Saturday, December 5, 2009

Torque and Angular Momentum

Rhett Allain has a very nice blog post about angular momentum featuring the precession of a bicycle-wheel gyroscope that he demonstrates here:

I find wearing a long-sleeve shirt with shorts to be an interesting touch.

In addition to my comment on his blog, I'll add the following about how I introduce it in my classes:

Based on experience as a student and an instructor, I think it is usually best to present the prediction before doing the experiment. However, in this case I generally interleave the two.

As with most intro textbooks, mine packages angular momentum along with the cross product definition of torque in its own section so it is easy to omit completely. I integrate tau=rxF into my initial introduction of torque and the various ways of calculating it, but then stick with tau = I*alpha until I get to L.

As soon as I introduce L, I go into the generalized second law as tau = dL/dt (pretty much the way we jump from F = ma to F = dp/dt once momentum is defined). After connecting this to tau = I*alpha, I then ask "So don't you wonder if that cross product in the definition of torque is real? Is torque really perpendicular to the force?".

Then I do the demo, quickly, just enough to see the rotation.

WTF? At this point I do the detailed calculation, exactly as shown in Rhett's blog, and then REPEAT the demo. This time, however, I slip an "L" arrow onto the handle so they can see it precess.

What if I hold it by the opposite handle? What is tau now? Aha, it goes the other way!

What if L = 0? Ah, so "falling" is actually rotation in this case.

... and finally ...

What force keeps the center of mass from falling with L is not zero?

The string! Now if I could only measure the force on the string during the demo with L not zero and compare it to the force when L is zero ....

But to summarize: In this case I think they need to see a taste of the phenomenon to understand why I would bother with such a detailed calculation. It also means that I end up doing the demo itself several times, and I use the wheel with the L arrow on it when doing the drawings, since they are not yet experienced at getting a 3-D image out of two projective views. Few have had a drafting class or Calc III.


BlackGriffen said...

"Now if I could only measure the force on the string during the demo with L not zero and compare it to the force when L is zero .... "

What, they don't have spring scales in your department?


Doctor Pion said...

The real problem is that I hadn't thought about doing that additional measurement until I blogged about it!

However, I don't think the scales we have around will do the job. (1) I think the wheel I use is more than 20 N, which is the limit for the spring scales we have. (2) The time scale is really short when it is just dropped, and it is hard to see both things at the same time. Tough enough when I drop a 1 kg mass along with a scale, and they only have to see that the scale goes to zero.

I think the only way to do it is with one of the Pasco load cells we have lying around. They go to 50N, which might be enough, and give decent enough time resolution. The down side is they require setup time and can only be used in the rooms where I have a scsi card in the classroom computer.

Cleon Teunissen said...

Hi CCPhysicist,

we had a bit of conversation on the comment section of one of Rhett Alain's blog entries.

I have no problem understanding the bicycle wheel demonstration, but what you wrote in your blog doesn't tell me how you present the demo to your students. I mean, what you wrote in the blog is kind of shorthand, and I can't follow it.

There is an MITopencourseware video on Youtube featuring the bicycle wheel demonstration at about 35 minutes into the video; Walter Lewin spins the wheel so fast that the precession has a period of about 10 seconds.

I have a website with physics articles, illustrated with animations.
I've just finished an article about gyroscopic precession.
Gyroscope physics

I hope I can persuade you to check out my gyroscope stuff. The presentation there will make clear, I expect, what I had in mind when writing the comments on Rhett Alain's blog.

Cleon Teunissen

Doctor Pion said...

I don't think you understand it, because the torque you apply is what makes it "fall", not the torque due to gravity. (Explained in Allain's comment section.) You can see this by pushing the other way (helping it precess), which will make the wheel pitch up. When you push sideways on the handle, you add a delta-L that points vertically (out of the paper in Allain's diagrams) and makes L precess upward or downward.

By the way, what you refer to as "unsightly rocking up and down" on your web page is not unsightly at all. It is called "nutation" (nodding motion, from the motion of a top) and is the tasty frosting on an already interesting problem.

Cleon Teunissen said...

I am aware of nutation, of course, and I do plan to add material about nutation on my website. The present article is the first of hopefully many, covering nutation, Euler angles etc.

I agree of course that if I would manually increase the precession rate then the wheel will pitch up. That is straightforward.
Conversely, if I manually decrease the precession rate (down to zero precession rate) then I decrease tendency to pitch up. When there is no more precession-correlated-tendency-to-pitch-up then gravity is free to pitch the wheel down. Clearly we are interpreting the same phenomenon differently.

Please contact me, so we can figure out what it is we disagree about.

Teaching physics is my passion, and my website has been included in collections of recommended websites, such as the physicical sciences resource center, a service provided by the AAPT.

Cleon Teunissen

Cleon Teunissen said...

Correcting the misspelled contact information in the previous message:
My contact page

Cleon Teunissen

Doctor Pion said...

When you write "I agree of course that if I would manually increase the precession rate then the wheel will pitch up." you appear to be taking a passive view of what is taking place. The wheel does not decide to pitch up. You have applied a net torque that changes the angular momentum in the direction of that torque.

This is confirmed when you write "Conversely, if I manually decrease the precession rate (down to zero precession rate) then I decrease tendency to pitch up. When there is no more precession-correlated-tendency-to-pitch-up then gravity is free to pitch the wheel down." This statement is fundamentally wrong. Gravity does not pitch the wheel down unless the wheel has zero angular momentum. YOU are pitching it down by applying a torque that adds a downward pointing vector to L.

If this is not clear in words, I'll have to pull together a blog with diagrams like were used on Allain's web site.

Cleon Teunissen said...

Hi Doctor Pion,

I do not take a passive view of what is taking place; as far as I can tell you are attributing a point of view to me that isn't mine.

My focus is on identifying cause and effect in terms of first principles. Angular momentum is (compared to linear momentum) a compound entity, just as moment of inertia is (compared to inertia) a compound entity.

In classical mechanics visceral understanding can play an important role. Why can a hammer drive in a nail? For a visceral understanding one visualizes the abrupt deceleration of the hammerhead on impact with the nail. In thinking that is focused on understanding gyroscopic effects I use direct entities (momentum, inertia), and not angular momentum.

I believe that is the junction where babylonian confusion sets in. Clearly, your thinking revolves around angular momentum, and you are interpreting my words strictly in those terms. But I'm not talking about angular momentum. In my mind I visualize the wheel as a ring-shaped assembly of point masses, at the ends of spokes, and I follow those point masses, helped by the symmetry of the setup. Each spoke is rigid, through the spoke each point mass confers some inertial effect to the hub of the wheel. I hope I can invite you to a discussion that is aimed at finding out why we disagree.

I specialize in the physics of rotation. Professor Joe Wolfe has kindly added a link to my site on his Foucault pendulum page, and so has Professor William Tobin.

Cleon Teunissen said...

Complementing my earlier comment:
Recently Rhett Alain posted on his blog a discussion of buoyancy. Why does a helium balloon float? Rhett presented an explanation of the buoyancy of a helium balloon from first principles. A comment poster wondered if it would have sufficed to invoke Archimedes principle. I replied that I thought not.

I think the discussion of balloon buoyancy illustrates well the importance of digging down to first principles.

Doctor Pion said...

The Babylonians were never confused. They had the best idea ever for a calendar, one with 360 days divided up into 12 regular months plus either 5 or 6 days of New Years holidays for religious rites like football.

You are welcome to work every problem from first principles using only forces, but the result will still be a moment of the force (torque) acting on a rigid body that resists that moment with its rotational inertia. And, no matter which way you describe it, the reason the wheel pitches up or down is because you applied a force with the appropriate moment to the wheel.

And you do give the wheel a mind of its own when you talk about IT having a "tendency" to do things. It does not pitch up or down or precess. You or gravity make it pitch up or down or precess.

Cleon Teunissen said...

Science blogs such as yours and the one by Rhett Alain are aimed towards education.

In applied physics the goal is to think efficiently, so you use rich, compounded concepts. Obviously in applied physics it would be pointless to work each and every problem from first principles. However, in education ability to reduce to first principles is key.

A phrase such as 'a rigid body resists that moment with its rotational inertia' could give the impression that you attribute volition to the spinning wheel. It is clear of course that you didn't intend to imply that. My point is: language is like that, most of our expressions are anthropomorphisms, if you were to take them literally.

The expression 'gives rise to a tendency to' has the advantage of emphasizing the neutral, mechanical nature of the process; it's the difference between a 'tendency to' and an 'intention to'.

I hope the digressions have been sufficiently addressed now. Can I persuade you to discuss gyroscope physics?

Doctor Pion said...

My expression was chosen to match the world view of the person I was teaching, and I stand firm on the viewpoint that torque and the moment of inertia tensor provide the simplest way to address a simple rigid body problem of this type. It is silly, especially if doing applied physics or engineering, to work from the much more complicated expressions involving forces and masses.

Cleon Teunissen said...

Hi Dr Pion,

Back in december, in the comments to Rhett Alain's posting about the bicycle wheel demonstration, I posted a final clarification, it's
comment number 13

I also proceeded to try and contact you here, on your own blog. Knowing that you're a teacher my expectation was that your response to me would be something like: "Well, if you are that confident, then by all means show me."

I was surprised by the actual development: while you did keep approving my postings, no dialog developed. The last message I submitted was around february 7. Either that message never reached you, or you didn't approve it.

Although I'm disappointed that I didn't get a dialog going, there is at least the final clarification in comment 13 on Rhett's blog.

Doctor Pion said...

Blog comments on an article that is almost three months old is not a practical way to carry on a "dialog". I would never see it if not for moderation intended to keep spammers out of old threads.

Feel free to publish what you think is interesting in your own blog. I have work to do.

CarlBrannen said...

I don't have any of those fancy "Newton" scales; I have some cheap fish scales from the local hardware store. They go to 50 pounds and are very entertaining to the students.