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## Rolling

## Discussion

Rolling without slipping is a combination of translation and rotation where the point of contact is instantaneously at rest.

When an object experiences pure translational motion, all of its points move with the same velocity as the center of mass; that is in the same direction and with the same speed

v(r) = | v_{center of mass} |

The object will also move in a straight line in the absence of a net external force.

When an object experiences pure rotational motion about its center of mass, all of its points move at right angles to the radius in a plane perpendicular to the axis of rotation with a speed proportional to the distance from the axis of rotation…

v(r) = | rω |

Thus points on opposite sides of the axis move in opposite directions, points on the axis do not move at all since ** r = 0 **there…

v = _{center of mass} | 0 |

and points on the outer edge move at the maximum speed…

v = _{outer edge} | Rω |

When an object experiences rolling motion the point of the object in contact with the surface is instantaneously at rest…

v = _{point of contact} | 0 |

and is the instantaneous axis of rotation. Thus, the center of mass of the object moves with speed…

v = _{center of mass} | Rω |

and the point fathest from the point of contact moves with twice that speed

v = _{opposite the point of contact} | 2v = 2_{cm}Rω |

### triskelion

The wheel is an extension of the foot.

### roulettes

**cycloids**- cycloid
- prolate cycloid
- curtate cycloid

**epicycloids**- epicycloid
- cardioid
- nephroid
- ranunculoid

**hypocycloids**- hypocycloid
- astroid
- deltoid

### rolling resistance

rolling resistance