**Rotating Systems**

**Rotational Motion**

In this class I introduced you to methods for drawing a free body diagram of a purely rotation system. The method follows that for translational systems closely, with the following substitutions.

Rotational Systems |
Translational Systems |

Angular Displacement q (rad) |
Linear Displacement x (m) |

Angular Velocity ω (rad/s) |
Linear Velocity v (m/s) |

Angular Acceleration α (rad/s ^{2}) |
Linear Accelertation a(m/s²) |

Flywheel (Moment of Inertia) J (kg-m ^{2}) |
Mass M (kg) |

Torque τ (N-m) |
Force f (N) |

Friction B (N-m-s/rad) |
Friction b (N-s/m) |

Springs K (N-m/rad) |
Springs k (N/m) |

Gears | Levers |

Power=τ·ω | Power=f·v |

Potential Energy=½K·θ² | Potential Energy=½k·x² |

Kinetic Energy=½J·ω² | Kinetic Energy=½m·v² |

When two objects come in contact and move without slipping (e.g., gears or a rack-and-pinion) we take this into account by introducing a contact force between the two elements. The direction of the force is not important, but you must insure that the directions are opposite each other across the interface between the two touching elements.

**Levers, Gears, Transformer**

We showed that for a lever with two arms whose lengths are d1 and d2, with displacement x1 and x2, with forces f1 and f2 (as defined in class) that:

x1=x2·d1/d2,

v1=v2·d1/d2, and

f1=f2·d2/d1.

Looking at these equations you can see that a lever is simply the mechnanical analog to a transformer. A lever trades velocity (or distance) for force. A transformer trades current for voltage. In both cases, power in is equal to power out. In particular

f1·v1=f2·v2

Gears and transformers obey a similar set of ratios.