(a) Object is rolling without sliding on a rough surface. What kind of friction is acting between the object and the surface?
(b) Object is moving such that the velocity of the center of mass is $V_{\omega}=R\omega$ ($R$=radius,$\omega$ =angular speed) then what kind of friction will act between the object and the surface?
The two answers of the above questions are respectively
A. Static friction, Rolling friction
B. Static friction, Kinetic friction
C. Rolling friction, Kinetic friction
D. Rolling friction, Static friction

Hint: Static friction is the maximum frictional force to keep the body at rest. When the object is moving it has already overcome the static friction and is moving under kinetic friction. If the object is rolling without sliding, then it has overcome the static friction and undergoes rolling motion influenced completely by the rolling friction. This is also called pure rolling motion.If the object is moving such that it satisfies the given relation $V_{\omega}=R\omega$ ($R$=radius, $\omega$ =angular speed), then also it has already overcome the static friction and is undergoing motion due to kinetic friction.

Complete answer:
Static friction:It is defined as the maximum force exerted by a surface on a body so long as it remains stationary is called the force of static friction.Kinetic friction is defined as a force that acts between moving surfaces.
Sliding friction:The minimum force required to keep the body moving over a surface such that it moves equal distances in equal intervals of time.
Rolling friction:The minimum force required to roll a body on a surface.

Now we know that if the object is rolling without sliding, then it has overcome the static friction and undergoes rolling motion influenced completely by the rolling friction. This is also called pure rolling motion. There is no sliding in this case. Hence the friction acting here is rolling friction.

If the object is moving such that the relation $V_{\omega}=R\omega$ (R=radius,$\omega$ =angular speed) holds, then also it has already overcome the static friction and is undergoing motion due to kinetic friction. Hence the friction acting here is kinetic friction

Hence option C is correct.

Note: We should carefully analyze the type of motion given in the question. Kinetic friction is a broader term that gets classified further into sliding and rolling frictions depending upon the motion of the object. The relation is satisfied in pure rolling motion as well but according to the options the correct answer in accordance to the first part is kinetic friction.