Foundations — Friction — static (maximum), kinetic, rolling

Before you can read the parent note Friction, every symbol in it must mean something to you as a picture, not a letter. This page builds them one at a time, each resting on the one before.

1. Force, and the arrow that draws it

Look at the figure: the box gets three arrows. A long orange arrow pointing right = a strong rightward push. A short one = a weak push. The picture is the force. Whenever the parent note says "push with $40\text{N}$", imagine an arrow of a certain length.

The unit of force is the newton, written $\text{N}$. One newton is roughly the weight of a small apple resting in your hand.

2. Mass $m$, gravity $g$, and weight $mg$

Put these together and you get weight:

Why does the topic need this? Because friction is proportional to how hard surfaces press together, and on a flat floor it is weight that does the pressing. No weight → no press → no friction.

3. Normal force $N$ — the floor pushing back

Here is the symbol the parent note leans on hardest. Picture the box sitting still on the floor. Gravity pulls it down with $mg$. Yet it does not sink through the floor. Something must push up on it, exactly cancelling gravity.

In the figure, the violet arrow $N$ points straight up out of the floor, at $90^{\circ }$ to the surface. On a flat floor with nothing else pushing vertically, $N$ must exactly balance the weight: $N=mg(\text{flat floor, no vertical push})$

4. The friction force $f$ and its direction

Look at the figure. You push the box right with force $F$. The box tends to slide right, so friction $f$ (magenta arrow) points left — always the opposite of the sliding tendency. The normal force $N$ still points up; weight $mg$ still points down.

The parent note splits $f$ into three names depending on the situation:

• $f_s$ — static friction: surfaces stuck together, not yet sliding.
• $f_k$ — kinetic friction: surfaces sliding.
• $f_r$ — rolling friction: a wheel rolling.

The subscript is just a label saying which situation.

5. Proportionality $\propto$ and the coefficient $\mu$

The parent writes $f\propto N$. Read the symbol $\propto$ as "grows in step with."

The figure shows the straight-line picture: plot friction $f$ against normal force $N$, and you get a straight line through the origin whose steepness is $\mu$. A steep line = grippy surfaces = big $\mu$. A shallow line = slippery = small $\mu$.

The parent uses three flavours of this same number:

• ${\mu }_s$ — static coefficient (sets the ceiling $f_s^{max}={\mu }_{s}N$).
• ${\mu }_k$ — kinetic coefficient (sets sliding friction $f_k={\mu }_{k}N$), always ${\mu }_k<{\mu }_s$.
• ${\mu }_r$ — rolling coefficient, tiny.

6. Inequality $\le$ — why static friction is special

7. Angles, and the tools $\tan$ and $\theta$

The parent note's incline section uses an angle $\theta$ (Greek "theta") and the function $\tan$.

Why does the topic need $\tan$ specifically? At the angle of repose, the sliding force $mg\sin \theta$ just equals the max friction ${\mu }_{s}mg\cos \theta$. Divide one by the other and the $mg$ cancels, leaving $\frac{\sin \theta }{\cos \theta }=\tan \theta ={\mu }_s$. So $\tan$ is exactly the tool that answers "at what tilt does it slip?" — it packages both the driving and pressing effects into one ratio. (The deep dive on incline geometry lives in Inclined Plane Problems.)

8. Newton's Second Law — the master rule tying it together

This is why friction matters: it is one of the arrows we add up to find $F_{net}$. See Newton's Second Law and the bookkeeping tool Free Body Diagrams for drawing all the arrows at once.

Prerequisite map

Equipment checklist

Test yourself — reveal only after answering out loud.