I showed this demo in class and I was surprised at how cool the students thought it was.
They actually thought it was some kind of trick. It is not a trick. Instead, this is an example of the angular momentum principle. If you want to try this yourself, I guess you are going to have to find some type of wheel. I attempted to get this to work with a small Lego wheel, but it wouldn’t spin fast enough. You should be able to do this with one of those toy gyroscopes though. Anyway, here is the angular momentum principle:
Or, if you prefer it without a derivative it could be written as:
Here Tau is the torque on the object (about some point) and the vector L is the angular momentum as the object rotates around an axis through that point. Wow. I was going to link to a previous post where I talked about angular momentum and torque. However, it seems I have never done that. Ok – short version: Torque is like the rotational force, it can be defined as:
The second line is the magnitude of the torque. The vector r is a vector from the point of rotation to the point where the force is applied to the object. The angle theta is the angle between r and the force. Remember that torque (just like force) is a vector. Direction matters. If you don’t know what the cross product is, just think of the magnitude expression. However, you do need to know the direction of the torque for this example. The torque must be perpendicular to both the force and the r vector. There will be two vectors that meet that criteria. Choose the one such that when the fingers of your right hand cross r and then F, your thumb is in the direction of the torque.
What about angular momentum? For an object rotating about a particular axis, the angular momentum can be defined as:
I is a scalar quantity called the moment of inertia. It is best to think about this as the “rotational mass”. Just like normal mass makes it more difficult for a force to change the velocity of an object, rotational mass (moment of inertia) makes it difficult for a torque to change the angular velocity. The moment of inertia depends on the mass of an object and how that mass is distributed about the axis of rotation. Note: things can actually get much more complicated than this. The angular momentum can be in a different direction than the angular velocity – but in this case, I is not a scalar quantity.
Here is the short scoop on the angular momentum principle. If you recall the momentum principle, it basically says that forces change the momentum (linear momentum) of an object. The angular momentum principle says that the net torque changes the angular momentum of an object.
Now back to the bike wheel. Here is a diagram showing the wheel while it is spinning.
I left off the vector representing the torque because it would be out of the screen. Here is a diagram from above.
What if I look at what happens after a short time interval, delta t? In this case, I could write:
And I can represent this with the following diagram:
Ok, one problem. I have two angular momentum vectors, but the wheel is still rotating in the direction of the first angular momentum. Oh well, I think you get the idea. But, in case you didn’t here are the keys:
- Angular momentum is a vector
- Torque is a vector
- The net torque CHANGES the angular momentum
- The new angular momentum will be in a different direction
- This is really could be made way more complicated, but I left some stuff out.
The last point – why do people like this demo so much? I think it is because their everyday experiences say this wheel should fall and it doesn’t. If you look at this in terms of forces, the tension from the string and the gravitational force add up to the zero vector, so motion of the center of mass doesn’t change. If I have time, I will do a more detailed calculation of this demo.
Update:
As pointed out by Dr. Pion, I made a mistake. The two above diagrams showing the torque should be from BELOW the wheel.