# Negative friction force, positive normal force

I realize that formulas don't always tell the whole story, but there seems to be some information missing from the equation: $F_f=\mu F_N$

Imagine we say up and right is positive. Then if an object is moving across a surface to the right, the friction force should be negative (acting against the positive applied force), and the normal force should be positive (acting against the gravitational force).

Let's say: $F_N=882\ \text{N}$ and $\mu=0.600$, then: $F_f=0.600\times822=529\ \text{N}$, which is positive, when it should be negative.

I realize that this formula converts between axes, so there can't be a reliable way to relate the signs, but then what is the convention? Does this formula just provide an absolute value for friction, and you can set it positive or negative, depending on the situation?

• The sign comes from the unit vector part of the force vector, which is a magnitude and a direction. – honeste_vivere Feb 2 '16 at 0:09
• @honeste_vivere Thank you. OK. I understand where the sign comes from, I'm just wondering, if there is anything wrong with just changing the sign, if I know what it is supposed to be? – tristanslater Feb 2 '16 at 0:16
• Um, I am not sure what you are asking. If you know the sign, why would you need to change it? – honeste_vivere Feb 2 '16 at 0:17
• @honeste_vivere, the sign is obvious, because it acts against the applied force, but the equation gives you the opposite sign, hence you need to change the sign. Mathematically, it doesn't make sense, but contextually it does. I am just trying to figure out if that is acceptable in this particular type of case. – tristanslater Feb 2 '16 at 0:22
• No, the equation does not give you the opposite sign (it cannot otherwise it would not be an equation). You should write the equation in terms of vectors, not magnitudes. – honeste_vivere Feb 2 '16 at 0:30

It would probably be wiser to state the friction law as:

$$|F_F|=\mu |F_N|$$

where $$|F_N|$$ denotes the modulus of the Normal force.

Now consider the following diagram:

Both blocks and slopes are identical.

1. Left: some net force on the block causes an acceleration $$a$$ (left and up). The friction force $$F_F$$ points in the opposite direction: it opposes relative motion.
2. Right: some net force on the block causes an acceleration $$a$$ (right and down). The friction force $$F_F$$ points in the opposite direction: it opposes relative motion.

So the friction force opposes relative motion between the sliding surfaces.

• That's what I was asking about absolute value. Then, you would actually get $F_f=\mu F_N\ or\ F_f=-\mu F_N$, in which case, you choose the one that makes sense. Positive for left example, and negative for right example. – tristanslater Feb 2 '16 at 1:15
• @tristanslater You would never get it with a minus. The formula is as it is, empirically derived and containing non-negative variables. You seem to wish to determine the direction even when the formula only gives magnitude. Direction is not included in this formula – Steeven Feb 3 '16 at 19:05

In the relation $F_f = \mu F_{N}$, $F_f$ and $F_{N}$ are magnitudes of the frictional force and the normal force, and they are both positive. (I assume you are considering kinetic friction.)

The direction of the frictional force is in such a way that it opposes the relative motion between the two surfaces. This cannot be inferred from $F_f = \mu F_{N}$.

• This is starting to get me somewhere. Why is it that the friction force cannot be negative? Can you not use sign to denote a direction? I've seen this done many times. – tristanslater Feb 2 '16 at 1:01
• $F_f$ is by definition "the magnitude of the frictional force." It can't be negative. – higgsss Feb 2 '16 at 1:05
• But isn't force a vector and therefore has a magnitude and direction. Can't we use the sign to denote direction? – tristanslater Feb 2 '16 at 1:08
• Force is a vector, but its magnitude is a scalar. The equation $F_{f} = \mu F_{N}$ relates the magnitudes. The vector equation $\vec{F}_{f} = \mu\vec{F}_{N}$ does not hold; $\vec{F}_{f}$ and $\vec{F}_{N}$ are not parallel. – higgsss Feb 2 '16 at 3:18
• I think you are getting hung up on notation, and missing the point. I might need to restate my question... – tristanslater Feb 3 '16 at 18:01

No, the minus sign is not used to show direction.

A vector is just a direction and magnitude and cannot be negative. But it may be subtracted from another vector. This minus sign does not show direction but just a mathematical procedure.

But it turns out that subtracting a vector $\vec A$ form another vector $\vec B$: $\vec A-\vec B$ is the same as adding the vector $\vec A$ to the reversed vector $\vec B$: $\vec A+(-1)*\vec B=\vec A+\vec B_{reverse}$

Thus is fundamental (and pretty obvious), but this is where the issue starts. The minus sign is not defining a direction of the vector, but if it appears in our math it cab be interpreted to flip a vector around.

In general in physics, especially with forces: If you ever happen to get a minus sign in a result of the magnitude of a force, it just means that the vector is flipped around compared to whatever direction you defined to begin with (if that direction somehow is included in the calculation). If you do not get a minus sign, it doesn't mean anything at all about direction.

After thinking over the answers and discussion, I believe that the problem is just that using sign to denote direction only really holds up in one dimensional problems (unless you're using complex numbers). Therefore, the solution is to use the formula $F_f=\mu F_f$ to deal with magnitude only, and if using sign notation, you'll just have to apply the sign to the resulting magnitude as it makes sense for the scenario.

## Update

After thinking on this a little more, the arrow notation isn't really any better for solving this situation. You'll be stuck with the same decision of applying the arrow based on direction of motion.

I think what I didn't make clear in my original question was that I wasn't asking for clarification about the situation, I just wanted to check on best practice / convention, since I am unexperienced.