Which Coordinate System Should Be Used for Force in Torque Calculations?

[alextex]

Hi!
I have been working on a rigid body subject quite a long. But till now there is an unresolved question for me. When we calculating the torque acting on the rigid body we use the following definition of the torque: t = r X f, X - is a cross product. So if I calculate the torque in a body-fixed system I use r in the body-fixed system too, but for force f - I am not sure. In which coordinate system should it be expressed - in the body-fixed or in space-fixed. Are this forces different in both systems? Sorry, for probably stupid questions. Thanks

 

[alextex]

Sorry, I've found - we really need to do all in the same coordinate system.

But now, I have some other question - probably, more interesting. The torque acting on the rigid body (expressed in the body-fixed frame) is given by: t = -0.5*S(q)*(dU/dq), gde U - potential, q - is a quaternion, ans S(q) - is a matrix such that: dq/dt = 0.5*S(q)w, where w - is a 4-dimensional angular velocity. Can anybody explain how to get that formula for the torque?

 

[astrokreng]

Hello there!

First of all, there is no such thing as a "stupid question" in science. It's important to ask questions and seek clarification in order to fully understand a concept.

To answer your question, the force should be expressed in the same coordinate system as the position vector r. This is because the torque is a vector that is perpendicular to both the position vector and the force vector. Therefore, in order to accurately calculate the torque, both vectors should be in the same coordinate system.

In a body-fixed system, the force and position vectors will have different components than in a space-fixed system. However, the physical force acting on the body will be the same regardless of the coordinate system used to express it. This is because forces are independent of the reference frame in which they are observed.

I hope this helps to clarify your understanding of torque in rigid body dynamics. Keep asking questions and exploring the subject - that's what being a scientist is all about!