# Is the Instantaneous Axis of Rotation in rolling without slipping defined only in non-inertial frame about the point the point in contact?

Consider a body rolling on rough surface. And its rolling without slipping. Is the Instantaneous Axis of rotation about the point in contact defined only in non-inertial frame? I mean do we consider point ‘A’ as part of the rotating body in contact with the surface and then define the Instantaneous Axis of rotation or Can we define instantaneous axis about a point B at infinitesimally small distance from the point in contact which (B) (ie a inertial frame) is a part of the surface on which the body is rolling.

The instantaneous centre of rotation (ICR) is not a material point, and it does not belong to any object. It a geometric object defined by measuring the motion of a body from some reference frame (origin + basis vectors), such that the measured velocity of a body at that location is zero (in 2D).

What is the velocity of whatever particle happens to be under the location in space I am interested in? Each velocity measurement is like having a magnifying glass and measuring the velocity of what is on the focal point of the glass.

The point where the magnifying glass is focused on is just a location in space, and not attached to the object we are measuring.

The ICR is exactly like that, except it is the special point in space where the velocity of the object on this location happens to be zero.

If the velocity measurements are against an inertial frame, then we're looking at the absolute ICR of a body, and if we attach a measuring frame to a moving body then we're look at the relative ICR of a body.

You can rotate a body many ways. The axis of rotation can pass through the body or be entirely outside it. The axis is a mathematical line. It isn't part of the body.

To see the point of the instantaneous axis of rotation, imagine two identical bodies. One is rolling on a plane. It passes over a line x. The other is placed next to the line so it is tangent to the surface of the body. This body rotates around x. There is an instant where the two bodies are moving exactly the same way.

For that instant, you can use the 2nd body to figure out things about the first body. They have the same kinetic energy and angular momentum. The speed of each point in the body is the same. And so on.

Note that only inertial frames at rest with respect to the plane were used in this example. Of course we have to keep changing to different frames with a different 2nd body if we want to keep up with the 1st body.

At the contact point A , you have the tangent vector $$~\vec t~$$ and the normal vector $$~\vec n~$$ to the curve $$~f(x)~$$ . $$(~\vec n\perp\vec t)~$$

thus, the Instantaneous Axis of rotation is $$~\vec t\times\vec n$$

The "Instantaneous Axis Frame" is then

$$F_{Ai}=[\vec t_{A_i}~,\vec n_{A_i}~,~\vec t_{Ai}\times\vec n_{A_i}]$$