Signum function and friction

Question

There've been some times where I've seen people including the signum function, $\text{sgn}(v_x)$ in equations of motion to account for the instant when the horitzontal velocity component, $v_x$, changes signs and so friction needs to be in the opposite sense.

Take as an example a particle in a bowl with a rough surface: when it slides down (from left to right), $v_x$ points to the right and kinetic friction upwards to the left. Once it is stationary at the other site, $v_x$ changes its direcion to the left and viceversa with friction. Another example would be an horitzontal spring oscillating a mass on a ground with friction.

Where I'm getting at is when do we need to include this function and when there's no need to? Is it only for kinetic frictions which are constant? What about linear and quadratic drag? If we're talking in one dimension, we'd have $v=v_x$, so the linear one would include the sign itself, but not the quadratic one, because it is squared. What about in $2$ or $3$ dimensions? The drag $\propto v=\sqrt{v_x^2+v_y^2}\ $ or $\ v^2=v_x^2+v_y^2$, which are positive. I guess the signum function in this case would depend on $v_x$ and $v_y$ too.

Answer 1

1. Friction and drag forces $${\bf F}~=~-F(v)~ \frac{\bf v}{v}, \qquad v~:=~|{\bf v}|, $$ are often modeled as being opposite the velocity vector ${\bf v}$.

2. The magnitude $F(v)$ is typically a quadratic or linear function of speed $v$ for drag, while the magnitude $F(v)=\mu N$ is a constant/independent of speed $v$ for kinetic friction.

3. In 1D with a 1D vector ${\bf v}$ having a single component, we can identify $\frac{\bf v}{v}={\rm sgn}({\bf v})$.

4. A 1D example: https://math.stackexchange.com/q/4634046/11127