# Rolling Motion

**Rolling** is rotation about an axis that is translating.

### Spinning, Skidding, Slipping

**Spinning** occurs when the rotational speed of the surface of an object is too high in comparison to the translational speed of the axis about which the object rotates. *Ex: Truck in mud*
$v_{cm}<rω$

**Skidding** occurs when the translational speed of a rolling object is too high compared to the rotational speed of its surface. *Ex: Drifting without wheels moving*
$v_{cm}>rω$

**Slipping** = spinning or skidding

### Rolling Without Slipping

If an object neither spins nor skids as it rolls, its motion is called *rolling without slipping* (also called ideal rolling).
$v_{translational}=v_{rotational}$
$v_{bottom}=0$
We have the following formulas, where $v_{cm}$ and $a_{cm}$ can be derived by taking the derivative.

$s_{cm}=rθ$ We can say the above only because rotation speed is the same as translation speed. I am having a hard time wrapping my head around this. Visualize below, the CM is covering the same distance as the arc s. $v_{cm}=rω$ $a_{cm}=rα$

When we looked at Rotational Kinematics and Dynamics, we often thought of the problem of a fixed rotation axis. But now, consider that the rotation axis is moving.

For stationary, $v_{cm}=0$, so the formulas below still apply

$v_{top}=v_{cm}+ωr$ $v_{bottom}=v_{cm}−ωr$ $v_{top}=v_{bottom}+2ωr$ Physics book, page 313.

### Rolling as Rotation about the Moving Centre of Mass

Assuming no slipping, $v_{bottom}=0$ $v_{cm}=v_{bottom}+ωr=ωr$ $v_{top}=2ωr$

### Friction, Rolling horizontally vs on an inclined surface

**Horizontal surface**

- If pure rolling has not started, need at least one external torque to initiate rolling
- If pure rolling has started, friction no longer needed If an object is moving horizontally, there is no static friction.

**Inclined surface**

- Friction non-zero

### Rolling Friction

Rolling friction can be quantitatively approximated using a coefficient of rolling friction. It results from deformation of object from surface.

### Misc. Knowledge

When you add spin angular momentum, it stabilizes.

Billiard ball thing#to-review Rolling without friction