Concept of Rotation

Consider a point on a cylinder (of radius $r$ ) that is purely rolling forward with angular velocity $\omega$ . Now follow its motion as the cylinder rolls. What is the translational velocity of this point relative to the surface it is rolling upon? $v = r \omega + r \omega \cos \theta$ where $\theta$ is the angle between the upwards vertical and your chosen point, measured clockwise. When this point is the point of contact between the cylinder and the ground, $\theta = \pi.$ Therefore $v_0 = r \omega + (-1)r \omega = r \omega - r \omega = \boxed{0}$ Note that this does not actually depend on the radius or angular velocity, as in this special case, it cancels out.