# Block on an inclined plane. 1 solution or 3?

There are actually an infinite number of solutions. You have just been assuming there is no static friction, or that static friction is equal to its maximum magnitude$$^*$$. But really all we have for the static friction magnitude is $$0\leq F_\text{fric}\leq \mu N$$ and then the friction force could be acting up or down the ramp, depending on the value of the mass (i.e. you could have in your equations $$-\mu N\leq F_\text{fric}\leq \mu N$$)

So really your mass should be able to take on any value between $$m_1$$ and $$m_3$$ (assuming your work is correct, which I have not checked).

If you want to check for another one of the infinite solutions you can do one of two things. The first thing you can do is just pick some other valid friction force value and direction (for example, $$F_\text{fric}=\frac12\mu N$$) and determine what the mass needs to be to prevent sliding. The other way is to pick a new mass $$m_1 and then determine what $$F_\text{fric}$$ needs to be in order to prevent sliding. If $$F_\text{fric}\leq \mu N$$ for that mass $$m_4$$, then you have found another valid solution.

$$^*$$Don't fall into the common intro-physics misunderstanding that $$F_\text{fric}=\mu N$$ for static friction. The only time you can use equality this is if you know that the object is right on the verge of slipping. In general all you can say is $$0\leq F_\text{fric}\leq\mu N$$, and so it is often incorrect to replace $$F_\text{fric}$$ with $$\mu N$$ in certain problems.

In the way you interpreted the problem, you will not only find 3 solutions, but an infinite number of them. This is because the friction can take any value between 0 and $$\mu N$$ in both directions.

The way I interpret the problem though, is that the force is the minimal force necessary to prevent the block from moving. The block will move if the component of gravity is larger than $$\mu N$$, so the solution would be your number 3.