Axis of Rotation for Rigid Body: How to Identify & Conditions

[andyrk]

for a perfectly rigid body, how can one identify what is the axis of rotation of the rigid body? What is the condition required for an axis to be called the axis of rotation?

 

[SteamKing]

This is a pretty vague question. It depends on what forces and moments are acting on the body.

 

[andyrk]

There is a solid sphere of radius R on a frictionless ground and we give a torque to it by a force F acting tangentially to its surface and parallel to the ground. We take torque to be FR because we take axis of rotation as the Centre of Mass of the sphere. Why is it that we take axis of rotation as the Centre of Mass of the sphere?

 

[voko]

If the torque is effected by a force applied at one point, the resultant motion will not be pure rotation, it will be rotation and translation. The instantaneous axis of rotation will not be in the center.

For the record, the axis of rotation is defined as a locus where the velocity is zero.

 

[andyrk]

Yes so in case of rotation and translation of a sphere the centre of mass doesn't have zero velocity. Yet it is considered as axis of rotation. Or is it that it is considered axis of rotation just when we are applying torque and not after it because then it has gained some velocity? So that's why we take the bottom most point in the sphere rolling without slipping as the instantaneous axis of rotation? So what would be the axis of rotation if a sphere is rolling with slipping?

 

[voko]

Yes so in case of rotation and translation of a sphere the centre of mass doesn't have zero velocity. Yet it is considered as axis of rotation.

This can be correct only in a reference frame co-moving with the sphere. In any other frame, the axis is elsewhere.

Or is it that it is considered axis of rotation just when we are applying torque and not after it because then it has gained some velocity?

When the entire body is stationary, you do not have any rotation, so there is no axis to it.

So that's why we take the bottom most point in the sphere rolling without slipping as the instantaneous axis of rotation?

Correct.

So what would be the axis of rotation if a sphere is rolling with slipping?

Depends. It may even be outside the sphere.

 

[andyrk]

This can be correct only in a reference frame co-moving with the sphere. In any other frame, the axis is elsewhere.

What would be the axis of rotation in an inertial frame of reference? Like ground?

 

[SteamKing]

The sphere is an object which doesn't have any preferential axis of rotation: it is entirely symmetric about its center because all points on the surface of the sphere are, by definition, located exactly r units from the center.

 

[voko]

What would be the axis of rotation in an inertial frame of reference? Like ground?

If not slipping, then you answered this above.

If slipping, then, as I said, it may be anywhere, the condition is that the velocity there must be zero.

 

[D H]

For the record, the axis of rotation is defined as a locus where the velocity is zero.

Better: The axis of rotation is that locus where the instantaneous velocity as expressed in some reference frame is zero.

Yes so in case of rotation and translation of a sphere the centre of mass doesn't have zero velocity. Yet it is considered as axis of rotation.

You're looking at things from a frame of reference that differs from the frame in which the axis of rotation is arbitrarily defined in this particular example.

A thought experiment: Hold an arm straight and point it straight down, with your upper arm tucked against your torso. Your elbow should be near your waist. Keeping your upper arm and elbow tucked against your torso, bend your elbow to make the fingers on that hand touch that shoulder. Now consider the point in time through this operation when your forearm is horizontal. Which is the correct way to describe the motion of your forearm and hand at this point in time:

• As a pure rotation about your elbow, or
• as a combination of translational motion of the forearm's center of mass and rotation about that center of mass.

The answer is that asking which description is correct is an invalid question. Both descriptions are valid. One description happens to make more sense in this case.

Here's an alternate definition of an axis of rotation: Pick an arbitrary reference point in some body-fixed frame, a frame of reference in which the rigid body is neither moving or rotating. At any point in time, the motion of the rigid body in some other frame of reference can be described in terms of a translation of that reference point plus a rotation of the rigid body about an axis passing through that reference point. This is Euler's rotation theorem. That axis is the instantaneous axis of rotation.

Note very well: Euler's rotation theorem is special to three dimensional space. It doesn't make sense in two dimensional space and it is not true in higher dimensions. This distinction is a bit irrelevant as we apparently live in a three dimensional universe. Also noteworthy is that key word "arbitrary". The axis of rotation is, by definition, going to pass through that arbitrarily selected reference point.The choice of that arbitrary reference point is up to you. It's arbitrary.

One consequence of Euler's rotation theorem is that you can always find a reference point such that the instantaneous motion is pure rotation about that point. The equations of motion simplify greatly if that instantaneous fixed point remains fixed throughout the motion, as is the case where an arm pivoting about a joint. It makes sense to describe the motion as pure rotation about this pivot point because this selection simplifies the equations of motion and because this selection is physically motivated.

In the case of your sphere that may be subject to external forces and torques, finding the point at which motion is instantaneously one of pure rotation doesn't make a lick of sense. Here arbitrarily choosing the center of mass as the reference point makes a good deal of sense because this choice decouples the translational and rotational equations of motion. The equations of motion get quite ugly if you choose some other point.