How Does the Rotation Axis Affect Equilibrium in Torque and Center of Mass?

[modulus]

This is about a little thing that I have a lot of trouble understanding, and which always bugs me when studying equilibrium. I've often tried to apply the concepts of torque in this context, but nothing ever seems to make sense.

I consider a uniform rod which is initially at rest, and is not pivoted to any point by anything, to suddenly be acted upon by a force (or forces) - impulsive or otherwise - not passing through it's center of mass in a single plane. The center of mass of the rod should accelerate according to the net force acting on the rod in that plane. But I have a lot of trouble understanding how the rod would rotate, i.e., about which axis it would rotate.

I'll break down my question by taking two cases here:

1. The first case is when the net of the forces is zero. I'm assuming the rotation axis would then be through the center of mass, because it is at rest (right?).
2. Now, this second case gets me really confused: what if the net of the forces is not zero?? What would be the rotation axis then. I tried assuming the rotation axis would be the one about which the net torque is zero, and I applied that idea in some questions, but they all went wrong...

Please help.
Thank you.

 

[D H]

In a very real sense, one center of rotation is just as valid as any another one center of rotation in terms of describing the motion as a combination of translation and rotation. For a free body, it is convenient to select the center of mass as the center of rotation because this choice uniquely results in translational and rotational motion that become decoupled in the case of no external forces and torques. Choose some other point as the center of rotation and translational and rotational dynamics become intrinsically coupled. In other words, choosing something other than the CoM will result in something that's a bit messy.

For a body that is not free such as a link on a robotic arm, it is often convenient to choose some other location such as the joint as the center of rotation. As the joint is not located at the center of mass, rotation and translation are definitely coupled, but at least the constraints are (somewhat) easily expressed. Another example where choosing something other than the CoM as the center of rotation is a rocket that is firing its thrusters and is also rotating. The center of mass does not have a fixed location within the rocket due to the burning of fuel. Choosing a fixed point on the rocket as the center of rotation can once again simplify the equations of motion.

For your free body (presumably a rigid body of constant mass), the center of mass is the natural choice for the center of rotation. A force applied at some point away from the center of mass is equivalent to the identical force acting through the center of mass plus a torque (aka moment) about the center of mass given by the cross product r×[/b]F[/b], where r is the displacement vector from the CoM to the point at which the force is applied.

 

[AlephZero]

You seen to be trying to find "the point that doesn't move when the body rotates" and calling that "the rotation axis". That isn't a very useful idea, unless it is obvious where the point is from the kinematics of the problem (e.g. a wheel rolling without slipping).

A better way is to remember that you can always split the motion of a rigid body into two parts:
1. Translation of the center of mass, and
2. Rotation about the center of mass.
In other words, if you resolve the forces and take moments about the CM, you can then deal with the linear and angular acclerations separately.

 

[modulus]

Yes, that all helps a lot from the perspective of problem-solving.

But, I'm more interested in knowing about the actual thing. What will actually happen? About which axis will rotation take place. For example, if I need to find the angular momentum of the rod, I'll find it by taking the product of the rod's moment of inertia, and it's angular velocity.

And, the angular velocity will have to be around the real rotation axis, as will have to be the calculated moment of inertia...where will that axis be located??

 

[D H]

And, the angular velocity will have to be around the real rotation axis, as will have to be the calculated moment of inertia...where will that axis be located??

What "real" rotation axis? While there is a unique direction to rotation, there is no unique choice for placement of the axis of rotation.

At any point in time, there does exist an instantaneous axis of rotation (direction and location) such that the motion can be instantaneously described as a pure rotation (no translation whatsoever) about this axis. However, for a free body this is not a very useful idea and has zero physical significance.

For free constant mass rigid bodies, there is one choice, having the axis pass through the center of mass, that is more convenient than any other choice. Choosing the axis thusly is the choice that uniquely decouples rotational and translational motion.

Note well: That does not mean that either this pure rotation axis or an axis passing through the center of mass is the "real" rotation axis. There is no such thing as the one and only "real" rotation axis.

 

[modulus]

OK, thanks...that helped, especially the instantaneous axis part...thanks, again!