Physics Trouble

[amd52289]

I've been trying to write a paper in my physics class about the physics of tossing a flag in Color guard. More specifically I'm trying to describe a one and a half concert toss. I have notes on mechanics and thermodynamics but I'm having trouble making sense of it.
If you can help I'm not quite to the college level of understanding. Any help is greatly appreciated.

[Gil Hamilton]

I would recommend studying the following related topics; torque, angular velocity, center of mass, moments of inertia, and other general mechanics topics.

For example, consider that you need to throw a flag such that it makes one and a half revolutions. What does the angular velocity of the flag have to achieve this in the time it takes for the center of mass of the flag to rise and fall. You'll need to know how long the flag remains airborne before it is caught again. Be sure to state any approximations you make, for example, an airless environment (non-trivial a problem when it comes to something with a banner hanging from it, but physicists make weirder simplifications) or that you are catching the flag at the same spot and coordinate.

Also, consider the structure of the flag. Can you approximate its moment of inertia as a rod (thus, simple) or is it more complicated with the banner hanging from it? Where will be the center of mass in relation to the where you throw it and does that affect how you throw it?

The most important thing to remember is this: you KNOW the physics. In your body, you know how to throw the flag, assuming that you do it in Color Guard. I recommend you start by throwing a flag exactly as your problem describes and honestly thinking about everything you do. Once you have a physical picture of what is happening, take it one step at a time and start thinking about what tools you need to describe that picture. When doing any science or engineering task, never let the magnitude of the problem deter you from starting, because solving the problem are a hundred little steps that are easy.

Good luck.

[amd52289]

Thanks for the help.

[Feil]

For just throwing the rod: Take its moment of inertia (I=integral(r^2)dm | rod = 2*(1/3)*(m|half)(L|half)^2 = (mL^2)/12 where m is the total mass, L is the total length. Measure its angular velocity (w = 2*pi*revolutions per second) and the height h the center of mass reaches above the point where it leaves your hand. Work is W = .5Iw^2 + mgh, where g is acceleration due to gravity, ~9.8m/s^2.

Do you have any skill with MATLAB, R, or another technical computing language? A computer can simulate physical systems and output pretty animated results, if you give it the right input information. The above (rotational work, vertical work, moment of inertia, mass, location of center of mass) are all you need to simulate the rod.

[Gil Hamilton]

Though, remember, with Feil's suggestion, you MUST note the difference in the structure of the rod. There are three rod scenarios (though I suppose you could argue two are extreme cases of the one); a hollow tube where you are treating the thickness of the material as insignificant, a tube with significant thickness, and a solid rod. You should look at your flag and see what case it is.

[amd52289]

Not really familiar with computer stuff. This is my first year taking physics and I'm still trying to grasp the concept. I have to write a paper that's less then 10 pages so I know my teacher doesn't expect a novel or immense detail of every physics thing to do with tossing a flag. And the pole is a hallow one. I'm having a little trouble understanding torque and how to form an equation to match what I'm doing. It could be that I'm over thinking it, any advice?

[Feil]

You need to ask yourself a few questions. These are fundamental to any problem solving in the physical sciences, not just physics, and definitely not just tossing a flag.

-What do I want to know?
-What does that mean about what my system is? (A system is all the stuff you want to know the physics of over a given time.)
-What can I ignore about the system? What do I need to include?
-How can I simplify the system into something that I can do math on?
-How, and under what conditions, can I justify to simplify it like that? (This is an important step! You can't just go throwing away information without a good reason!)

Now you have reduced the system - an insoluble mess of variables and unquantifiable values - to a model. A model is an abstract concept that retains the important physical characteristics of the system for the conditions you have specified, but is easy to apply math to. Next you ask a few more questions.

-What math do I need to do to learn what I want to know from this model? (This means equations, which you or Friend Computer can solve.)
-Do I know how to do this math (or how to make a computer do this math?)

If the answer is yes, you are done with science, and all you have left is crunching equations.

If the answer is no, it's back to science for you, because your model isn't model-y enough, yet. Ask yourself a few more questions:

-Okay, do I really need everything I included in the model before? Can I get rid of anything else?
-What if I can't - is there anything that I want to know that I can find out by making this model even simpler?
-Or, can I make some good estimations of the things I want to know by making the model simpler?
-Can I put together the results of two or more simple models to make a more complex model that is still correct?

Once you have reduced the model to something you know how to do the math for, you start generating information that you didn't have before. You have done science! You have conjured information out of the bottomless pit of despair that is the soul of the student of the physical sciences, and eternal glory shall be yours!

A potential example of this process, relating to your problem:

-I want to know where every point on the rod of a flag is at all times for a particular flag toss, from when I release the flag to when I catch it again.
-That means my system is the flag and the medium it passes through, from the time I release it to when I catch it again.
-It's passing through air. The air resistance on a moving body significantly more dense than air for low velocities and small cross-sectional areas, is negligible. The flag might have some contribution to air resistance, but I observe that it is behind the flagpole by the time it leaves my hand, and follows the flagpole around without catching much air. I am therefore going to pretend that the air isn't there.
-Now I have a flag on a flagpole in a vacuum. That flag is a problem: I can't possibly do the math required to calculate all the movements of that non-rigid body. But the flag stays out at a straight angle when the flag pole goes around, so I can treat the flag as a rigid square attached to the flagpole.
-The flagpole itself is basically a cylinder. It has a few dents and scratches, but those aren't going to have a significant effect on the behavior of the system, because the change in mass and moment of inertia they cause on that local area of the flagpole is tiny with respect to the total mass and moment of the flagpole.
-Now I have a square of mass m1, negligible thickness, and area mass density D1 attached to a hollow rod of mass m2 of volume mass density D2. Good model! I can do math on this.
-What about the vertical and rotational movement? That could be messy. I know that an object rotates around its center of mass, and I know that the curve followed by a projectile in free-fall describes the movement of the center of mass. I'll just take those one at a time.
-The projectile path is super easy. Apply gravitational acceleration to an initial velocity and the total mass, and I am done. I can leave initial velocity and mass as variables, and add values for them later to fit whatever physical system I want to describe.
-Now I need to know the rotational movement. There's no torque in effect, here, since the torque on a rigid body is proportional to the curl of a force field operating on it, and earth's gravitational field is constant over the system I've defined. That means all I need to figure this out is an initial angular velocity, since that's not going to change, and the location of the center of mass that the body is rotating around.
-I can just go out and measure both of those.
-Now I have equations with respect to time for where the center of mass is, and where everything else is with respect to the center of mass. To identify the location of every point, all I need is a little bit of vector algebra: add the position of the center of mass to the position relative to the center of mass, and I am done.

There is plenty more you can know about something than where it is, of course - but I'll leave that up to you, your teacher, and Friend Computer.

[Gil Hamilton]

Not really familiar with computer stuff. This is my first year taking physics and I'm still trying to grasp the concept. I have to write a paper that's less then 10 pages so I know my teacher doesn't expect a novel or immense detail of every physics thing to do with tossing a flag. And the pole is a hallow one. I'm having a little trouble understanding torque and how to form an equation to match what I'm doing. It could be that I'm over thinking it, any advice?

Torque is the analog to force when doing a rotation. When you chance the angular momentum of something (say, causing your flag to spin) then you are applying a torque. Just like a force is a the change in momentum with time, a torque is the change in angular momentum with respect to time.

Let L be the angular momentum and set it equal the moment of inertia and the angular velocity. The directivative of that is torque. So, if you know what your moment of inertia is, you can figure out what angular velocity you can achieve for a given torque.