Hello everyone, I cannot understand the logic behind determining the directions in the rolling problems. In all examples I have seen has a logic. Let us assume a rolling cylinder is moving to right as accelerating. If the cause of the acceleration is a torque the direction of the friction force is right. I can understand it as it is the only force exist and it has to be in the same direction as the acceleration. In the examples I have seen another assumptation is that if the acceleration is because of a force (e.g. gravitional force on a inclined plane.) the friction force is in the opposite direction of this force. I don't understand why. 1) Let us assume I exert a force shown as the red. How can we know which one of the yellow ones will be the frictional force. Is there a mathematical way to prove this? (Maybe something about non-sliding condition but I want to see it mathematically) 2) Also I wonder how would I determine the direction if both a force and a torque existed. 3) Let us say this time the rolling ressistance is not neglected and I am not trying to accelerate but deccelarate. (In the below image the red arrow shows the force I exert and the blue one shows the rolling deformation.) How can I determine the direction of the frictional force? 4) The last question is about the non-sliping condition. If I know the ressistance and friction forces and the coefficient of friction between the surface and the cylinder, how can I determine the maximum force I can exert without causing it to slip? Thanks in advance!