# Why does the bottommost point of a rolling body have a radial acceleration in the ground frame?

Let's say a solid cylinder of radius r is rolling on a stationary horizontal surface with linear velocity v, angular velocity $$\omega$$, linear acceleration a and angular acceleration $$\alpha$$

Bottom most point is P

Now I require to find the acceleration of the point P (on the cylinder).

To do this first I could see it in the COM( Centre of Mass ) Frame. I observe it to have an $$\omega ^2 r$$ centripetal acceleration and a leftward $$r\alpha$$ tangential acceleration.

Coming to the ground frame I would have to add the acceleration of COM to it which cancels the $$r\alpha$$ component (as it is rolling) leaving $$\omega ^2 r$$ upward acceleration.

However I can't get the intuitive understanding as to why there exists a vertical component.

In the ground frame the velocity of P is 0 and so is the tangential acceleration. Due to no velocity I can't directly assume it to have a radial acceleration.

Could someone explain intuitively why it does? Or maybe how I could derive the same by completely solving it from the ground frame?

Consider just a simple wheel that is spinning with constant angular speed $$\omega$$. Imagine there is a particle on the edge. Recall that velocity is speed with a direction.