Box being pulled up a slope with friction

[alikim]

If I solve a problem of a box sliding down a slope or standing still on the slope, the force of friction is directed up along the slope.

What happens if there is a force F pulling the box up along the slope, but it's unknown if it overcomes gravity and box is moving up or it only slows the box down and it is still sliding down.

Since the force of friction is directed opposite the movement, is it directed up or down?

If the box is moving down:
ma = mg sinθ - F - μmg cosθ, a >= 0

If the box is moving up:
ma = mg sinθ - F + μmg cosθ, a < 0

Is there a way to write one "universal" equation to solve? Or do I check a conditions first, and then choose which equation to use?

 

[PeroK]

If I solve a problem of a box sliding down a slope or standing still on the slope, the force of friction is directed up along the slope.

What happens if there is a force F pulling the box up along the slope, but it's unknown if it overcomes gravity and box is moving up or it only slows the box down and it is still sliding down.

You have to calculate the net force to determine the direction in which motion is possible, before including the forces that resist motion.

Since the force of friction is directed opposite the movement, is it directed up or down?

If the box is moving down:
ma = mg sinθ - F - μmg cosθ, a >= 0

If the box is moving up:
ma = mg sinθ - F + μmg cosθ, a < 0

Is there a way to write one "universal" equation to solve? Or do I check a conditions first, and then choose which equation to use?

You have to solve the problem in two stages. Note that there is the intermediate case where the net force is insufficient to overcome friction, in which case the box remains at rest; and, the friction force is not the maximum.

Moreover, it's often the case that the coefficient of static friction is greater than the coefficient of kinetic friction, which adds another layer of complexity - and another step in the solution.

 

[john1954]

If I solve a problem of a box sliding down a slope or standing still on the slope, the force of friction is directed up along the slope.

What happens if there is a force F pulling the box up along the slope, but it's unknown if it overcomes gravity and box is moving up or it only slows the box down and it is still sliding down.

Since the force of friction is directed opposite the movement, is it directed up or down?

If the box is moving down:
ma = mg sinθ - F - μmg cosθ, a >= 0

If the box is moving up:
ma = mg sinθ - F + μmg cosθ, a < 0

Is there a way to write one "universal" equation to solve? Or do I check a conditions first, and then choose which equation to use?

Initially assume the box has zero acceleration and the friction is an unknown quantity. Solve for the friction force using the equations of statics. This is the friction force needed to prevent slip. Its sign will tell you which way the friction force has to point. if the magnitude of this force is greater than the max friction available then do a sliding analysis with the max friction available but pointing in the direction you just found.

 

[john1954]

If I solve a problem of a box sliding down a slope or standing still on the slope, the force of friction is directed up along the slope.

What happens if there is a force F pulling the box up along the slope, but it's unknown if it overcomes gravity and box is moving up or it only slows the box down and it is still sliding down.

Since the force of friction is directed opposite the movement, is it directed up or down?

If the box is moving down:
ma = mg sinθ - F - μmg cosθ, a >= 0

If the box is moving up:
ma = mg sinθ - F + μmg cosθ, a < 0

Is there a way to write one "universal" equation to solve? Or do I check a conditions first, and then choose which equation to use?

Write the general equation with the friction force input as unknown variable eg F_f. It does not matter which way F_f points as it is an algebraic variable that can have a + or - value. Perform a static analysis with all derivatives =0. Solve this equation for F_f. The answer will be the friction force needed to prevent motion. If the answer is positive you drew it in the correct direction on your FBD. If negative it points the other way. Take the absolute value of F_f and compare with mu R. If less than mu_static R the block sticks of greater, set F_f to +- mu_static R and solve the time dependent problem. This methodology applies to all friction problems.