How to show that motion of a rigid body = translation + rotation

[martin_blckrs]

We all learn in the introductory mechanics class that the motion of a rigid body can be composed of a rotation and a translation. But how can one prove this? I mean: Let us have some rigid body in two configurations in space, how can I show that I can transform one configuration to another by just a translation and rotation?
I could take the two configurations to have the same center of mass or any other point by a translation. But how do I then find the rotation (constructively)?

 

[dE_logics]

Configuration?...what do you exactly mean by it?

 

[martin_blckrs]

Well, I mean position, i.e. I have two copies of the same rigid body somewhere in the space and I want to find a transformation (composed of translation and rotation) taking the first copy to the second.

 

[monty37]

do you mean 2 different bodies(rigid) in space in different positions?
and what do you mean rotation constructively?

 

[0xDEADBEEF]

Go into the center of mass' system, so it doesn't move. Try to show that for the body to be rigid, the distance of a point to the center of mass must stay the same. Then you can show that this is only the case for rotations around the center of mass. Finally show that two points must rotate around the same axis and the same angle so their distance doesn't change either --> this implies that all points rotate around the same axis and angle.

This is what I would do.

 

[martin_blckrs]

do you mean 2 different bodies(rigid) in space in different positions?

yes

and what do you mean rotation constructively?

The question was how do I find the rotation constructively = how do I construct such a rotation.

 

[martin_blckrs]

Go into the center of mass' system, so it doesn't move. Try to show that for the body to be rigid, the distance of a point to the center of mass must stay the same. Then you can show that this is only the case for rotations around the center of mass. Finally show that two points must rotate around the same axis and the same angle so their distance doesn't change either --> this implies that all points rotate around the same axis and angle.

This is what I would do.

Well, this is nice, but the problem is that you just show that if there exists such a transformation, it must be a rotation. What I'm trying to see is that such a rotation actually exists.

 

[dE_logics]

You're asking to transform motion of one copy to the other...right?

Or is it that you're assuming one of the copies to have rotational motion, and the other translational?...and THEN trying to transform the frames.

 

[martin_blckrs]

You're asking to transform motion of one copy to the other...right?

I don't want to transform motion. I want to transform space configuration (i.e. position) of one copy to another, i.e. given the two copies I want to find a transformation = rotation + translation such that it brings me from one copy to another.

 

[Born2bwire]

Maybe using matrix operations could help. You can represent the rigid body as a collection of particles with a rigid bar connection between the particles. This bar can be represented as a vector in space and the particles as coordinates. From here, it is easy to construct equivalent translation (addition of a vector) and a rotation (matrix product with a rotation matrix).

So the question is, given a simple rigid body, an arbitrary 3D vector pointing from the origin to some point, show that you can decompose any general transformation into the the combination of a translation and rotation. The problem isn't unique though but perhaps you can setup the appropriate matrix equation (Ax=b) and show that the system has a solution?

 

[tiny-tim]

We all learn in the introductory mechanics class that the motion of a rigid body can be composed of a rotation and a translation. But how can one prove this? I mean: Let us have some rigid body in two configurations in space, how can I show that I can transform one configuration to another by just a translation and rotation?
I could take the two configurations to have the same center of mass or any other point by a translation. But how do I then find the rotation (constructively)?

Hi Martin!

Take any three non-collinear points A B and C, fixed in the rigid body.

Move A to new-A. New-B lies on a sphere of the correct radius with centre at new-A, so use any rotation of that sphere to move B to new-B. New-C lies on a cylinder of the correct radius with axis new-A-new-B, so use any rotation about that axis to move C to New-C. Finally confirm that any other point is in the right place.

 

[Andy Resnick]

We all learn in the introductory mechanics class that the motion of a rigid body can be composed of a rotation and a translation. But how can one prove this? I mean: Let us have some rigid body in two configurations in space, how can I show that I can transform one configuration to another by just a translation and rotation?
I could take the two configurations to have the same center of mass or any other point by a translation. But how do I then find the rotation (constructively)?

The first step is to recognize that the rigid body in two different configurations is equivalent to a coordinate transformation- going from a 'reference' set of coordinates (x,y,z if you like) to a set of coordinates that move with the body. The 'transformation matrix' that transforms one configuration to another has certain specific properties including noncommutivity (AB != BA), invertibility, orthogonality, and a few others- an important one is that reflections (coordiante inversions) are not allowed.

Euler's theorem then states that the general displacement of a rigid body with one point fixed is a rotation about some axis. This plus a linear translation, then gives the most general displacement of a rigid body.