What determines the direction of static friction forces?

[rhotonsix]

My understanding of static friction is that it is a resistive force in response to an applied force. I recently read the following example. A rope of uniform density, length L, is fastened to a plane with incline angle “theta” along its length. The question asks for the tension at the top of the rope where it is fastened to the plane. The solution states that the direction of the static friction depends on how the rope arrived on the plane - if it is simply placed on the plane and fastened the static friction points up the plane, while if the rope was dragged up the plane and fastened, the static friction force points down the plane. This in turn will affect the tension. I get why if one drags the rope up the plane there is a resistive kinetic friction force in the opposite direction. But it seems to me that once the rope is fastened on the plane and is at rest, the forces it experiences are the same regardless of how it got there and hence the static friction would be the same. Where am I wrong? Thanks in advance.

 

[PeroK]

Summary:: Why does the direction of static friction on an object depend on how the object was placed on the surface?

My understanding of static friction is that it is a resistive force in response to an applied force. I recently read the following example. A rope of uniform density, length L, is fastened to a plane with incline angle “theta” along its length. The question asks for the tension at the top of the rope where it is fastened to the plane. The solution states that the direction of the static friction depends on how the rope arrived on the plane - if it is simply placed on the plane and fastened the static friction points up the plane, while if the rope was dragged up the plane and fastened, the static friction force points down the plane. This in turn will affect the tension. I get why if one drags the rope up the plane there is a resistive kinetic friction force in the opposite direction. But it seems to me that once the rope is fastened on the plane and is at rest, the forces it experiences are the same regardless of how it got there and hence the static friction would be the same. Where am I wrong? Thanks in advance.

It can depend on whether the rope is itself under tension. The static friction makes up the difference between the other forces on each segment of the rope.

A clearer example would be a block attached to a spring. If the spring is at its natural length, then the block is trying to slide down the incline. If, however, the spring is extended it will be trying to pull the block up the incline. Static friction on the block will be in opposite directions in those two cases.

 

[Dale]

Summary:: Why does the direction of static friction on an object depend on how the object was placed on the surface?

My understanding of static friction is that it is a resistive force in response to an applied force.

Both the magnitude and the direction of static friction are whatever is needed to prevent slipping.

 

[vela]

But it seems to me that once the rope is fastened on the plane and is at rest, the forces it experiences are the same regardless of how it got there and hence the static friction would be the same. Where am I wrong?

You're neglecting the fact that the rope will stretch as you drag it up the slope. When you stop dragging, it's not guaranteed that all parts of the rope will continue to move far enough to get the rope back to its unstretched length.

 

[A.T.]

But it seems to me that once the rope is fastened on the plane and is at rest, the forces it experiences are the same regardless of how it got there and hence the static friction would be the same.

But which forces would that be? There are many combinations of friction and fixation force that would keep the rope in equilibrium.

See also:
https://en.wikipedia.org/wiki/Statically_indeterminate

 

[Lnewqban]

... But it seems to me that once the rope is fastened on the plane and is at rest, the forces it experiences are the same regardless of how it got there and hence the static friction would be the same. Where am I wrong? Thanks in advance.

I would replace the idea of anchoring the rope with clamping the top end of the rope against the plane, only to avoid manipulation of the rope along the surface of that plane.

If we drop the rope on the plane, it will naturaly tend to slide down the plane.
If it doesn't, we have enough static friction force and we don't need the clamp: friction force vector is pointing up the plane.

If it does slide down, we have dynamic friction opposing that movement; then, we clamp the top end of the rope and stop the sliding movement, which increases the value of the friction force from kinetic to static, but without changing the up-slope direction of its vector.

Imagine that the plane-oriented component of any external force is appied to the center-line of the rope, while friction force is applied onto the bottom surface line.
There is a minor but real elastic deformation of the fibers of the rope.
Because of that, in this case, any paired point of the top surface line will be a little displaced down the slope respect to its pair at the bottom surface line.

 

[rhotonsix]

Thank you for all the insightful comments. Lnewqban, I initially approached the problem with the thought process you describe but when the solution included the scenario where the rope is dragged up the plane and fastened is where I got confused. As Vela said, I neglected the stretch associated with dragging the rope up the plane, which I think is determined by both internal factors and external forces (g and friction, which is initially static, then kinetic as it is being dragged, and then static again once at rest) and the residual stretch is the tension force experienced by the top of the rope where it is fastened to the plane. At least this makes sense to me based on my novice conceptualization lol.