What is the direction of the frictional force acting on a rolling wheel? Let's keep things simple and assume that the wheel does not slide. Assume you are in a car that moves with constant speed. Obviously, the wheels exert a tangential force to the surface on the road in the points of contact, P. According to Newton's third law action equals minus reaction so the road exerts a force to the wheels in the direction of motion (please, see fig. 1). So the friction acts in the direction of motion of the car. It seems reasonable since if there were no friction then the wheels would be turning but the car would not move. However, there is one thing I can not understand. Let's now imagine a wheel rolling (without slipping) on a smooth surface with initial velocity V0 (here I mean that t). The same argument should hold (the wheel exert a tangential force to the surface on the road in the point of contact, P. According to Newton's third law action equals minus reaction so the road exerts a force to the wheel in the direction of motion). So the frictional force should again act in the direction of motion. Nevertheless, the wheel slows down with time which means that frictional force should point in the opposite direction to the direction of motion. Could someone give a QUALITATIVE explanation of the seeming paradox? I would also ask to use JUST the "language" of forces (not to apply the concept of the torque since it makes the explanation less vivid). 
 A: This may help you a bit
"Obviously, the wheels exert a tangential force to the surface on the road in the points of contact, P"
No!!! You must understand that the wheel is not capable to exert a force in any direction it wants or you want it to.  If the wheel is performing pure rolling motion, it will only exert a tangential force if it is accelerating and no tangential force is possible in case it's speed remains constant.
Here,you must also note that, it does "not"  depends on whether the surface is rough or smooth(frictionless), no tangential force is involved with the condition that it's speed remains constant(uniform rolling motion).
"I think"  you are relating rolling motion with the mechanism of walking. Don't do. You(because of your muscles and all that) can press the floor in any direction you want and at any time but  it is not possible for the wheel.
Now, you might be thinking why no tangential force(or friction) comes into play for uniform rolling motion.For this you must be knowing that
1-> The point of contact has zero relative velocity with respect the ground and also it does not even try to have a relative velocity(which it would try if accelerating due to an external torque)
2->Friction always acts so as to prevent relative slipping(or relative motion) between surfaces,i.e., it will come into play if either the surfaces are slipping relatively or are trying to slip.
Since in uniform pure rolling motion there is no "tendency"  of the point of contact to slip so no Friction (tangential force) and so no paradox.
You can also see how friction is not possible in "uniform" pure rolling like this->
If there is friction acting in forward direction (in direction of motion) then it must accelerate the wheel in forward direction besides it will also provide an external anticlockwise toque. So just imagine this situation -it no longer remains in uniform rolling motion(it's linear speed increase +
angular velocity decreases, and so it will also start slipping,i.e.,rolling stops). But, initially we supposed that the wheel to be rolling with uniform speed. This itself leads to a paradox and so, you can nicely conclude that friction does not act in uniform rolling motion(as it will not no remain rolling if friction acts).
You can now wonders that how a  car is able to accelerate forward then!!!But wait! it is not "uniform" pure rolling.  The car is accelerating! So static friction now comes into play. But how! Ok it is accelerating but it is not slipping, it's still rolling, so no relative velocity so no relative slipping(or relative motion) so no friction.
If you are thinking like above then the point you missed out is that I said that the it is "static" friction that comes into play not kinetic friction. And also when the car's engine apply a torque on the wheels the wheels(the point of contact of wheel) "try to slip" relatively to the ground. So static friction(our hero) comes,acts in forward direction and prevents that slipping and also accelerates the car forward. Now after travelling some distance the car stops accelerating and moves with uniform speed. Now that static friction vanishes as the now the point does "not even try to slip" relative to the ground.
At last, if you think this answer is good, then give it a second and upvote.
A: As the wheel is rolling without slipping the frictional force on the wheel is zero. This is because friction acts when objects slide over each other or when objects try to slide over each other. however at the contact point the velocity of the wheel relative to the ground is zero so there is no slipping so no friction acts. The declaration of the wheels translational and angular velocity is due to a concept called " Rolling Resistance"
A: If a car is gaining speed, the force from the road is forward.  If it is breaking, the force is backward.  If the speed is constant, the force is forward and large enough to overcome various sources of drag.
