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The London Eye
This Ferris Wheel is along the Thames in London. The original Ferris Wheel was designed and constructed by George Washington Gale Ferris, Jr. as a landmark for the 1893 World's Columbian Exposition in Chicago.
How many machines using circular motion are needed to keep the passengers moving in a circle?
OK, lot's and lot's of rides use circular motion. . . .
How fun is that?
Is that the only use of circular motion? Fun?
Transportation
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?
References
http://www.ideafinder.com/history/inventions/wheel.htm
http://www.smithsonianmag.com/science-nature/A-Salute-to-the-Wheel.html
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
Rotational Vectors
The Right Hand Rule
If it is rotating clockwise, the rotational vector is pointing AWAY from you.
If it is rotating counter-clockwise, look out, because it is pointing at you.
Wiki Quizzes: Angular Displacement
A jogger is running on a circular track with a radius of 50.0m.
What is her angular displacement when she runs 20.0m?
How far has she run when her angular displacement is 5.0 radians?
What is her angular displacement when she has run 1.0 km? (Answer in both radians and revolutions.)
An automobile tire is guaranteed for 40,000 miles.
If the diameter of the tire is about 3.0 feet, how many revolutions will the tire make in it's lifetime?
What will it's angular displacement in radians be (assuming that it doesn't go in reverse).
The Cartesian coordinates of a point are (-2.5, 5.9).
What are the polar coordinates?
In Cartesian coordinates, the equation of a circle is x^{2} + y^{2} = r^{2}.
What is the equation using polar coordinates?
The radius of a circle fits around the circle exactly 2*PI times.
Create a conversion factor to change from radians to degrees.
Create a conversion factor to change from degrees to radians.
The pizza delivery person brings a 24 inch (diameter) pizza to your school for the radian physics party.
How many wedge shaped pieces can be cut with a crust length equal to the radius of the pizza.
What is the crust length of the remaining piece?
When angles are very small, the difference between the arc length and the chord length (b) becomes so small that we can ignore the difference.
In other words, if the difference is smaller than we can measure, we can assume that the two are equal.
This approximation makes a lot of scientific formulas much easier to deal with. It may seem like cheating, but it's not cheating if you cannot calculate the difference, and this situation happens in the real world often enough to use small angle approximations when necessary.
Wiki Quizzes: Small Angles
A submarine sailor using his persicope sees a ship on the surface. From past experience, he knows the ship is 15m tall. The periscope contains divisions that tell the submariner that the ship's height takes up 2.0 degrees of arc.
How far away is the ship?
Small angle approximations are usually used when the sine of an angle is pretty much the same as the angle itself.
If you measure an angle to be 1.1 radians (2 significant figures), can you use small angle approximations for problems that use this angle?
If you measure an angle to be 0.010 radians, can you use small angle approximations for calculations that use this measurement?
How fast does it go around the circle.
Well, how fast does it go aroung the circle?
Some gifted people can run around a 400m track in one minute. And, they can do it over and over (at least 4 times in a row).
That means they are running the track with an angular frequency of ONE rpm - one rotation per minute.
What do I mean when I say frequency?
Frequency means HOW FREQUENTLY, HOW MANY TIMES DOES IT GO ROUNDY ROUNDY; completely around! Each time it goes all the way around is called a REVOLUTION.
Now, think carefully. If it goes around 1 time every minute, then the frequency is 1 rpm or one rotation per minute. But wait, a rotation is 2*PI radians, so the frequency tells us how fast the object is rotating.
In other words, it is an angular velocity! It may not be in radians, but it still lets us know how fast is it spinning!
If you run around a 400m track 4 times in 4 minutes, how long does it take you to go around once?
The answer is obviously 1 minute. The PERIOD of time that it takes to complete one cycle, or one revolution is called the period (because it's the period of time.) See if you can figure out how the period is related to the frequency.
A cycle is a completion of a repeating pattern.
The pattern in circular motion is one complete A revolution. So in circular motion, one revolution is one cycle. (There are other kinds of repeating motions that are not circular.)
65 cycles means 65 revolutions in circular motion.
One cycle per second has a special name - it is known as one Hertz (or 1 Hz).
Our electrical power grid revolves 60 times per second - 60 cycles per second - 60 Hz.
Angles can be measured a lot of different ways. You probably learned degrees first. Then you learned rotations (how many times does it spin) and now you know about radians.
All of these usually show up as a way to describe angular velocity (frequency).
Common Angular Displacement Units
SI (mks) | radians | |
revolutions | rev | |
Cycles |
Common Angular Velocity Units
SI (mks) | radians/sec | s^{-1} |
rpm | revolutions / min | min^{-1} |
Hz | Cycles / sec | s^{-1} |
Common Angular Acceleration Units
SI (mks) | radians/sec^{2} | s^{-2} |
Wiki Quizzes: Angular Velocity
A spinning bicycle wheel is rotating counter-clockwise (CCW) when you look at it.
Which way is the angular velocity vector pointing? Up, down, left, right, toward you, away from you, or something else?
Do all points on the spoke of the wheel have the same linear velocity?
Do all points on the spoke of the wheel have the same angular velocity?
Your car engine's crankshaft is spinning at 700 rpm as shown on the tachometer.
What is the angular velocity of the engine in radians per second?
What is the frequency of the engine in Hz?
A satellite in a circular orbit has a period of 10h.
What is the satellite's frequency in revolutions per day? What about in Hz?
A car makes 2.5 laps around a circular track with a radius of 85m in 3.0 min.
What is the car's average angular speed?
What is the car's tangential speed?
Wiki Quizzes: Angular Acceleration
A merry-go-round with a diameter of 11 meters rotates once every 8.0 seconds.
If it takes 3.5 seconds to get up to speed (from rest), what is the average angular acceleration during startup.
After the ride is complete, the merry-go-round slows to a stop. If it rotates 3/4 of a revolution while stopping, what was it's average angular acceleration during this time.
Wiki Quizzes: Rotational Kinematics
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.
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).
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.
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.
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!
Circular motion like nothing you have ever seen (unless you're my age).
What? What is forcing them to move in a circle? Which way does it push them?
Le Rotor
Oh my gosh! These people stick to the walls too, just like the last video. Where can you buy these kinds of walls. How do they make them?
Physics books like to talk about something called uniform circular motion.
It sounds like its a mystery or something when it's said like that.
So let's keep it simple.
A particle moving in uniform circular motion moves in a circle at a constant angular velocity.
That means that an object that is spinning at a constant angular velocity has all of its molecules moving in uniform circular motion (the same angular speed).
It simply means going in a circle at a constant angular velocity! Not too hard.
Not a bad video. It is incorrect on the meaning of uniform circular motion, but the rest is pretty good!
Wiki Quizzes: Centripetal Acceleration
In uniform circular motion, there is
(a) a constant velocity.
(b) a constant angular velocity.
(c) zero acceleration.
(d) a non-zero tangential acceleration.
If the centripetal force on a particle in uniform circular motion is increased,
(a) the tangential speed will not change.
(b) the tangential speed will decrease.
(c) the radius of the path will increase.
(d) the radius of the path will decrease.
In a washing machine, the spin cycle is used to remove water from the clothes.
Explain how this works!
In the picture (mouse over) we have two jars with fishing bobs (floating) in jars partially filled with water. We are going to start the arms of the machine spinning.
Which way will the floats move? Will they move? Does it matter which way we spin the arms?
A psycho physics teacher swings a cup of water around on a plate attached to a rope as shown in this video. Assume the rope is 0.50 m long.
If he swings the ball around 2 times per second, what is the centripetal acceleration of the ball?
What if the rope was only 0.25 meters long. What would the centripetal acceleration be then?
OK, back to the rope being 0.50 meters long. If the nutso teacher swings the rope twice as fast (4.0 times per second) what is the centripetal acceleration?
Wiki Quizzes: Centripetal Force
For the car in this picture (mouseover) to move in a circle, there must be a force pushing it inward - the centripetal force.
Where does the centripetal force come from that allows the car to turn?
If the seats are slippery, the passengers feel like they are being pushed to the outside of the turn. Why do they feel this? Is there a force that is pushing them outward?
If the car moving at 83 km/hr goes around a turn with a 40.0 meter radius of curvature, what is the car's centripetal acceleration?
If the coeffecient of static friction between the tires and the road is 0.65, will the car make it through the turn without slipping?
In the previous problem, there had to be friction between the car and the tires in order to push the car inward toward the center of the circle. Sharp curves on highways and racetracks are often designed with banked turns (inclined) so that there will be a source of centripetal force even without friction.
Draw a free body diagram of a car in a banked turn on a frictionless surface (look at the front or the back of the car and have the bank going up to the left or up to the right).
15. Find the centripetal accelerations of (a) a point on the equator of Earth and (b) the North Pole, due to the rotation of Earth about its axis.
Why does the chimney break as it falls?
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.
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.
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 A NET FORCE
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.
HORIZONTAL CIRCLES AND FREE BODY DIAGRAMS
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.
VERTICAL CIRCLES - INSIDE AND OUSTIDE LOOPS
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.
TOO STRONG FOR THE ALLOWABLE CENTRIPETAL FORCE
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.
Blue Angels and Torque
Longer download time! Maybe you should read the lesson before watching this one.
The spins and banked turns performed by these jets require precise control of the torque which causes them to rotate about their longitudinal (the long one) axis.
Quizzes: Basic Torque
Wow, I caught a fish. The fish is pulling on the line with a force of 175 N.
This fishing reel is of the type shown. The string pulls on the outside of the circular reel.
If the radius of the fishing reel is 3.5 cm, what is the torque being applied to the reel?
In the picture above (at the top), a mechanic pushes on the propellor with a force of 200 N. Assume the diameter of the propellor is 16 feet.
If he pushes it sideways (to his right and perpendicular to the propellor), what is the the torque that he applies to the prop?
If he pushes at an angle of 45 degrees to the direction of the propellor blade (to his left), what is the torque he applies to the prop?
If he causes the propellor on the previous question to have an angular an acceleration of 0.25 rad/s^{2}, what is the propellor's moment of inertia (that's the I in the formula). Don't forget UNITS!
Moment of Inertia Hoops and Cylinders
The video demonstrates how the distribution of mass affects the moment of inertia causing some things to spin easier than others, even when everything else seems to be the same.
Wiki Quizzes: Moment of Inertia
A 0.5 kg ball is attached to a string that is being used to spin the ball in a horizontal circle with a diameter of 2.82m.
What is the moment of inertia of the ball about the axis of rotation (the hand holding the string).
If the ball is swung in a vertial loop instead of a horizontal one, what is the moment of inertia?
A 5.4 kg mass is at position (5m, 2m) using cartestian coordinates.
What is the moment of inertia about the x-axis?
What is the moment of inertia about the y-axis?
If I roll a spherical ball and a cylinder of the same mass and radius down a slope.
Which one reaches the bottom first?
Wiki Quizzes: Torque and Newton's Laws
An airliner lands with a speed of 50.0 m/s. Each wheel of the plane has a radius of 1.25 m and a moment of inertia of 110 kg∙m^{2}. At touchdown, the wheels begin to spin because of the torque caused by kinetic friction. The coeffecient of kinetic friction is 0.40.
If one wheel supports a weight of 1.40 × 10^{4}N, what is the torque applied to the wheels at touchdown?
A uniform spherical ball has a mass of 20.0 kg and a radius of 0.20 m.
What is the magnitude of the torque needed to make the ball spin with an angular acceleration of 2.0 rad/s^{2}?
A 10.0 kg solid disk of radius 0.50m is rotated about an axis through it's center point (perpendicular to the plane of the disk).
What torque is require to make the the disk attain an angular speed of 3.0 rad/s in 0.50s? Assume the disk starts from rest.
In the setup shown here (mouse over), m1 is 8.0 kg, m2 is 3.0 kg, and the angle of the inclined plane is 30 degrees. The radius of the pulley is 0.10 m and the mass of the pulley is 0.10 kg.
If you neglect friction, what the acceleration (linear) of the masses?
If the friction causes a torque of 0.60 Nm, what is the acceleration (linear) of the masses? (Hint: With friction, the tension in the 2 parts of the string is NOT the same! Think about why this is so.)
Calculate rotational kinetic energy.
Example Kii need to help here.
Lots of problems with rotational kinetic energy.
Simple Machines
(Hint: The angular equivalent of Wnet = FΔx = 1/2 mvf2 – 1/2 mvi2 is Wnet = τΔθ = 1/2 Iωf2 – 1/2 Iωi2. You should convince yourself that this relationship is correct.)
Pulley
Lever
Wedge
Wheel & Axle
Inclined plane
Screw
How much of DOT Force or how much of DOT torque?
You got dot? or not?
rolling advanced
There are things that move in circular paths (centripetal force), there are things that spin (torque), and there are things that move along a path and spin at the same time!
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.
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.
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.
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.
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.
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.
and Work done by a Torque
And there is power too!
How much energy can you deliver in a specific amount of time?
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.
We use the letter l.
Since omega is directly related to linear momentum, we can also calculate angular momentum like this.
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.
Balance Rock - Garden of Gods Colorado
The rock is balanced. It is in equilibrium, but it doesn't appear to be too stable does it. Even though it has been sitting there for aeons of time, if you disturb it enough, it will fall away and will not return to this position. It is an unstable equilibrium.
Kaliuwaa Falls - Oahu
Unstable rock formations may have been in place for millenia, but when the right conditions come along, they will topple over as they lose potential energy with no way to recover it. The base of the falls above is littered with large boulders that have fallen from the cliffs above.
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!
Equilibrium
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
Since this course is primarily about teaching you how to solve problems. I will tell you right now what some of the most important methods are.
First and formeomost, have a pencil/pen and paper. As you study and as you attempt to solve the problems, DRAW, WRITE, DOODLE, WRITE, DRAW, ERASE, CROSS IT OUT AND DO IT AGAIN!!!!
I cannot overemphasize the importance of you trying to picture the process by using your hands and a pencil or pen. Have lots of scratch paper handy. It is vitally important that you learn to form your own understanding and absolutely nothing serves as a substitute for you putting the pencil on the paper and drawing and writing what you understand. Even if it's wrong at first. ESPECIALLY IF IT'S WRONG AT FIRST!
Each section has quick (wiki) quizzes full of examples designed to walk you through the math involved in solving basic problems. The answers to these examples are available by dragging your mouse over the question.
BEFORE YOU LOOK AT THE ANSWERS, TRY TO DO THEM ON YOUR OWN! That is the point of example problems. If you cannot do them, then read through the solution. After you understand the solution, TRY IT AGAIN ON YOUR OWN, and keep trying until you get the hang of it.
For the written homework assignments that you will hand in, I insist on the following 6 step process. This is not the only problem solving process that you will find in physics books, but it is the one I want you to use for now. As you go through the course, you will develop your own skills and procedures that may work better for you. That is the whole point. Nonetheless, whenever the 6 step process is requested, you must use it. After you have completed this class, devise your own method and use it.
SIX STEP PROBLEM SOLVING
Those are the six steps. I'll go through them one at a time.
Step 1. What is the question? What is being asked? You should read through the question and determine exactly WHAT it is you are supposed to answer. Word problems can be long and sometimes confusing. In step one, you need to dig through all of that and get to the heart of the matter. When you have, WRITE IT DOWN! Don't write a paragraph, or even a sentence, just write down what it is that you are supposed to answer in as few words as possible. You have now focused yourself to the task at hand. You will come back to this step when you are done to see if you actually answered the question. This may be the most important step. It will allow you to cut to the chase in every subject, not just physics.
Step 2. Draw a picture. You need to picture what is going on. It is not about artwork, it is about coming up with some simple sketches that illustrate the geometry and the coordinate system of the problem. YOU HAVE TO FIGURE THIS OUT! This is where you will work out your attack. This is often where the question will make sense or not. Do not draw small, irrelevant pictures just to check off this step and do not attempt to conserve paper. Your mind needs the space on the paper for you to think it through!
Step 3. List the given information. Comb through the problem, the picture(s), the graph(s) to find information that you may need to solve the problem. It is best if you can label the picture with this information, if you cannot, put it here in step 3. If you decide your picture is not going to work, redraw it (go back to step 2) so that it will!
Step 4. Write down the formulas. After step 3, you should have some idea of how to proceed. If not, go back to step 3. In this step, write down any formulas that you may need to solve the problem. Do not try to solve it yet, just write down the formulas. Leave room for more formulas in case you realize later on that you forgot one or two. When you get this far, you have probably understood the physics necessary to solve the problem. Next comes the math.
Step 5. Solve the problem. Using the formulas you listed in step 4, work through the necessary algebra to find the solution. Try to find an algebraic solution (using letters instead of numbers) first if you can. It is easier to check your work for errors if you will use letters instead of numbers. Plug in numbers and units only when you have to in order to complete the probelm. This step is more math than it is physics. Your math skills will improve quite a bit during this course if you will follow these directions.
Step 6. Check your work. Go through the following checklist carefully. Write the word on the paper, think about it, and physically check it off.
Don't always say it is reasonable if it surprised you! You will not learn/discover anything if you are never surprised.
Circular Kinematics
Problem Set A (MS Word Document)
Centripetal Acceleration and Force
Problem Set B (MS Word Document)
Dynamics - Torque Energy Momentum
Statics
Grading Codes
Lettered Codes on your homework have the following meanings:
- A: Neatness
Your work must be neater. Period. - 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. - C: Units
Once you put numbers on the paper that demand units, you must have the units. - 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. - E: MATH SKILLS PROBLEMATIC
These will be coming more often as the course wears on. Your ability with math is ALL IMPORTANT. - 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. - 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. - 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.
Battery Touch
Chair Lift
MIT Lectures
Labs and Activities
Free Body Diagram Drawing
Cart Acceleration
Friction acceleration
Inertia labs activities demos