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 Post subject: Seriously Non-AV:T Physics Question #1
PostPosted: Mon Aug 30, 2010 1:47 pm 
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High-school physics question that I can't wrap my head around (showing why I shouldn't be allowed to play with anything more dangerous than a pencil-and-paper game):

A rod lies on a perfectly frictionless surface.* The rod's mass is evenly distributed and it does not flex at all. You kick one end of it perpendicular to the long axis. What happens?

I cannot for the life of me figure out how to calculate this. My gut tells me that you impart a velocity to the whole rod in the direction of the force you applied. I also imagine that you would spin the rod, but my gut tells me that it's not spinning on the center of mass, because there's no pivot point holding it in place and you're only applying force to one end.

*Or in free-fall vacuum, whatever.

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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Mon Aug 30, 2010 2:17 pm 
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I'm sure someone will come along with a math- and force model-based answer, but I think the rod will begin moving in the direction of the force, and rotating about its center of mass. The speed of the rotation and the lateral speed are related, in that for a constant force, they'll maintain the same ratio. A force imparted closer to the center of mass will impart more lateral and less rotational force, while a force closer to the end, farther away from the center of mass, would impart less lateral and more rotational force.


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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Fri Sep 03, 2010 3:25 pm 
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I'd wonder if the rod would start rotating around its long axis as well.

So you have the rod moving across the surface, the rod rotating around an axis vertical from the surface, and around its own axis. Have fun with the math, I'll be munching popcorn.

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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Mon Sep 06, 2010 9:40 am 
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The phrase to search for here is "rigid body dynamics".


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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Sun Sep 12, 2010 9:15 pm 
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Location: Calgary
Ethan McKinney wrote:
High-school physics question that I can't wrap my head around (showing why I shouldn't be allowed to play with anything more dangerous than a pencil-and-paper game):

A rod lies on a perfectly frictionless surface.* The rod's mass is evenly distributed and it does not flex at all. You kick one end of it perpendicular to the long axis. What happens?

I cannot for the life of me figure out how to calculate this. My gut tells me that you impart a velocity to the whole rod in the direction of the force you applied. I also imagine that you would spin the rod, but my gut tells me that it's not spinning on the center of mass, because there's no pivot point holding it in place and you're only applying force to one end.

*Or in free-fall vacuum, whatever.


Find someone with an air hockey table, substitute a piece of balsa wood that will float on the air table an approximates a rod by being longer than it is wide, and perform the experiment.

Failing that, a force applied to a rod is resolved into two components:

1) a translational component that accelerates the rod

2) a torsional component that sets the rod spinning about its rotational center of inertia

At this point, you write the formulas for linear and angular momentum and linear and rotational energy. The kick is defined as a force applied through a distance. Set up the equations for the total momentum and total energy and use the Lagrangian* and conservation laws to solve for the translational movement and spin. I have forgotten how to formulate the lagrangian*, which is really annoying as it is not difficult to do, and it is only a few lines of algebra, once it is set up (the last time I did was sometime in 1988). All I remember was that calculating the motions of a disk rolling without slipping down a wedge free to slide on a frictionless surface was, at the time, a really easy problem.

* It was either the Lagrangian, or the Hamiltonian (like I said, it was back in 1988)

PS: this would never be a high school physics question, as I learned how to do it in a second year university course for physics majors and solving it by applying f=ma is too difficult.

PPS: an alternate form of the experiment is to lay a broomstick on an ice rink and hit it with another stick (I was going to suggest kicking it, but that requires balancing on one leg while standing on ice). It will move with some combination of (depending how far away from the center of mass that you hit it) move along the ice and spin in the plane of the ice about its center. If your stick hits it above, or below, it long axis (or the stick is moving vertically as it contacts the broomstick), it will spin about its long axis, too.

If I find my copy of the Feynmann Physics Lectures, I will check to see if it covers Hamiltonians/Lagrangians.

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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Sun Sep 26, 2010 12:28 am 
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Just to prove Richard wrong, I'm going to try to work this out from first principles, using nothing more than high-school physics.

DISCLAIMER: I have no idea if this is right! I am not a trained physicist! Do not bring these results to NASA and make a rocket that relies on them! It will not go well and NASA will sue you for all of your pencils!

The setup: A thin rod with length x and mass m lies at rest on a frictionless surface. An impulse p is applied to one end, perpendicular to the rod. Predict the subsequent motion of the rod.

The translational component is perfectly straightforward. Applying an impulse p gives the rod linear momentum of p, so its velocity (at the center of mass) is just v = p/m. This ALWAYS works.

The rotational component is a little trickier. An impulse of p applied at right angles to one end of the rod (x/2 from the center of mass) will increase the rotational momentum by L = px/2 around the center of the rod. A quick table lookup will tell you that the moment of inertia of a thin rod rotating in a plane around its center is I = (mx^2)/12. Thus, the angular velocity w = L/I = (px/2)/((mx^2)/12) = 6p/mx. This is clearly (6/x)v, so the rod rotates 6 radians (a little less than one revolution) every time it moves through its own length. It also gives the ends of the rod a speed of ((6/x)v)(x/2) = 3v relative to the center, so the far end of the rod is actually moving backwards quite quickly at the instant of impact. This seems surprising but it makes sense if you think of the rod as very slightly flexible. The initial impulse bends it into a bow-shape, and when this shape springs back to straight it launches the far end backward to start its spin.

The surface is frictionless, so in answer to Todd's question, no, it will not start rolling around its long axis. There is no torque to start it spinning in that direction. Unless your impulse is off-center, of course. Then it gets really complicated because precession comes into play. I don't even want to think about that.


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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Mon Sep 27, 2010 11:11 am 
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Paul, if you're right about the rotation, why have reaction jets on opposite sides of a vehicle? Unless it's seriously non-homogeneous, it will always rotate around its center of mass, regardless of the placement of a single jet.

That said, is it possible to have sustained rotation of an homogeneous object that's not about the center of mass?

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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Mon Sep 27, 2010 12:14 pm 
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Ethan McKinney wrote:
Paul, if you're right about the rotation, why have reaction jets on opposite sides of a vehicle? Unless it's seriously non-homogeneous, it will always rotate around its center of mass, regardless of the placement of a single jet.

That said, is it possible to have sustained rotation of an homogeneous object that's not about the center of mass?


A single jet will not only start the craft rotating, it will impart a translational velocity (back to the rod given an offcenter impulse). Paired jets will cancel their translational components. As there will also be pairs of jets to cancel out spins, the jets can be fired to cancel rotational components and impart strictly translational vectors

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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Mon Sep 27, 2010 7:35 pm 
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Richard Bell wrote:
Ethan McKinney wrote:
Paul, if you're right about the rotation, why have reaction jets on opposite sides of a vehicle? Unless it's seriously non-homogeneous, it will always rotate around its center of mass, regardless of the placement of a single jet.

That said, is it possible to have sustained rotation of an homogeneous object that's not about the center of mass?


A single jet will not only start the craft rotating, it will impart a translational velocity (back to the rod given an offcenter impulse). Paired jets will cancel their translational components.

This is exactly why. You could get away with just one cluster of attitude jets, if you don't mind careening all over the sky whenever you try to turn. Firing one attitude jet is EXACTLY the situation we started with: the rod spaceship just fired one of its nose-mounted pivot thrusters for an instant, and in addition to pivoting it is now sliding sideways relative to its previous course.

Even if the ship is non-homogeneous, its motion can still be described as a pure translation + a pure rotation around its center of mass. It's just that its center of mass won't be in the same place as its center of volume, which makes a difference if there's some kind of drag or something.

Richard Bell wrote:
As there will also be pairs of jets to cancel out spins, the jets can be fired to cancel rotational components and impart strictly translational vectors.

Although assuming that your spaceship is relatively dense i.e. has a lot of mass for its length, this probably isn't a very effective way to provide propulsion. Notice that in our example above the rod is spinning quite a lot faster than it's moving, which suggests that attitude jets just sufficient to swing the ship around probably aren't sufficient to give it any useful net velocity in a reasonable amount of time. This effect will be even more noticeable if your ship has some other geometry than a thin rod: a real spaceship might have much of its mass at the center with the attitude jets on a long boom, giving it a much lower moment of inertia relative to the jet's effective radius.

With the attitude jets sufficiently remote from the main mass of the ship, you could actually get away with just one cluster, because the translational components it provides are negligible. This is how AV:T ships work: the pivot jets are all out at the engine, on the end of the mast, while most of the mass of the ship is in the main part. Even assuming that an al-Rafik is a hollow sphere with the mast of negligible mass, I = (2/3)mr^2, but the thrust is acting at about 5r from the center of the sphere, so the same calculation as above gives the al-Rafik rotating through 7.5 radians (about 1 1/5 full revolutions) every time it travels through its own 16-meter radius! Clearly at this point the translational effects are completely ignorable for anything short of docking.


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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Tue Sep 28, 2010 11:25 am 
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Paul Zagieboylo wrote:
With the attitude jets sufficiently remote from the main mass of the ship, you could actually get away with just one cluster, because the translational components it provides are negligible. This is how AV:T ships work: the pivot jets are all out at the engine, on the end of the mast, while most of the mass of the ship is in the main part.


Actually, it's not. If you look at the illustrations you'll see that all of the ships have attitude thrusters mounted up near the nose as well. (They're blocky with big circular holes in them.)

Also, I don't think that anyone was advocating the attitude thrusters imparting tactically useful translational velocities, just that you can generate some translation.

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 Post subject: Re: Seriously Non-AV:T Physics Question #1
PostPosted: Fri Oct 08, 2010 4:23 pm 
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As a physics Ph.D. student, I think I may be able to contribute something here.

Mr. Zagieboylo looks to be going about this in the correct way. I didn't check all the algebra, but treating the impulse translationally and rotationally separately is the way to go.

One could certainly use Lagrangian or Hamiltonian formalism to solve this problem, but it would be overkill.


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