I am having trouble coming up with a consistent method of determining the direction of static friction.
So far the best I have come up with is: it should oppose the relative acceleration the contact point would have parallel to the surface of contact in its absence.
This seems to work up to a point:
A block sits on a rug. The rug is pulled to the left. The relative acceleration of the block to the rug is to the right. So static friction must act (in the frame of the rug) to the left.
A cyclist makes a turn to the right and leans to the right. The torque that his weight puts about the contact point will make it accelerate to the left relative to the ground in the absence of friction. So the ground sees static friction pointing to the right.
A coin rotates with a turntable. In the absence of static friction, the table sees a centrifugal force on the coin that will make it accelerate away radially from the centre of rotation. So friction must act radially towards the centre of rotation, in the frame of the table.
But then I run into some difficulties.
(already solved; see first answer) A cyclist makes a turn to the right. As a result he leans to the right. The resulting torque around the centre of mass causes the point of contact to slide to the left in the absence of static friction, so static friction must act to the right. However, now the cyclist as a whole must make circular motion with static friction acting as the centripetal force. As there is no torque, this means the contact point must move along with the centre of mass and therefore act as something with an acceleration to the right. So according to my understanding of static friction, another layer of it must now be added acting to the left. Why is it not?
A car enters a banked curve with speed $v$. Suppose the curve is banked at the right angle for the car to make circular motion in the absence of friction. In the absence of any static friction, therefore, the car's acceleration points towards the centre of the banked curve. However, in this case the car's acceleration still has a downward component along the slope. My understanding of the direction of static friction means therefore, when it is added in, it should act up the slope. But in reality no static friction is produced in such a case.
Can someone explain what is wrong with my method of finding the direction of friction? Or if there is nothing wrong, how have I misapplied it in this case to create this paradox?