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# Geometric bonds

Carrying a spinning toy top around the room on a tray doesn’t affect its spinning. The top’s translation and rotation are not connected or coupled. But speed up on a bike and the spinning wheels will necessarily spin faster. Wheel translation and rotation do depend on each other – are coupled – in this case.

In fact, it seems that

• the curving wheel-surface follows the linear road-surface exactly. Also,
• the curving paint-roller surface follows the linear wall-surface and
• the curving yo-yo axle follows the linear string etc.

Different shapes with different geometries can be “bound” where their surfaces meeting. When rotation and translation are connected like this, let’s call it a geometric bond. How fast $v$ you move on your bike is somehow bound to the angular speed $\omega$ of its wheels – knowing $v$ should make it possible to figure out the matching $\omega$, and vice versa.

And what is this bond? If there is wheel-and-road contact, the distance $s$ passed over the road corresponds to the angle $\theta$ passed on the wheel. $\theta$ in radians is ‘a number of radius-lengths’,1 so if the wheel turns $\theta=2$, then 2 full radius-lengths have been passed. $s$ must be equal to 2 full radius-lengths, $s=2r$.

Our general geometric bonds are thus (equivalent for speed and acceleration – see Proof 22):

$$s=r\theta\\ v=r\omega\\ a=r\alpha$$

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