Question about torque and center of mass

[AGC]

I am a beginner and this is my first post.

We know torque $\vec T = \vec r × \vec F$ Now, should r be fixed in magnitude? I mean can I calculate torque only if the particle is rotating at a fixed distance around a fixed axis? I am having this idea because torque is explained (from where I read) using an analogy with doors.

Second question. I read that generally a system of forces acting on a rigid body can not be reduced to a single force equal to the vector sum of the forces. Then I get confused because in case of weight, we replace gravitational forces on each particle of a rigid body with a resultant force through the center of mass.
Will someone please explain?

 

[Maged Saeed]

about your first question , your question is how can we deal with torque when r is variable 'not constant in magnitude' ?

 

[Maged Saeed]

About your second question , [As I know] we treat the rigid body as a particle-like object so that the gravitational force will act only on its center of mass.

 

[AGC]

about your first question , your question is how can we deal with work when r is variable not constant 'in magnitude' ?

I think my first question was not clear enough. I asked if the concept of torque is applicable only if the particle (or body) is rotating about an axis? And should it rotate in a fixed radius? Sorry if the question is of very intro level.

 

[Maged Saeed]

No , [As far I know] The concept of torque can be applied if the rigid body is rotating about either a fixed point or a fixed axis , where the point 'similarly to the axis of rotation ' is allocated on an axis vertical to the xy-plane.

 

[Maged Saeed]

See this vedio: