Why does a free rigid body rotate only about its COM when F(ext) is applied?

[rahaverhma]

Why a free rigid body with mass m, when applied with an external force F, translates with center of mass and all particles of body having same linear acceleration = F/m, but it always rotates only about its center of mass, but not about some any other point on the body? And, thus if the line of action of force passes through COM, then it doesn't rotate, only translates. I think the condition for linear acceleration of COM of the body as per Newton's second law is just the condition for translation, not the condition for the summation of accelerations of all the particles due to rotation of the body about CoM to follow so that it becomes zero. There should be a different reason.

 

[Orodruin]

It is unclear what you even mean by the question. The rotation of the object is a property of the object as a whole, it does not rotate in particular about the CoM. What is true is that the CoM will move accelerate according to the external applied force and that if you apply the external force through the CoM, then no rotation will be induced. The easiest way of seeing this is to go to the accelerated CoM frame, where there is no net force, but there will be a torque of the force does not act through the CoM. This separates the motion of the CoM from the rotational motion.

 

[A.T.]

but it always rotates only about its center of mass, but not about some any other point on the body?

You can use any other point as center of rotation, but using the center of mass often gives you the simplest math.

 

[Lnewqban]

The center of mass describes the shortest possible path.

 

[A.T.]

The center of mass describes the shortest possible path.

With forces being applied, there can be points fixed relative to the body, that move less than the center of mass, when viewed in an inertial frame.

 

[rahaverhma]

The free rigid body is said to rotate about COM, here, all the frames other than COM are non-inertial as they have tangential acceleration produced by the torque of external force. Only COM is an inertial one. We need to subtract acceleration of such frames vectorially from the acceleration of particle to obtain its acceleration w.r.t. an inertial frame which is asked in most of the questions.

So, why the center of mass frame is an inertial frame ( moving without any acceleration due to rotation ) as also observed by an inertial observer (in real life cases) outside the rigid body system except where it sees the acceleration of COM imparted to it only due to the linear forces as per Newton's 2nd law.

Please show how the acceleration imparted to COM ( i.e. of whole rigid body system ) due to the torque of force is zero mathematically and thus, prove that the COM frame is an inertial frame.

 

[Orodruin]

Please show how the acceleration imparted to COM ( i.e. of whole rigid body system ) due to the torque of force is zero mathematically and thus, prove that the COM frame is an inertial frame.

This sounds an awful lot like you want us to solve your homework for you. It is also not compatible with your initial question where you soecified that there was an external force applied - in which case the CoM rest frame is not inertial.

 

[rahaverhma]

Is it wrong to break the free rigid body dynamics into pure translation and pure rotation parts? The COM has only linear acceleration, not any acceleration due to torque of force, it means as other particles on the rigid body has angular acceleration, alpha = F*r/I . ( F : external force, r : moment arm, I : moment of inertial ) then they have tangential acceleration = x*alpha. x is distance of any particle from center of rotation. And, torque due to internal forces will cancel out in pairs, not giving any angular acceleration. Then their accelerations' weighted average will give the total acceleration of COM. This acceleration of COM is due to torque of force that I am talking about.