Physics 211
4-3-09
Friday
The average on the homework was an 8.3; I have not gotten to the extension requests.
If we have time we'll go over questions on it.
We've been talking about angular momentum, now we are going to start talking about spinning
and how it relates to conventional linear quantities.
I'm not going to be here on Monday, but your TA Dave will be here to talk about torque.
Chapter 10
Say you have a CD in your player and this is a
practical problem if you are designing players. We
want to find the angular position. You want to
know what the orientation of the CD is so the way
we measure where something is, in linear
quantities we measure position relative to
something. So we have to define an arbitrary zero.
If we want to know the angular position of a given
point on the CD, the angular position then is just defined as this angle, the distance between
the reference position and the point is that you are interested in is called the angular position.
We measure it in radians.
There are two pi radians in a circle. So two pi radians= 360 degrees.
Student: So the zero is the reference of the frame we're in?
You're right, its whatever you want it to be. It is whatever is most convenient for the problem
you are doing as long as you are consistent it doesn't matter what arbitrary zero you use.
If you want to figure the total distance you travel across a CD, recall an arc length is equal to
the angular position there, theta, times r, where r is the distance from the center of rotation.
We call this center of rotation, it rotates around a singular point in 2 dimensions. In three
dimensions it is a line of rotation.
Angular Velocity:
Similar to the way velocity is the change in position over time, this is the change of angular
position over time
This comes out in radians per second usually. You can do degrees per year.
If you want to figure the average angular velocity, you can take the position, you can take two
positions and two times and divide it by the final time minus the initial time. The average
velocity is the change in angular position over change in time. For a CD then, this is two
radians.
Remember when we talked about velocity, it is a vector. It has a magnitude and a direction.
What could you define as a direction for an angular velocity? We'll use this concept a lot.
There is a vector quantity angular velocity. We are rotating in an angular plane. The
direction of the angular velocity is perpendicular to the plane. We use the right hand rule.
You wrap your fingers around the direction of motion with your right hand, and your thumb will
point in the right direction. Make sure you use your right hand.
We go back and define a 3D coordinate system. The angular velocity will be in the positive k
hat direction, towards you in z.
Angular Acceleration
When you have a CD, it starts from rest and as the rotation increases, the angular velocity
increases and we find that:
This angle is the radians per second squared.
If you need to figure the average acceleration:
The direction if we have an angular acceleration vector, we define it to be perpendicular to the
plane and in the right hand direction.
If this CD is speeding up, then the vector is still in the same direction, but getting bigger. The
angular acceleration will be the same direction. If the CD were slowing down, it would be in
the opposite direction.
In your exam, use the right hand rule. It will help.
There are a lot of similarities between linear and angular acceleration. I'm going to make a
table:
We'll continue to look at this relationship over the next chapters. They are not the same. The
units are different. They are very similar, they have the same functional form, but they are not
the same thing.
Student: Why is it theta? Why isn't it S? Theta is just an angle.
Right. So if you want to transfer between positions, you have to convert it to a linear distance.
Its a matter of where the orientation of the rigid body that is rotating with respect to this quantity.
Recall back to chapter 4, we talked about linear motion. We do the same thing for angular
quantities. If you have a ball and you are dropping it in that gravitational field, you should be
able to derive expressions for angular velocity as a function of time.
So its like chapter 4 only we are making it rotational.
Say you have a wheel and it starts from rest and is accelerating 3.50 radians per second
squared. What is the angular position after two seconds?
So we use the equation over there;
The second question is how many rotations is that?
One rotation is 2 pi radians, or 360 degrees.
So angular position is the same whether you go all the way around once or just start here. But
if you want to find velocity, you want to use the full wrap around. If you just want to find
position it doesn't matter. If you are asked for an angular acceleration, you need to use the full
wraparound. If webassign makes it matter, let me know.
There are 6.28 radians per revolution. That's why the 7 radians is just over one revolution.
So what is the speed?
Who has a tachometer in their cars? Let's convert this to rpm as if it were the tachometer in
your car.
So as we were talking about earlier, say you want to move between angular and linear position.
We go back to our CD. Say we need to figure how far part of the CD travels. We have an
angular position theta, and want to know the linear position. We need to know r/, the distance
between the center and the arc length. The velocity is equal to the angular velocity times r/.
How do we figure the linear acceleration, then? If only it were so
easy. Any object in circular motion has a centrifugal force toward the
center and a tangential acceleration. We need to find both and add
them together.
We use the Pythagorean theorem.
I want to go over an example from the book. Back to the CD
player. Sometimes when you run your CD player, the frequency
of the hum is different. The way the CD stores information is there
is individual bits located in a circle on the CD. Some are very reflective, some are not. If it
receives a signal back, that's a one. If it doesn't, that's a zero. It reads in bits per second.
To read the same number of bits, recall that to shove the most information you can, it shoves
the same bits per centimeter as you go around. The final data rate is equal to the density of
bits per second times the velocity, which is omega times r. If omega is always the same, then
when r is small, you are reading fewer bits per second. What limits the speed of your CD is
not the rate of spin, but the limit is the electronics and how fast they can read the bits per
second. You have to spin it faster to read the inside. CDs are filled from the outside in. The
reason is that it has to spin slower.
So the question it asks is how fast does the disc have to revolve when read from inner to the
outer track?
If it takes 3 seconds per revolution for the innermost, what
speed would it have to go for the outermost? I think these
numbers are too slow. But maybe that's just data, not
songs.
Let's convert this to an angular velocity. We'll start with
one revolution in 1.3 seconds and we said before that in one revolution there is two pi radians.
That ends up being 6.28...
Student: Do vinyls have to speed up as they go to the center?
For a 33 and ⅓ record, it uses the same rotation rate, but the data is tighter towards the center.
It doesn't effectively use the area very well. It gets omega constant, and changes the storage
rate. The same song is thicker near the center on a CD. It uses the opposite assumption. It
uses constant angular velocity.
Kinetic energy of a Spinning Object:
If you want to figure the kinetic energy, it needs to be the same as if you took the kinetic energy
of all the parts and added it up.
The angular velocity and angular acceleration of every individual part is all the same. Omega
is the same for each of these particles. i is the total number of particles. On Monday this will
be an integral. I'm going to break this up:
Rotational Inertia
This is also known as the moment of inertia. The greater this i value is, then the kinetic energy
rotational is equal to
If you can store kinetic energy in a rotating object, you should be able to build a car where you
have a steel cylinder that you spin up fast enough and bleed that energy off over time. You
probably have to plug it in to spin it up. You have to turn this fly wheel really, really fast. And
if you crash, your fly wheel would come off and start killing people.