Suppose a cone is purely rolling (no slipping) around a fixed axis. I mean, it is revolving around a fixed axis perpendicular to the ground and passing through its vertex and also rotating, so the vertex is stationary. (sorry this might be a bit confusing but I hope you understand what I mean). Something like this, but rolling on plane surface instead of another cone: rolling cone http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Rolling_cone.pdf/page1-1024px-Rolling_cone.pdf.jpg
Now the instantaneous axis of rotation (IAR) of the cone is the 'line' that is touching the ground right? So how do you find the velocity of any other point using that? I mean, in the rolling wheel, you multiply the angular velocity by the distance from the IAR to get the velocity. Is it the same here?
If it is, then consider the center of base of the cone. If the height of cone is $h$ then its distance from the IAR is clearly $h\sin x$ where $x$ is the cone's half apex angle. So its velocity should be $ah\sin x$, where $a$ is the angular velocity with which the cone is rotating. Is this right?
Now we can also analyse the cone's motion by considering it in two parts: rotation + revolution, right? So again considering the center of base of the cone, it has no velocity by virtue of rotation (since the cone is rotating about an axis through the center), right? And by virtue of it rotating in a circle (of radius $h\cos x$) around the axis passing through its vertex, it has velocity $bh\cos x$ , where $b$ is the angular velocity with which the cone is revolving.
Now these two must be same, so we get $b=a \tan x$.
But Wikipedia states here that the ratio is $\sin x$.
And at the same time, this video (which I found in the external links section of the Wikipedia page) states that $a=b\cot x$ which is same as what I got.
So I'm really confused. Is everything that I did right? If not please correct me. Thank you.
Edit: OK so as carl commented, I'm also confused about how the instantaneous linear velocity of the center of the cone's base is different from the velocity of the center of a rolling disk.
Edit 2: How to find velocity of any point on the cone? There ought to be two approaches , one using IAR and another by considering motion as rotation + revolution but I'm not able to do it.