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When I first studied friction I faced f = $\mu$ N where f is force of friction, $\mu$ is coefficient of friction for the surface considered and N is the normal force for the body on surface. Now f is a force so it must be a vector which means it has a direction. N is normal force so it has direction too. But both f and N are mutually perpendicular to each other! And $\mu$ is a constant, so how can we write f and N in one equation? They do not have the same direction!

I think this is not clear to me because I have never read about the origin of this equation. From where do we get it?

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The greek letter mu, which i intended to write looks like m. sorry! – Pallavi Roy Mar 17 '13 at 17:39
You can edit you question and write $\mu$ writing \mu between two dollar symbols. – PML Mar 17 '13 at 17:40
Thank you! No i meant to write it without the minus sign. – Pallavi Roy Mar 17 '13 at 17:48
@PML Er ... the frictional force is generally not colinear (positive or negative) with the normal force. Indeed it is generally perpendicular to it. You might want something like $\vec{f}_f = (- \hat{v}) \mu N$, but you rarely see it written out that way. Most treatments content themselves with saying something like "against the direction of motion" and leaving it at that. – dmckee Mar 17 '13 at 21:51
@dmckee Now this is a great example of what comes from not reading the answer carefully. Yes you're completely right.Thank you – PML Mar 17 '13 at 22:18
up vote 4 down vote accepted

The friction force $F \leq \mu N$ is just an idealized macroscopic law and makes no reference to any microscopic details. At the end of the day, you know that all interactions in nature are mediated by the 4 fundamental forces (gravitational, electromagnetic, weak, strong), and friction primarily so by electromagnetic interaction, but what the friction equation says is: screw all that microscopic details, let's just model the friction to be what is is, up to a max of $N$ times a constant of proportionality $\mu$.

This is akin to the case of an Ohmic resistor where we say the response (current) to an applied voltage of the device is $I = V/R$, where $R$ is some constant. Note that $R$ makes no reference to any microscopic details - whether it is made of silicon, germanium etc.

So you really shouldn't take the friction equation seriously and think that there is some deep connection between $\vec{N}$ and $\vec{F}$. Treat it more as a recipe for obtaining the approximate friction force of a macroscopic body, where the rule is $F \leq \mu N$, where $F$ and $N$ are the magnitudes.

At this point, notice that I've been writing $F \leq \mu N$ and not $F = \mu N$. This is because the friction force can take values between $0$ to $\mu N$ and is NOT EQUAL $\mu N$ in general. This is a common mistake by many young students of physics.

Now you are right, of course, that friction force should be a vector. So what's missing in this recipe is the direction, which I think most textbooks do not state very explicitly. Here it is:

1) for kinetic friction, the frictional force is maximal, and the direction points away from the velocity that the body is travelling in. I guess in this case you could 'write' it as one single equation: $\vec{F} = -\mu N \frac{\vec{v}}{|\vec{v}|}$ but this is kind of artificial.

2) for static friction, the frictional force is less than or equal to maximal, and the direction is in the direction in which it would decrease the acceleration if the system was frictionless. Basically you have to guess, but usually the physical situation is simple enough to deduce which direction it should be in. For example, say you have a book and you are pressing it against the wall, normally. If the book doesn't fall, obviously the frictional force points up, not down. If there was no friction, the book would fall, so force pointing up reduces the acceleration of the book which is the correct direction.

Note that there are many subtleties with the rules I've described. For example, let's say you have a cylindrical pot which is upright and you spin it in air, about the $z$ direction, and then you let the base touch the ground. Obviously the object will stop spinning due to the kinetic friction. But what is the 'velocity' of the body? The velocity of the center of mass is $0$. But for points that are radially away from the $z$ axis, they are travelling at different velocities! The rule needs some supplementing! It's an interesting question to think about. :) Cheers

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Good answer.... – Fabian Mar 17 '13 at 19:57

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