Foundations — Rolling without slipping — v = Rω condition

This page builds every symbol and idea the parent note Rolling without slipping — the v = Rω condition leans on — starting from nothing. If a term made you pause up there, it gets earned right here.

1. The picture we are always drawing

Before any symbol, fix the scene in your head: a circle (the wheel), sitting on a flat floor, moving to the right.

Everything below is a label on this picture: the dot in the middle, the arrow to the rim, the angle it turns, the speed it moves. Keep glancing back at it.

2. $R$ — the radius

• Plain words: "how big the wheel is, measured from middle to edge."
• The picture: the spoke drawn from center to rim.
• Why the topic needs it: the rule $v=R\omega$ literally has $R$ as the glue between forward speed and spin. A big wheel ($R$ large) covers more ground per turn; a tiny wheel covers less. $R$ is what encodes that.

Units: metres (m).

3. Angle in radians — $\theta$

Here is the first symbol that trips people up, so we build it slowly.

• Plain words: $\theta$ is "how much the wheel has rotated."
• Why radians and not degrees? Because radians are defined by arc length, they make the arc formula ridiculously clean: $\text{arc length}=R\times \theta$ No messy conversion factor. Degrees would force an ugly $\frac{\pi }{180}$ into every equation. We pick the tool (radians) precisely because it answers the question "how much rim did I unroll?" with a plain multiplication.

4. $s$ — arc length (the unrolled rim)

• The picture: the teal arc hugging the rim in figure s02.
• Why the topic needs it: "no slipping" means the rim laid onto the road (length $s$) equals the ground distance covered. That single equality is the entire derivation of $v=R\omega$. See Instantaneous axis of rotation for why the contact patch never slides.

5. $x_{cm}$ — position of the center

• Plain words: "where the middle of the wheel is right now."
• The picture: horizontal distance the center dot has slid to the right.
• Why the topic needs it: the no-slip magic is the statement $x_{cm}=s$. The center travels exactly as far as the rim unrolled — nothing lost to skidding.

6. Speed and $v_{cm}$ — how fast the center moves

• Plain words: "how fast the wheel is going forward."
• The picture: the length of the forward arrow on the center dot.
• Units: metres per second (m/s).

But how do we get speed from position? That needs one new tool.

7. The derivative $\frac{d}{dt}$ — the "rate of change" machine

• The picture: the slope (steepness) of a position-vs-time graph. Steeper line = faster = bigger derivative. A flat line = not moving = derivative zero.
• The two facts we use:
  • $\frac{d{x}_{cm}}{dt}=v_{cm}$ (rate of change of position = speed).
  • $\frac{d\theta }{dt}=\omega$ (rate of change of angle = spin rate — next section).

Because $R$ is a constant (the wheel doesn't change size), it slides straight through the derivative: $\frac{d}{dt}(R\theta )=R\frac{d\theta }{dt}.$ That is the single step that turns $x_{cm}=R\theta$ into $v_{cm}=R\omega$.

8. $\omega$ — angular velocity (spin rate)

• Plain words: "how many radians the wheel spins each second."
• The picture: a curved arrow around the center; longer arrow = spinning faster.
• Units: radians per second (rad/s).
• Why the topic needs it: $\omega$ is the spin half of the pairing. See Angular velocity and angular acceleration for the full story.

9. $\alpha$ — angular acceleration (spin speeding up)

• Plain words: "how quickly the spin rate ramps up."
• The picture: the curved spin arrow growing over time.
• Units: radians per second-squared (rad/s²).
• Why the topic needs it: differentiate $v_{cm}=R\omega$ once more and you get $a_{cm}=R\alpha$, the rule that governs Rolling down an incline.

10. $a_{cm}$ — the center's (linear) acceleration

Paired with $\alpha$ by the same constant-$R$ derivative trick: $a_{cm}=R\alpha$.

11. Vectors and $\vec{\omega }\times \vec{r}$ — the arrow toolkit

The parent note writes ${\vec{v}}_{point}={\vec{v}}_{cm}+\vec{\omega }\times \vec{r}$. Let's decode every mark.

• Why this tool? We need to add the "swinging" motion of a point to the "sliding forward" motion of the whole wheel. The cross product is the exact machine that turns "spin rate + where you are" into "which way and how fast you're being carried around." See Instantaneous axis of rotation.

12. Friction — the invisible enforcer

• Why the topic needs it: no-slip rolling relies on static (not kinetic) friction. Because the contact point isn't moving, static friction does zero work — mechanical energy is conserved. Contrast static vs sliding grip in Static vs kinetic friction.

How the foundations feed the topic

Equipment checklist

Test yourself — can you answer each before revealing?