Difficulty over a Problem of Friction about Sleeves

How can I solve this problem below:

The asking problem:
A safety device used by workers climbing ladders fixed to high structures consists of a rail attached to the ladder and a sleeve which may slide on the flange of the rail. A chain connects the worker's belt to the end of an eccentric cam which may rotate about an axle attached to the sleeve at C. Determine the smallest allowable common value of the coefficient of static friction between the flange of the rail, the pins at A and B, and the eccentric cam if the sleeve is not to slide down when the chain is pulled vertically downward.

Note: I included the sleeve drawing with this file.
The problem was taken from a book, so I had all the data for this problem.

Here I will give you the answer of this problem for your verification of your resolution: Coefficient of static friction = 0.0533

A 17.2 kg box is released on a 38.1 degree incline and accelerates down the incline at 0.269 m/s^2. Find the friction force impeding its motion, and the coefficient of friction.

A box is given a push so that it slides across the floor. How far will it go, given that the coefficient of kinetic friction is 0.190 and the push imparts an initial speed of 4.20 m/s?

1.) A horizontal force of 3.00 N is applied to a 1.00-kg radio, initially at rest on a table with a level surface.
a.) Will the radio move if the coefficient of the friction is 0.45?
b.) What is the coefficient of friction if it just begins to move?
c.) What is its acceleration if the coefficient of the friction is only 0.0

It's not a problem but I need a good explanation please because I really don't understand:
I need someone to explain why when we have (for example) one block on another, why when we represent the friction force, its direction can sometimes become, in the same sense, not only direction but also acceleration? How is this possib

Question: A 16-kg sled is being pulled along the horizontal snow-covered ground by a force of 24 N. Starting from rest, the sled attains a speed of 2.0 m/s in 8.0 m. Find the coefficient of kinetic friction between the runners of the sled and the snow.

If an object slides directly down an inclined place, at a constant velocity of 6 m/s, how large is the coefficient of kinetic friction if the plane make an angle of 25 degrees with the horizontal?

A slide-loving pig slides down a 37° incline (Fig. 6-24) in twice the time it would take to slide down a frictionless 37° incline. What is the coefficient of kinetic friction between the pig and the slide?

A 1.2ft diameter ball is thrown onto a rough surface, such that it has a velocity of 6ft/s and a backspin angular velocity of w. If it is to stop backspinning at the same instant as the velocity is 0, the backspin w in rad/s is what? HINT: The answer is independent of the mass and coefficient of friction.
a. 17.8
b. 22.0
c. 2

Find the force P necessary to push a 50.6 kg block down an incline of 32.4 degrees with an acceleration of 1.78 m/s^2. P is parallel to the incline and the coefficient of friction between the block and the incline surface is 0.805.
Picture is attached.