Angular Momentum Example

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.


As pointed out by Dr. Pion, I made a mistake. The two above diagrams showing the torque should be from BELOW the wheel.

More like this

You know what's funny? I watched the demo and it didn't even occur to me that the "cool" thing about the demo is that the wheel doesn't fall (until I read the rest of the post). I guess I've just done too many demos.

The demo my students thought was the coolest was me sitting on a stool holding a spinning bicycle wheel and changing my direction and speed of rotation by turning the wheel.

By Chris Goedde (not verified) on 03 Dec 2009 #permalink


My students also think the stool thing is awesome. I am with you, I have seen cooler stuff.

San Francisco's Exploratorium and (if memory serves) the Cabot Science Center in Balboa Park in San Diego both have these demos as hands-on exhibits.

It is amazing what some students think is cool. My 3rd period conceptual-level class was awed when I swiped an embroidery hoop from under a dry-erase marker, letting it drop straight down into a glass bottle. Some of my AP students are fascinated by a rubber band stretched between two support stands. That can keep 'em occupied for a good 10 minutes (unless I nudge them to get a move on and collect some data!).

A dramatic illustration of angular momentum:

On a rifle range, two paper targets are set downrange, one at twenty feet, the other twenty feet farther, directly in the line of fire. The shooter fires at the front target.

Observers expect the second target to be holed, like the first, but that does not happen. Instead, the second target remains intact.

Why? A high-velocity .22 centerfire, such as a .223 Remington, fired down a barrel rifled to one turn in 12 inches, exiting the muzzle at 3000 feet per second, is spining on its long axis at 180,000 RPM. When the bullet integrity is disrupted by contact with the fragile target paper, the bullet fragments, breaking outward, and whatever particles were near the axis and so had very little angular momentum, they have such high aerodynamic drag that they fall to the ground less than twenty feet from the impact.

Cooler in person. A bike wheel, a handle made from a rolling pin (the kind used for making pie), and an office chair (the kind that swivels) and they can do it themselves.

I think it would be very interesting to include the following in the demonstration: prevent the precession of the spinning bicycle wheel.
If you prevent the precession then the wheel will drop down just as hard as when it isn't spinning.

A better option than @7, which applies a second torque:

Include the case when L1 = 0 in your explanation. In that case, L2 is in the direction of tau which results in the wheel rotating around that axis -- hence "falling".

I'll put some comments in my own blog about how I fit this in with the demo and the full explanation. I find this (along with magnetic forces) to be the best way of showing that the cross product is "real".

Responding to @8:
I agree that preventing the precession involves applying a second torque.
However, the second torque is different: the torque from gravity follows the instantaneous orientation of the wheel; the second torque has a fixed orientation.
Preventing the precession from commencing has as side effect that the torque from gravity becomes simpler; it's orientation becomes stationary too.

Preventing the precession demonstrates that it's the precession that keeps the spinning/precessing wheel from toppling over.

Haha, so *that's* how that's supposed to work.

In Spectroscopic Elucidation, as we were discussing proton NMR and the precession of the protons involved in it, the professor tried to do this with a bicycle wheel as an example. He'd just taken the front wheel off of his bike, though, and lacked any sort of handle to get a good hold on it, so he couldn't get it spinning fast enough to do, nor have any way of holding one end of the axle strongly enough to support it like that.

I just saw your comment on my blog and came back here to see your observation. I think it explains why you don't see the relevance of the L=0 case.

The second torque, the one you apply, causes the wheel to "fall" (its center of mass rotates around a horizontal axis), not gravity. In the case of that applied force, rxF points down so the delta-L precesses L toward the floor. If you tried to "help" the precession, the wheel will rotate up (above horizontal).

PS to Allain:
The two diagrams are backwards if they are supposed to be from above. Tau is towards the wall, not away from it.

@CCPhysicist - holy cow! you are correct. Sorry about that.

The following 37 second YouTube video (uploaded by Glenn Turner), offers a good display of spinning gyroscope observations.

In the first 20 seconds you see Glenn gently moving the gyroscope wheel. He pushes with his fingertips, swiveling the gyroscope wheel clockwise and counterclockwise 10 or 20 degrees. Swiveling one way the gyroscope wheel pitches one way, swiveling the other way the gyroscope wheel pitches the other way. That is: to reverse the direction of pitch the direction of swivel must be reversed. Also you can see that the gyroscope wheel swivels readily, it just adds a motion component: to the swiveling motion a pitching motion is added.

Now the setup that CCPhysicist and I have been discussing (comments 8, 9 and 11):
Start with the setup displayed in Rhett's video. I'll call the torque from gravity 'the first torque'. Precession sets in. As demonstrated in the Glenn Turner video: the precessing motion gives rise to a tendency to pitch up. That tendency to pitch up keeps gravity from pitching the wheel down.

- What will happen when you apply a second torque, around the swivel axis, in such a way that the existing precession rate increases? Answer: the bicycle wheel will pitch up.
- What will happen when the second torque is applied in such a way that it decreases/nullifies the existing precession rate? Answer: then the precession-correlated-tendency-to-pitch-up is reduced/nullified, and the torque from gravity is free to pitch the wheel down.

CCPhysicist has argued that the second torque should be regarded as the cause of pitching down, and not gravity. I think the following analogy applies: think of a satellite in orbit, firing retro-rockets: the altitude of the satellite will decrease. I prefer to say that gravity is the cause of the descent, and that firing of the retro-rockets created opportunity for gravity to pull the satellite down.

Returning now to my original observation in comment 7:
If you prevent the precession from continuing then gravity will cause the bicycle wheel to flop down, just as fast as when it's not spinning.

please tell me what your branch of physics is ?
thanks alot.

By mohammasd (not verified) on 15 Dec 2010 #permalink