We were mostly talking about single objects moving in circles.

In this section we are going to talk about spinning things.   Centripetal force is not going away, but we are going to assume most of the time that the internal bonds inside of the spinning object are strong enough to keep it from breaking apart.

Having said that, let's move on.    

We learned that an object in motion with a constant velocity (or at rest - remember it's the same thing) will not change unless there is some force from the outside that  causes it.  The same is true for rotational motion.  To get something to spin, or to cause it to speed up or slow down its spinning motion, we need some kind of rotational force; we call that rotational force torque.

To get something to spin, or to cause it to speed up or slow down its spinning motion, we need some kind of rotational force; we call that rotational force torque.

Just like we did in the kinematics section, we're going to use the linear formulas to help us understand the rotational ones.  So here's Newton's Second Law for rotating objects compared to the same law for linear motion.

Newton's Second Law for Linear Motion (Left)
and for Rotational Motion (Right)

Remember that mass was something that resisted a force trying to cause acceleration.  Bigger masses just meant you had more inertia (more resistance).  In the same way, we are going to make up something like mass for rotation.  The letter I in the formula is called the moment of inertia; you can think of it as a kind of rotational mass - a resistance to being rotated (old science guys like to use the word moment when they talked about rotating things).  You already know what the letter alpha stands for (go back to the kinematics tab if you don't), so the formula basically says that a net torque will cause angular acceleration.  How much acceleration it causes depends on the moment of inertia.

Time to back up a little.  All this talk about torque and making things spin when we haven't even learned what would possibly make something spin. What is torque anyway? 

To spin an object, you have to apply a force to it, but not just any force, the force has to be some distance (at least a little bit) away from the place you are trying to spin it around (otherwise known as the axis of rotation or fulcrum).  Try to open a door by pushing on the hinges.  It won't rotate! You are not creating torque that way.  In order to get it to rotate, you need to apply the force a finite distance (that means not zero) away from the axis or hinges.

That distance is called the lever arm; it creates leverage, and we use the letter r to represent it in the definition of torque.

Definition of Torque

We've seen these kinds of formulas before.  The torque depends not just on the size of the lever arm (r), but also on the size of the force and the angle between the force and the lever arm.  You create the greatest torque when you push perpendicular to the lever arm (the sine of 90 degrees is ONE) and you create zero torque if you apply the force parallel to the lever arm (the sine of zero and 180 is ZERO).

We'll use this formula quite a bit from here on.  It's going to be one of the formulas you need when you are calculating net torque using Newton's Second Law for rotational motion (above).

So, let's make sure we got the idea.

Torque is a "spinny" force.  It makes things spin - which is circular motion.

But torque is different than the forces we have been learning about, because torque depends not only on how big the force is, but WHERE it is.  Where do you need to push to make something spin?

... torque depends not only on how big the force is, but WHERE it is.  Where do you need to push to make something spin?


Calculating Moment of Inertia

Moment of inertia is like reverse leverage.  It may be easier to make something rotate by increasing the lever arm (pushing with a force farther away from the axis of rotation), but some of that scenario depends on just where the mass is located.  If you build some kind of weird door with most of its mass near the hinge, it will swing open and closed (rotate) pretty easily, but if you put the mass farther away from the hinge, it will become more difficult to rotate.

Objects resist torque not only based on how much mass they have, but also where the mass is located.  When the mass if farther from the axis of rotation, it is more difficult to rotate than when it is closer to the axis of rotation.  The moment of inertia (resistance to torque) increases as the mass gets farther from the axis of rotation and decreases as the mass gets closer to the axis of rotation.

And of course, there's a cool formula to calculate all this.

Formula(s) for Moment of Inertia

This formula is for the moment of inertia of several discrete (individual) masses.  The r in this case is the distance from the axis of rotation (its a distance, so it's always positive).

If you want to know the moment of inertia for a single solid object, you would have to add up the  moments of inertia for all the atoms and molecules in it.  You cannot do that right now, but when you learn calculus, you will understand how can approximate it.  For now, you'll just have to believe what I tell you about the moments of inertia for solid spinning objects (physics people call them rigid bodies).  Here are some of them.  You don't have to memorize them, but you should look through them so that you understand what we physics types are talking about.

Moments of Inertia
The moments are relative to the axes shown.


Rotational Kinetic Energy

If an object is moving, it has kinetic energy.  Well, spinning or rotating is motion, so spinning/rotating things have kinetic energy.  And just like we've done so many times before, we are going to introduce you to the equation  by comparing it to the one for linear kinetic energy.  Here it is.

Linear and Rotational Kinetic Energy

So remember, energy is given to things when forces do positive work on them.  It is taken away when forces do negative work.  You probably can guess where this is going, but you'll need to do some WORK to understand it.

Torque also does work by making things spin faster or slower! 

Oh, wait, I can guess what's coming next. 

We are going to have a formula for work done by a torque that looks like the one for work done by a force with the names changed from linear ones to angular ones, aren't we?

Yes we are, and here is that one.

Work done by a Force
and Work done by a Torque

And there is power too! 

How much energy can you deliver in a specific amount of time?

Power delivered by a Force
and Power delivered by a Torque

Although it looks like a bunch of new equations, it is not.  All we are doing is revisiting what we have already learned.

Oh, btw, all the energy theorems still WORK here! Why wouldn't they?

Can things have both linear (also called translational) kinetic energy and rotational (or angular) kinetic energy at the same time?

YES!  They can.

Angular Momentum

Rotating objects have momentum, even if they don't appear to be going anywhere.  Doing the whole comparison trick again, here is the formula for angular momentum.


Linear and Rotational (Angular) Momentum


We use the letter l.

Since omega is directly related to linear momentum, we can also calculate angular momentum like this.

Angular Momentum and
Linear Momentum


where r is the distance from the axis of rotation to the mass that is rotating!



In physics, a lever (from French lever, "to raise", c.f. a levant) is a rigid object that is used with an appropriate fulcrum or pivot point to multiply the mechanical force (effort) that can be applied to another object (load). This leverage is also termed mechanical advantage, and is one example of the principle of moments. A lever is one of the six simple machines.

Contrast torque (spinning objects) with centripetal forces!!!  Make sure it's clear!

There are things that spin (torque) and there are things that move in circular patterns (centripetal force).


Uniform Circular Motion

Define Uniform Circular Motion.  Do not relate it to sine and cosine functions until the waves units.



What if you don't want rotation or translation?

What if you don't want it to move linearly or to rotate, but you just want it to stay still?

Most architects need to know how to build something that doesn't move.  That means not at all!  Not linearly (translational) or angularly (rotation).

You want it to stay still.  Not to move, to be STATIC!


If an object is not accelerating linearly or angularly, it is in equilibrium.  In other words, if it's not getting faster or slower or spinning faster or slower then it is in equilibrium.

For the most part, we are going to be talking about two kinds of equilibrium; stable and unstable equilibrium. The words stable and unstable are used in a lot of different ways in science.  In this section, we are talking about it's mechanical stability (not it's radioactive or chemical stability).  It's probably easiest if we just take a look at some examples.

The left ball has the lowest possible potential energy for this situation.  It would have to gain energy somehow to move away from its position.  The ball on the right has lots of stored potential energy.  If you blow on it even just a little bit.  The potential energy will be converted to kinetic energy causing it to move far from its original location.

In this picture, there are two objects in equilibrium.  One of these is stable and one is not. 

Why is the left ball and rod more stable than the one standing up on the right?

The answer has to do with energy.  The more stable configurations have the lowest energy.

You can think of them as being kind of trapped because they don't have any energy to get away or to move anywhere.  If they had more energy, they might be able to use it to move (or change their configuration).

This one was easy.  We can all look at it and understand that the left picture is more stable, what about a more complex situation?




Let's take a look at a rigid body (a solid) that does not have all of its mass at a single location.  How can we tell if it is in a stable or unstable arrangement?

In this picture, we have a rectangle.  It could be just a piece of wood.  We are going to hang the rectangle from a point up near the top and push on it to see what happens.

Rectangle Hanging from a Pivot Point

The rectangle on the right swings back and forth around its original position. The center of mass swings upward (increasing its energy) when it leaves the bottom position but it will eventually fall back down. You can think of the center of mass as a ball in a well (picture on the left). You can bump it a little, but it won't go too far from its original location. This is a stable configuration.

Now lets try that again using a different pivot point.

Rectangle balanced on a pivot near its lower edge.

Now the rectangle is free to pivot around the lower point. 

If I bump it even a little, gravity will cause it to rotate until it is hanging from the pivot point instead of being balanced on top of it.  You can think of the center of mass as being perfectly balanced on top of a hill.  It will stay there until it gets a small disturbance.  After it rolls off the hill, it won't roll back up to its original location.  This is not a stable configuration.

Finally, we could pin the pivot point right at the center of mass.

Rectangle free to pivot about its center of mass.

If we do this, the rectangle will rotate.  It might come back to its original position and it might not. This is not really stable or unstable.  It is uncertain or indifferent.

So here's the deal with rigid bodies.

If you push on it, think about what will happen to the center of mass.

If a small (and I mean small) impulse will cause the center of mass to rise, then it is a stable equilibrium (like a ball in a well).  If the small impulse causes the center of mass to go down (like a ball on a hill), OOPS, you just released the potential energy and who knows what will happen (in other words, this was not a stable equilibrium.

If a small .... impulse will cause the center of mass to rise, then it is a stable equilibrium ...

If the small impulse causes the center of mass to go down ...  this was not a stable equilibrium.

Now the obvious question is . . .

When is something in equilibrium?  (Didn't you think that was obvious?)


Conditions For Equilibrium


The Conditions for Equilibrium



































Circular Motion Exam 1

Circular Motion Exam 2















Circular Kinematics

Problem Set A  (MS Word Document)

Problem Set A Answers

Centripetal Acceleration and Force

Problem Set B  (MS Word Document)

Problems Set B Answers

Dynamics - Torque    Energy    Momentum

Problem Set C


Problem Set D



Grading Codes

Lettered Codes on your homework have the following meanings:

  1. A: Neatness
    Your work must be neater.   Period.
  2.  B: Steps
    There are 6 problem solving steps.  If you skip one, it will be listed as B.1, B.2, etc.  The most common one to skip is B.6.  Don’t do it.
  3. C: Units
    Once you put numbers on the paper that demand units, you must have the units.
  4. D: Not a valid formula.
    This means you did not start with the formulas taught in class.  You started with a hybrid, one of your own invention, or one that had already been through several steps of transformation.
    These will be coming more often as the course wears on.  Your ability with math is ALL IMPORTANT. 
  6. F: Incorrect Answer.
    Not as big a problem as the other issues at the beginning of the course.  It will carry more weight as time goes on.
  7. G: Unfinished.
    This is just me noting that I do not believe you finished the problem.  Don’t let that discourage you if you did some or most of the steps, I will see that and adjust accordingly.
  8. H: More.
    The place I put this means you need more explanation of what you did.  I am not saying it’s wrong, I just don’t believe it drives to the conclusion for whatever reason.

You may receive other comments on your homework, but those above are so common that I do not have time to write them on everyone’s papers.






































































































Labs and Activities

Free Body Diagram Drawing

Cart Acceleration

Friction acceleration


Inertia labs activities demos



Circular Motion
Kamehameha Site

Answer:   0.40 radians

Answer:   250 meters

Remember that radians have no dimensions.  You can write the word to help you remember what's going on when you convert units, but they have NO dimensions.

Answer:  20 radians or 3.2 revolutions

Angles have no dimensions, but you can still convert from system of angles to another.  You just need to remember that they have no dimensions when you are converting to linear properties.

Answer:   22 million revolutions

Answer:   140 million radians

See the solution to the first question!

Answer:   (6.4,  2.7)

In polar coordinates the first coordinate is the radius.  The second is the angle.  We are measuring angles in radians so the second coordinate is in radians.

Much depends on your ability to picture this.  DRAW A PICTURE!!!!!

It depends on the radius of the circle.

If the radius is one, the equation is r = 1.

If it is 2, then the equation is r = 2.

If it is 3, then the equation is r = 3.

Think about this.  What is the equation for a horizontal line using Cartesian (x,y) coordinates?

Below are two possibilities.  Of course, there are many more.  Choose OR MAKE the one you like best!

The first one is one complete circle or revolution, the second one is only a half of a revolution.

OK, I can't believe I asked this question after the last one.  But what the heck? What should you do to the answers for the last question to answer this question?

Answer:   6 pieces

The 12 inch radius fits around the outside of the pizza circle 6 times before it doesn't fit anymore!

Answer:  3.36 inches

Since the radius fits about 6.28 (or 2*PI) times around the edge of the pizza, the last piece makes an angle of 0.28 (the other 6 were eaten already).

Answer:   430 m

Not that far for a submarine; not even half a kilometer.

The sub is at the  center of the circle (DRAW A PICTURE) which is why you are looking for the radius.

Since it is a small angle, the arc length (s) can be the same as the actual height of the ship (including the masts).

The sine of 1.1 radians is 0.89.

The angle is 1.1 radians.

The sine of the angle is more than 1.23 times larger than the angle itself.

NO! Do not use the small angle approximation!

(You can, but you will need to understand your error range by doing this.)

The sine of 0.010 radians is 0.010 (2 significant figures).

There is no apparent difference (with our accuracy) between the 2 numbers.

YES! Use the small angle approximation.

It is pointing toward you.  Use the right hand rule!

No.  The parts of the wheel spoke that ar farther from the center must be going faster because they travel a greater linear distance in the same time as the parts of the spoke that are closer to the center.

If you run in lane 8 of a track and I run in lane 1, you will have to run faster in order to finish in the same time.

Yes.  All points on the spoke complete their revolutions together so they have the same angular speed. 

If they didn't, spoke would stretch and probably break.

Answer:   73 rad/s

Answer:   12 Hz

Answer:   2.4 rev/day  or 28 micro Hz

Answer:   0.087 rad/s

Answer:   7.4 m/s or 27 km/hr

Answer:   0.22 rad/s2

Answer:   0.083 rad/s2

Answer:   (b)  constant angular velocity.

The linear or tangential velocity keeps changing (direction) it is not constant, but the angular velocity is.

Answer:   (d)  the radius of the path decreases.

The water is only spinning in a circle because some force is keeping it from flying out of the circle.

The force that is pushing it inward (toward the center) comes from the walls of the washer.

Since the walls have holes in them, some of the water will end up where the holes are and instead of being pushed back toward the center they will fly out (through the hole) with whatever linear velocity they had at that time.

The ball will float toward the inside of the circle no matter which way we rotate it.  Watch the video.

Content on this page requires a newer version of Adobe Flash Player.

Get Adobe Flash player

The ball is being pushed in the direction of the centripetal acceleration. It is toward the center.

We'll talk about how this works when we discuss apparent weight and accelerating reference frames.

Sorry, I forgot to video it going the other way, but it does the same thing.  Why wouldn't it? (Think about that.)

Content on this page requires a newer version of Adobe Flash Player.

Get Adobe Flash player

Answer:   79 rad/s2

Answer:   40 rad/s2

Answer:   320 rad/s2

Notice that this 4 times the first answer to this question.  Why is that?

The force is due to the static frictional force between the tires and the road.

When the car turns the tires (to the left) while it is moving forward, the jaggy forces in the road react to that and say NO! 

The jaggies push back on the turned tire which causes the car to TURN!

There is no force pushing them outward.

The word "outward" refers to the circle.

If you think the circle is "natural" then leaving the circle is not.

If you want to keep rotating in a circle, then you have to accept a force PUSHING (or) PULLING you into the circle.

On the other hand, from the perspective of one moving in a circle, objects that leave the circle APPEAR to be forced out.

The APPARENT outward force is not a real force, but if you believe in the circle, the APPARENT force is real to you.

It might even appear to be a form of GRAVITY!

Answer:   13 m/s2

Answer:   No.  There is no way.

The calculation below shows you that the acceleration that static friction can sustain is the coeffecient times g.  Since g is only 9.81 m/s2, there is no way a coeffecient that is less than one can cause an acceleration of 13 m/s2.

You should have figured this out from the Ki`i Learning.

Answer:  6.1 N·m  (CCW)

Answer:   490 N·m  (CCW)

Answer:   340  N·m  (CW)

Answer:   1400  kg·m2

Answer:   1.0 kg·m2

Just use the formula.  Don't forget to use the radius instead of the diameter.

Answer:   1.0 kg·m2

Nothing changes in the formula by swinging it in a vertical loop.

Answer:   22 kg·m2

2 significant figures.

Answer:   140 kg·m2

2 significant figures.

Answer:   The sphere.

Compare the moment of inertia for them.  The sphere has a smaller moment of inertia which means it will rotate faster and reach the bottom of the hill sooner.

Answer:   7000 N·m (CW)

You should be able to figure this out from the picture.

How did I come up with the frictional force in this equaiton?

Answer: 0.64 N·m

Look up the moment of inertia for a solid sphere.

Answer:   7.5 N·m