Foundations — Banking of roads — derivation

This page assumes you have seen nothing. We build each symbol, draw its picture, and only then let the parent note use it.

0. What a "force" and its arrow mean

Before any letter appears, fix the picture.

Why we need this: the whole derivation is nothing but three arrows (weight, normal, friction) that must add up to one specific arrow (the inward pull). If you cannot picture arrows adding, no equation later will mean anything.

1. $m$ — mass, and $g$ — gravity's pull

Picture: an arrow of length $mg$ pointing at the ground, glued to the car's centre. It never tilts, never changes — gravity does not care that the road is banked.

Why the topic needs it: $mg$ is one of the three forces in the free-body diagram, and it is the thing the road's push must balance vertically.

2. Speed $v$, radius $r$, and "circular motion"

Look at the figure: even when the speed $v$ stays constant, the direction the car points keeps changing. A change of direction is still a change of motion — so the car is accelerating, and by Newton's Second Law an acceleration always needs a force.

3. Acceleration $a$ — the rate of change of motion

Picture: an arrow, separate from the velocity arrow, showing which way the velocity is being nudged. For circular motion this nudge-arrow always aims at the centre — that is why the pull must be inward.

Why the topic needs it: by Newton's Second Law, force and acceleration are chained together as $\text{force}=m\times a$. We need $a$ before we can say how big the inward force must be.

4. The centripetal requirement $\frac{m{v}^2}{r}$

This is the single most important quantity in the topic, so we build it slowly.

Read the formula as a story:

• More mass $m$ → harder to swing around → more pull needed (top grows).
• Faster speed $v$ → sharper change of direction each second → much more pull (it grows with $v^2$, the square).
• Bigger radius $r$ → gentler, lazier curve → less pull needed (bottom grows).

Why the topic needs it: every banking formula is just the sentence "the sideways part of the road's forces must equal $\frac{m{v}^2}{r}$." This is the target the arrows have to hit. Full story in Centripetal force.

5. The banking angle $\theta$ and the tilted road

Picture: the road cross-section is a ramp. The outer edge is lifted; the inner edge (near the centre of the circle) is low, like the inside of a bowl. See the same geometry in Inclined plane and Conical pendulum.

6. Choosing our two axes

Before we split any arrow, we must agree on which two directions to measure things along.

Why this exact choice: if we measure forces along axes that match the acceleration, the bookkeeping in §9 becomes "vertical forces cancel, horizontal forces make $m{v}^2/r$" — the simplest possible statement. Any other axes would mix the two together.

7. The normal force $N$ — and why it tilts

Because $N$ leans, it has TWO useful parts (measured along our two axes from §6):

• an upward part, $N\cos \theta$, that fights gravity;
• an inward part, $N\sin \theta$, that points toward the centre — this is the piece that does the turning.

Why the topic needs it: this tilt is the entire trick of banking. A flat road's $N$ points straight up and can help turn nothing; a tilted road's $N$ donates its inward slice to the centripetal job.

8. Splitting an arrow: $\sin \theta$ and $\cos \theta$

We keep saying "$N\cos \theta$" and "$N\sin \theta$". Here is why those exact factors appear.

Why these tools and not others: the acceleration is purely horizontal (inward) and zero vertical. So the natural thing is to break every slanted arrow into a vertical piece and a horizontal inward piece. Sine and cosine are exactly the machines that read off those two pieces from a length and an angle. Nothing else does that job.

9. Friction $f$ and its coefficient ${\mu }_s$

Picture: an arrow lying flat on the ramp surface. It can point up the slope (when the car is slow and slipping inward) or down the slope (when the car is fast and flying outward). Its length can be anything from $0$ up to ${\mu }_{s}N$.

10. Putting it on axes — $\sum F=ma$ per direction

That single rule, applied twice, is the derivation. Everything above just earned the symbols it uses — including $a$, the two axes, $N$'s split, and the static ${\mu }_s$.

Prerequisite map

Equipment checklist

Connections

• Banking of roads — derivation — the topic these foundations feed.
• Centripetal force — the inward requirement $m{v}^2/r$ built in §4.
• Uniform Circular Motion — why turning is accelerating (§2).
• Newton's Second Law — the per-axis bookkeeping (§10).
• Friction — the range-setting force (§9).
• Inclined plane — same arrow-splitting geometry.
• Conical pendulum — identical $\tan \theta =v^2/rg$ setup.