Understanding the Concept of Axis of Rotation: Definition and Explanation

[andyrk]

Homework Statement

This is more of a conceptual doubt.
Why does the axis being called as the axis of rotation of a rolling body have to be at rest with respect to some frame of reference?

What is the definition of axis of rotation? When is an axis called an axis of rotation?

 

[Tanya Sharma]

Homework Statement

This is more of a conceptual doubt.
Why does the axis being called as the centre of rotation of a rolling body have to be at rest with respect to some frame of reference?

What you are referring to as center of rotation being at rest is called as Instantaneous axis of rotation .The axis of rotation doesn't necessarily have to be at rest .If the axis is at rest then the body is undergoing purely rotational motion .Otherwise you may decompose the motion of the body in translation motion of the center of rotation + rotation of the object about the center .

For example in case of a sphere rolling on a flat surface ,the body can be considered to be rotating about the center of the sphere (also CM), where the center itself is translating .Or,you may consider the body in purely rotational motion about the bottommost point of contact of the sphere with the surface .The bottommost point is instantaneously at rest .A line passing through this point and perpendicular to the plane of motion of the sphere is called as the Instantaneous axis of rotation (IAOR) .

 

[andyrk]

What you are referring to as center of rotation being at rest is called as Instantaneous axis of rotation .The axis of rotation doesn't necessarily have to be at rest .If the axis is at rest then the body is undergoing purely rotational motion .Otherwise you may decompose the motion of the body in translation motion of the center of rotation + rotation of the object about the center .

For example in case of a sphere rolling on a flat surface ,the body can be considered to be rotating about the center of the sphere (also CM), where the center itself is translating .Or,you may consider the body in purely rotational motion about the bottommost point of contact of the sphere with the surface .The bottommost point is instantaneously at rest .A line passing through this point and perpendicular to the plane of motion of the sphere is called as the Instantaneous axis of rotation (IAOR) .

So can any point on the whole sphere be treated as the centre of rotation through which the axis passes? What about the direction of the axis of rotation? I mean, say for example in the case of a rolling sphere we take the axis of rotation as the CM. And we represent the sphere in the plane of paper. So does the axis of rotation have to come perpendicularly out of the plane of paper? Or can it also make some angle with the normal to the plane of paper?

 

[andyrk]

Is anyone there?

 

[HallsofIvy]

You expect people to be waiting for you to post? It is hardly reasonable to expect a response in
less than an hour and a half!

So can any point on the whole sphere be treated as the centre of rotation through which the axis passes?

That sentence is not very clear. Perhaps you mean "any point on the whole sphere through which the axis passes can be treated as the centre of rotation".
There is NO such thing as the centre of rotation of a three dimensional body. Only a two dimensional figure can have a centre of rotation. If is true that if you were to slice a body in a plane perpendicular to the axis of rotation then the centre of rotation of that two dimensional figure would lie on the axis of rotation.

What about the direction of the axis of rotation? I mean, say for example in the case of a rolling sphere we take the axis of rotation as the CM. And we represent the sphere in the plane of paper. So does the axis of rotation have to come perpendicularly out of the plane of paper? Or can it also make some angle with the normal to the plane of paper?

I think what you are saying about the axis of rotation "have to come perpendicularly out of the paper" is just the converse of what I said about "slice the body in a plane perpendicular to the axis of rotation. But I am not clear what you mean by "represent the sphere in a plane of paper".

 

[andyrk]

I think what you are saying about the axis of rotation "have to come perpendicularly out of the paper" is just the converse of what I said about "slice the body in a plane perpendicular to the axis of rotation. But I am not clear what you mean by "represent the sphere in a plane of paper".

I meant that while solving the problem you can't possibly draw a 3D representation of a sphere with a pen, which is rolling without slipping. You will have to draw it in 2D (i.e plane of paper) by making a circle and say that it is a sphere. And you will take the axis of rotation through the CM of the sphere i.e its geometrical centre or the IAOR (Instantaneous Axis Of Rotation) in this case. So the axis would come out of this point PERPENDICULAR to the plane of paper. So my doubt was, can this happen only at the CM or the lowermost point OR in this case (and say for any rolling/rotating case) can the axis of rotation pass through any point on the whole sphere? And the second doubt was that is it necessary for the axis of rotation to come perpendicularly out of the plane of paper? Can it also come out at some angle to it? If not then why? If yes then how would we apply the torque equation then? For all the points on the boundary of the sphere the shortest distance between them and the axis of rotation would not be now equal to R as the axis of rotation is not perpendicular to the plane of paper and rather it makes some angle to it. So things turn out differently. Let's take the axis of rotation as the axis but not perpendicular to the plane of paper and through the CM. Like say a point on the surface of the sphere would not have velocity Rω but rather something like Dω as the perpendicular (shortest) distance of the point from the point to the axis of rotation has changed as the axis does not come out perpendicularly out of the plane of paper. So that shortest distance has changed from R to D. But this changes the entire motion of the sphere and this should not be so. So I got very confused here as it is nowhere mentioned what CONDITIONS are NECESSARY for an AXIS to be CALLED AXIS OF ROTATION. Are there any conditions for it? Basically, that's what I am looking for but haven't got a clue till now. If there aren't any conditions than any axis through the sphere in any direction can be an axis of rotation. But then if we apply the Rω concept using different axis of rotation then Rω also changes with each axis. But it should not be like this as the velocity of the point needs to remain same REGARDLESS of what axis we choose. So I think there must be some conditions which I am not aware about. An axis is called the axis of rotation when all the system of particles of the rigid body appear to be rotating about that axis. So according to the problems and theory the only axis about which the system of particles of the rigid body appear to be rotating is the IAOR(Instantaneous Axis Of Rotation) and the axis through CM and that too both of them being perpendicular to the plane of paper. So is there NO OTHER Axis of Rotation about which the system of particles of the rigid body appear to be rotating around? If they don't appear to be rotating around any other axis then what do they appear to do?