# Instantaneous axis of rotation for a cylinder

In Landau's Mechanics volume at section 32 solved problem no. 5, the instantaneous axis is chosen as the one which coincides with the line where the cylinder touches the plane. This is possible only if there is rolling without slipping. My question is, had it been rolling with forward slipping, the instantaneous axis should be somewhere placed outside the cylinder. Am I right?

PROBLEM 5. Find the kinetic energy of a cylinder of radius $$\:R\:$$ rolling on a plane, if the mass of the cylinder is so distributed that one of the principal axes of inertia is parallel to the axis of the cylinder and at a distance $$\:a\:$$ from it, and the moment of inertia about that principal axis is $$\:I$$.

• On the rotation plane, if you select two points on the cylinder (for example, point of contact $A$ and the furthest point from ground $B$) and draw velocity vectors $\vec{AA_1}$ (if there is no slipping $A_1\equiv A$) and $\vec{BB_1}$, then the intersection of lines $AB$ and $A_1B_1$ will give you the immediate centre of rotation $C$ (the axis will be perpendicular and passing through that point). Thus, it's straightforward to see that if $AA_1$ and $BB_1$ point in the same direction, then $C$ is outside of the cylinder and if vectors point in different directions, then $C$ is inside. Dec 4, 2021 at 18:54