Dot Physics

I saw this video on several places. It shows an astronaut playing with a CD player.

I wish I were an astronaut. I would probably not stop throwing up though. It would still be worth it. You can only throw up so much right? (I know the answer to this question). Anyway, this is a really cool demo. Look at the first CD player that is on. When the guy taps it, it doesn’t rotate but rather it wobbles. This is a rather difficult concept, but I am going to try to give a reasonable explanation.

I will start with angular momentum. Angular momentum is sort of like momentum (linear momentum). Momentum is the thing that changes when a force is acting on an object. In the same way, angular momentum is what changes when a rotational force is acting on an object (rotational force is also called torque). Linear momentum is mass times velocity. One (not always correct) definition of angular momentum is “rotational mass” times angular velocity. Rotational mass is usually called the moment of inertia. Note that angular velocity (using the symbol ?) is a vector. Convention puts the angular velocity vector along the axis of rotation. If you put your right hand so that your fingers point in the direction of rotation, your thumb will be in the direction of the angular velocity vector.

Not sure if this picture helps, but here is a spinning disk.

i-6ef4ada36724b07ab865180b87ae08f8-omega-1.jpg

Note that I used vpython to make my 3-d drawings. Normally, I use Apple’s Keynote software. Unfortunately, it does not really make things in 3-d. Just saying. Ok – so what about angular momentum? In most introductory textbooks, you see the following definition of angular momentum:

i-6dc5dd2b4b7a61c926bde4a3631bb0db-l1.jpg

This is a fairly useful definition of angular momentum (L vector). Here I represents the moment of inertia (rotational mass). It depends on both the mass of the object and how that mass is distributed about the axis of rotation.

One other thing is torque. Torque is like the “rotational force”. The problem with torque is that it is inherently 3 dimensional in that it depends on the vector cross product. Torque is defined as:

i-dcbcd45ef0f3c05869cf57390b5a56ea-torque-11.jpg

Where r is the vector from the center of mass to the location the force is applied (in this case that is how I will define it). F, of course, is the force applied. In the vector cross product, the resultant (in this case, the torque) is perpendicular to both the r vector and the force vector. And, what does the torque do? It changes the angular momentum:

i-55099944b98453ec66d4629c1442fca9-torqueangularmomentum.jpg

So, let me look at the case where the astronaut pushes on the CD play with the CD player off. He taps it right below the center of mass and initially it was at rest with zero (vector) angular momentum. His “tap” produces a short torque. Here is my 3-D picture of this:

i-dbebb55aec060c3d99b218dc74911a6b-torque-23.jpg

The green arrow is the r vector. The blue arrow is going in the direction of the tap (it’s the force). This produces a torque (red arrow) that points from the center of the disk to the right. Which is the same direction as the CHANGE in angular momentum. Since there was no angular momentum before the tap, the new angular momentum is in that direction. This makes the CD player rotate with an angular velocity in that same direction. Note also that this tap force changes the linear momentum of the CD player also and makes it move backward.

Now, what happens when the CD player is on? The tap produces the EXCACT same torque. It also produces the same change in angular momentum. (and the same change in linear momentum) The only difference is that it already has an initial angular momentum. The result is that the new angular momentum is slightly “off axis”. Here is a diagram showing how the torque changes the angular momentum:

i-5f1c26c85001935ae51641924872bcf5-delta-l.jpg

So the new angular momentum is not along the axis of the rotating CD. Here is where the weird part comes in. If you force the axis of rotation to be something particular, you can easily determine a scalar value for the moment of inertia (I). However, if it is a free object (all objects want to be free), then it can rotate about any axis. In this case, it is not a simple situation. Really, angular momentum should be written as:

i-5090e6bf7ffd7820f3e5d4f69df5acac-screenshot-11.jpg

Where I is a tensor, not a scalar. Basically, this means that L and ? do NOT have to be in the same direction. The operation of I on the angular velocity has to be in the same direction (and magnitude) as the angular momentum. The result is the complicated motion you see (well, you can’t see the CD spinning). What happens is the direction of the angular velocity of the CD player continually rotates around direction of the angular momentum. Really, this is sort of complicated mathematically – so I am just trying to describe it.

As a bonus, here is something really cool about freely rotating objects. First, for any object one can pick at least three axes about which the object could rotate and have the angular momentum and the angular velocity in the same direction. Sometimes, it is easy to pick out these three axes. Take something like a ruler, here are the three axes about which it could rotate with L vector in the same direction as the angular velocity:

i-a17206a491e67176512a375504a94919-ruler-i.jpg

Although you can rotate about these three axes and have L and angular velocity in the same direction, only two of these cases are stable. You should try this. Take a ruler (which is like the shape above) and toss it in the air to spin in the three different orientations (any rectangulary shaped object will do – like a hard drive). If you throw it and spin it around the red or blue green axes (from the drawing), it should work ok. However, if you try to rotate it around the blue axis, it will not stay that way. Again, this is something that may be a little complicated, but you can try it anyway.

Comments

  1. #1 Jon Snow
    March 26, 2010

    Hi Rhett,

    The last lines of your post are;

    If you throw it and spin it around the red or blue axes (from the drawing), it should work ok. However, if you try to rotate it around the blue axis, it will not stay that way.

    I suppose you meant to say green instead of blue which appears twice.

    Best regards

  2. #2 Rhett Allain
    March 27, 2010

    @Jon,

    Thanks for finding my mistake – I fixed it. Actually in that diagram, the red and green axes are stable and the blue is unstable.

  3. #3 Leif Nabil
    March 28, 2010

    I don’t see why the CD players were arranged at right angles? The angular momenta of each player are perpendicular. The only reason I can imagine is that the perpendicular arrangement seems more “science-y” or “engineer-y.” When two are taped together at 90°, the maximum angular momentum is sqrt(2)L, and when three are taped perpendicularly, sqrt(3)L. But they are stacked directly above each other, the maximum angular momentum will be 2L and 3L.