Physics - Dynamics - Combined Linear and Rotational quantities - Forum Discussion

By: DataBeaver - databeaver
file Rotation direction  
2006-03-19 00:00

From Physics - Dynamics - Combined Linear and Rotational quantities: "I think something is wrong with the sign since J should produce a greater velocity change if there is both linear and rotational motion, I assume its something to do with the direction chosen for positive rotation." 
 
You are somewhat right there. The root of the problem is that you have mismatched equations for impulse torque and the rotational component of linear speed. J x r for the former produces a clockwise rotation axis. However, w x r for the latter requires a counterclockwise one. When combined into (J x r) x r, this produces a vector in the opposite direction of J. 
 
My suggestion is to change the impulse torque to r x J, since counterclockwise is conventionally the positive direction (think about the cos and sin functions). This will change the combined expression to (r x J) x r, which will produce the expected result.

By: Martin Baker - martinbakerProject Admin
file RE: Rotation direction  
2006-03-19 10:57

Thanks very much for this. 
 
The reason that I've got clockwise motion as positive is that I have tried to standardise on: 
1) a left hand coordinate system (+x to right, +y up and +z toward viewer). 
2) left hand rule for rotation (left thumb points in positive direction of axis of rotation, curl of fingers gives rotation). 
 
So if we are considering rotation about the z-axis and if our left hand thumb points in the positive z-direction (toward viewer) then the fingers curl in a clockwise direction. 
 
However, if we choose a right hand rule for both coordinate and rotation (right thumb away from viewer) then we also get positive clockwise rotation. 
 
Its only if we choose left hand coordinates with right hand rotation, or visa versa, that we get anticlockwise rotation as positive. 
 
I agree that anticlockwise rotation is usually shown as positive, but I keep finding cases of conflicting standards and conventions in this subject, I guess that's what makes it so hard (or should I say such an interesting challenge) to work out. 
 
I think your right about using r x J instead of J x r, because the corresponding torque equation is usually r x F, although I think I've seen it the other way round. I guess I could just change it and make everything fit but having got to this stage I would really like to clarify these issues. 
 
I guess we need to go back to the equations: 
angular momentum of a particle = L = r x p 
torque = T = F x r (or rotational impulse = J x r) 
and find out if they are defined for left or right hand coordinates and left or right hand rotations? 
 
The vector 'x' product itself must have some left-or-right-handed-ness because we arbitrarily choose a direction for the result, we could just as easily chosen the other direction for the result of a cross product. 
 
I'm not quite sure how to take this forward? I would like the site to clarify exactly what standards and conventions are being assumed. 
 
I'll have to think about this a bit more I would welcome any more ideas. 
 
cheers, 
 
Martin

By: Martin Baker - martinbakerProject Admin
file RE: Rotation direction  
2006-03-19 11:09

Oops, A small correction to my last message, the site actually standardises on right hand coordinates and right hand screw rule (as does VRML/X3D and OpenGL), but I don't think that affects the issues discussed. 
 
for details see, 
https://www.euclideanspace.com/maths/standards/ 
 
Martin

By: Martin Baker - martinbakerProject Admin
file RE: Rotation direction  
2006-03-19 11:26

Sorry, my brain cant be working today! Having worked it out again, if both coordinates and rotations use right hand rule then the rotation will be anticlockwise - which is what you said in the first place! 
 
I'll change the page as you suggest. 
 
Martin


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