Intuition for why friction on rolling objects is in the same direction as motion? When we usually draw friction, it is opposite the direction of motion. However, in rotational motion problem when a cylinder is rolling on the floor, it is in the same direction as the motion. I did some math, and figured out that indeed, if you make it point to the left then it will have a negative sign. While I am convinced of its validity, I do not understand why this is intuitively true, and why rolling friction is then opposite to sliding friction.
EDIT: Upon request, here is a diagram. The friction at the bottom should be negative and point the other way, but I don't know why.

 A: Firstly, a more general advice(something that was told to my class by my professor):
The "opposite to the motion" direction of friction is not the best way to see it. In fact, nothing in physics should be viewed as being an absolute rule except from the very basic foundations of physics, which are its laws. One case in which friction is not opposite of motion(well, the one component of friction) is the case with a motorcycle making circles in a cylinder. In order for the motorcycle to be able to stay at a constant height, friction has to be applied upwards. So, here the direction of motion is tangent to the circle that the motorcycles does but one component of friction is perpendicular to that direction. So, use your intuition to find out where does the friction point. 
Now, for your particular example, you can think of it like the friction that "helps" us walk. In order for you to take a step forward, you need to push the Earth to the back. This can be achieved via the friction between your shoe and Earth.  The same applies for your example. In order for the cylinder to roll, it needs to push the Earth backwards, and so the Earth needs to push it forward. So, that forward force is the rolling friction. The sliding friction is the same case as if you had a box that was moving forwards and the friction was pushing backwards(here, it opposes the motion and indeed it is opposite of the direction of motion, where direction of motion is the direction of the sliding velocity).  
A: Frictional forces between two surfaces try and prevent relative movement between the two surfaces (static friction) or reduce the relative movement between the two surface (kinetic friction).
This does not necessarily mean that friction opposes motion.
If a ball of radius $r$ is rolling on horizontal surface without slipping $v = r\, \omega$ where $v$ is its translational velocity $v$ and $\omega$ is its angular velocity.
In such a state the frictional force due to the surface over which it is rolling is zero.  
Now consider a ball projected along a horizontal surface with a velocity $v$ and angular speed $\omega$ zero ie it is not rotating.  

If there are no frictional forces between the ball and the surface then the ball will continue to move a velocity $v$ and not rotating.  
Now suppose that there is a frictional force $F_{\rm bot}$.  
The no slipping condition $v=r\omega$ is obviously not satisfied and at the point of contact there is relative movement between the bottom of the ball and the surface.
At the point of contact the frictional force due to the surface acting on the ball affects the ball in two ways to try and get the ball to the no-slipping condition.  


*

*The frictional force reduces the translational velocity of the ball
as $F_{\rm bot}$ is acting in the opposite direction to the
translational velocity.

*The frictional force applies a torque about the centre of mass of the
ball $C$ such that the angular velocity of the ball is increased.  
So $v$ is decreasing and $\omega$ is increasing until you get to the no-slipping condition $v=e \omega$.
Si the kinetic frictional force has acting in a direction such as to reduce the relative movement at the point of contact between the ball and the surface.  

Now you should be able to state and explain the direction of the frictional force in this case where the ball is rotating and initially has zero translational velocity.  

A: Well, though it's true that direction of friction is not always in the opposite direction of the motion, above example isn't really accurate.
Rolling friction is actually a misnomer, actual term is Rolling Resistance and direction of Rolling Resistance is diffused as it's origin is different than 'Friction'. Rolling resistance arises due to deformation in contact surfaces and thereby loss of kinetic energy in addition to sliding friction in different directions tangent to the deformed surfaces.
Thus, there is one component of rolling resistance that is in the opposite direction of the motion.
If Rolling friction is assumed to be in the direction of motion as explained above in the figure, the ball will continue to accelerate since there will be no  linear stopping force.
