Need help understanding an equation for statics problem

[dlacombe13]

Homework Statement

The block resting on the inclined plane shown has a mass of 40kg. Determine the maximum and minimum value for P for which the block is in equilibrium. (f_s = 0.35 and θ=25°)

Homework Equations

ΣFx = 0
ΣFy=0
F_max=(f_s)(N)

The Attempt at a Solution

w_x = (392.4)(sin(25)) = 165.84
w_y= (392.4)(cos(25)) = 355.64
F_max= (0.35)(355.64) = 124.47

-P_max + F_max + w_x = 0
P_max = 290.31 N

-P_min - F_max + w_x = 0
P_min = 41.37 N

Okay so the problem isn't that I couldn't solve it: I followed my notes by my professor and got the answers right. My problem is my understanding of the concept behind some points in the equations. My questions are:

1) What exactly are Pmin and Pmax?
2) When solving for Pmin and Pmax, I understand that it is just the equilibrium equation for the whole system (in this case ΣFx because the friction is parallel to the surface) right?
3) If so, why does the F_max turn to negative when solving for the minimum? I can see why it is positive since it is in the positive direction (right), but why does it turn negative all of a sudden?

 

[CWatters]

For large P the block will try to slide up the slope so friction acts down it. For small P the block will try to slide down so friction acts up the slope. This is why the sign of Fmax changes.

In order to be "in equilibrium" the tension P in such a rope would have to fall within a range between Pmin and Pmax depending on which way friction acts (up or down the slope).

 

[dlacombe13]

Oh okay I get it now. Basically Pmin to Pmax is a range, where if the force P is below the minimum, it will not be enough force to prevent the block from sliding downwards, And if the force P is above the maximum, the force P will overcome the friction and move the block upwards. And for the force P within this range, the block remain still, or in equilibrium.

Thank you very much for clarifying! (: