Friction, Statics, Finding maximum angle and minimum coefficient of friction

[papasmurf]

Homework Statement

If the coefficient of static friction at B is μs = 0.3, determine the largest angle
θ, and the minimum coefficient of static friction at A, so that the roller remains self-
locking, regardless of the magnitude of the force P applied to the belt. Neglect the weight
of the roller and neglect friction between belt and vertical surface.

Homework Equations

ƩM=0
ƩF_x=0
ƩF_y=0
F_A=μ_AN_A
F_B=μ_BN_B=0.3N_B

The Attempt at a Solution

I have attempted the solution quite a few times a couple of different ways. I keep getting stuck with variables that I can't solve for, so I have come to the conclusion that I am probably not going about it the right way. My first attempt, I took the moment about A, and then summed up the forces in the x and y directions, but wound up not being able to solve for θ or μ_A. I attached an attempt at a free body diagram, perhaps I don't have the friction forces going in the right direction? I would appreciate any help!

 

[KataKoniK]

Thank you for your question. Solving for the largest angle θ and the minimum coefficient of static friction at A can be a bit tricky, but here are some tips that may help you:

1. Draw a clear and accurate free body diagram: Make sure to include all the forces acting on the roller, including the unknown forces. In this case, the forces would be P, the normal force at A (NA), and the normal force at B (NB).

2. Use the equations of static equilibrium: As you mentioned, the equations of static equilibrium are ΣM = 0, ΣFx = 0, and ΣFy = 0. These equations will help you solve for the unknown forces and angles.

3. Consider the maximum and minimum values: Remember that you are looking for the largest angle and the minimum coefficient of static friction. This means that you will need to consider extreme cases where these values are at their maximum or minimum.

4. Think about the direction of the friction forces: In your free body diagram, the direction of the friction forces should be such that they oppose the motion of the roller, which in this case is clockwise. This means that the friction force at A should be pointing to the left, and the friction force at B should be pointing downwards.

I hope these tips will help you in solving the problem. If you are still having trouble, you can also try reaching out to your classmates or your instructor for additional help. Good luck!