## Rotation theory

# Rotation

A Rotation is a circular movement of an object around a center (or point) of rotation. A three-dimensional object rotates always around an imaginary line called a **rotation axis**.

A rotation of a body is represented by an operator that affects a set of points or vectors.

A rotary motion is represented by vector angular velocity \ boldsymbol \ omega, which is a vector slipperiness, and it is located on the rotation axis. When the shaft passes through the gravity center mass is said that the body “turns on itself.” Other units that can be used such as Hertz (cycles per second) or revolutions per minute (rpm).

There are two parameters that are often used in rotating systems at constant speed:

The time it takes to make one complete revolution.__Period:__It is the period inverse.__Frequency:__

**Euler’s rotation theorem:** Any rotation or successive rotations set can always be expressed as a rotation about a single direction or main rotation axle. Thus, any rotation (or set of successive rotations) in three dimensional space can be specified through the axis of rotation vectorially equivalent defined by three parameters and a fourth parameter representative of the rotated angle. Generally they referred to these four parameters as degrees of freedom of rotation.

**FORMULA SUMMARY:**

Fundamental equation of dynamics of Rotation

(M = External momentum applied ; I = Momentum of Inertia of the solid about the axis ; α = angular acceleration of the solid)

Period and Frequency

Momentum of Inertia of Cylinder

Momentum of Inertia of a thin sheet

Discus

Solid sphere

Hollow sphere

Turning Radius

Steiner’s theorem

(I = Momentum of Inertia of the solid about one axis ; IG Momentum of Inertia of the solid about the axis which pass through its center of mass ; d = distance between both axes ; m = mass of the body)

Angular momentum of rotation

Angular momentum theorem

Principle of conservation of angular momentum

Rotational kinetic energy

Work-Energy theorem for the Rotation

Total energy of a body subjected to translation and rotation

Principle of conservation of Mechanical Energy

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