How can a body have two axis of rotation at the same time?

Question

I m not concerned with rotation of a body with two simultaneous axis but concerned with how we choose the axis,while going through pure rolling I have observed that there are two axis of rotation one is passing through the center of mass and the other is through the point in contact with the ground,my concern is how can there be any axis of rotation through the point of contact where as m very well finding the body does not rotate in that axis of rotation that is it very well rotates only through the center of mass.

Answer 1

A body will only have one instantaneous axis of rotation (Chasle's Theorem).

In your example, the center of mass translates, and thus it is not a center of rotation. The only center of rotation is the point that does not move (the contact point).

To find the location of the center of rotation relative to some point A where you know the velocity vector $\boldsymbol{v}_A$ for a body with rotational velocity vector $\boldsymbol{\omega}$, calculate the following

$$ \boldsymbol{r}_{\rm rot} = \frac{ \boldsymbol{\omega} \times \boldsymbol{v}_A }{ \| \boldsymbol{\omega} \|^2} $$

_{Here $\times$ is the vector cross product.}

So consider the motion of the center $v$ to the right, and the no-slip condition $v + \dot{\theta} R =0$. In vector form the motion of the center is

$$ \begin{aligned} \boldsymbol{v}_A & = \pmatrix{v \\ 0 \\0}& \boldsymbol{\omega} & = \pmatrix{0 \\ 0 \\ -\frac{v}{R}} \end{aligned} $$

The center of rotation is thus

$$ \boldsymbol{r}_{\rm rot} = \frac{ \pmatrix{0 \\ 0 \\ -\frac{v}{R}} \times \pmatrix{v \\ 0 \\0} }{ \| \pmatrix{0 \\ 0 \\ -\frac{v}{R}} \|^2 } = \frac{ \pmatrix{0 \\ -\frac{v^2}{R}\\0}}{ \left( \frac{v}{R} \right)^2} = \pmatrix{0 \\ -R \\ 0}$$

which is in fact the contact point.

Answer 2

... it very well rotates only through the center of mass.

This is not the case. In some applications, it's most convenient to describe the motion of a rigid body as a combination of translation by the center of mass and rotation about an axis passing through the center of mass. You appear to have fallen into the trap of thinking that this is the only way to describe the motion of the rigid body. In other applications, it's even more convenient to describe the motion as a combination of translation of some other central point and rotation about an axis passing through that point.

A rigid body can be viewed as having an infinite (uncountably infinite) number of axes of rotation. Suppose you know the velocity of some central point $c$ of the object and the object's angular velocity. The velocity of some other point of the rigid body $p$ is $\boldsymbol v_p = \boldsymbol v_c + \boldsymbol \omega \times (\boldsymbol r_p - \boldsymbol r_c)$. There's nothing special about the center of mass in this construction. You can pick any arbitrary point on, inside, or even outside the body as the central point.

Another way to look at it: Angular velocity is a free vector. It's the same for every point inside or on the rigid body.

Answer 3

It isn't so. You can't have two axes at once. You can combine rotation along two axes to give a effective rotation along a single axis and the other way round.