# Ramp friction error: Answer seems to depend on coordinate system

I am a junior level mechanical engineering student, and have taken physics, statics, dynamics, etc. so I know how to do this problem, though something seems to be tripping me up.

Given a standard ramp friction problem such as the one below, and given a coefficient of static friction $$\mu_s$$, and knowing it is not moving, we can create a system of equations to determine all forces in terms of the mass. $$\vec{F_g} + \vec{F_N} + \vec{F_f} = \vec{0}$$ If we define $$\hat{x}$$, $$\hat{y}$$ to be unit vectors parallel, perpendicular to the bottom surface of the ramp: $$-mg \hat{y} + \sin{\theta} F_N \hat{x} + \cos{\theta} F_N \hat{y} - \cos{\theta} F_f \hat{x} + \sin{\theta} F_f \hat{y} = 0 \hat{x} + 0 \hat{y}$$ By substituting $$F_f = \mu_k F_N$$, and seperating the above into an equation solely with the $$\hat{y}$$ components, we obtain the following: $$F_N = \frac{mg}{\mu_s \sin(\theta) + \cos(\theta)}$$

If we define $$\hat{i}$$, $$\hat{j}$$ to be unit vectors parallel, perpendicular to the top surface of the ramp, respectively: $$F_N \hat{j} - F_g \cos{\theta} \hat{j} = 0 \hat{j}$$ therefore, $$F_N = mg \cos{\theta}\,.$$

Selecting arbitrary values for $$\theta$$ and $$\mu_s$$ shows that these two expressions are not equal. It is obvious that a coordinate rotation should not change our answer, so what is happening?

• Both equations are correct, it's just that they determine both $N$ and $\theta$. Mar 10, 2023 at 20:44

Both equations are correct, it's just that they determine both $$N$$ and $$\theta$$, so you've got two equations that you can then solve for these two quantities. In other words, there is only a particular angle at which all of these things can be true at once.
Because the friction is static, your equation for friction is only the maximum possible static friction. There is no formula for static friction. In your first example (i.e., horizontal & vertical), leave $$F_f$$ as $$F_f$$. You must use both horizontal and vertical components to eliminate the unknown $$F_f$$.