Tuesday, August 14, 2012

Physics-Angular motion

Angular motion is when a rigid body rotates around some point or points. Before we go in too deeply into this we first must define the environment we'll be working in. Normally we describe motion with cartesian coordinates, x, y, and z to define the axis, and assume that the object behaves as a point mass. We can't really do that with angular motion. For example a ceiling fan is hard to describe with only one point. Thus we bring in the rigid body, where there is a definite shape to objects. We also want to work with something more convenient than cartesian coordinates. For example although we can describe a circle in cartesian coordinates with x2 + y2 = 1, its much easier to just say r = 1 for all values of θ. We define the old x and y as:
x=r cos (θ)
y=r sin(θ)

Now we replace the terms in normal motion with their angular motion counterparts.
v = δx/δt
ω = δθ/δt
velocity goes from the change in position over time to angular velocity, the change in angle over time. However, it is defined slightly differently. Velocity is defined with a vector pointing in the same direction as the motion of the object. Angular velocity is defined with a vector pointing perpendicular to the plane of the angular motion. To be more specific, the direction is defined by the cross product of the motion of a point on the edge of the object and the vector from the center of the object to the point.

a = δv/δt
α = δω/δt
Angular acceleration is also defined with a vector perpendicular to  the rotational motion of the object. This means that both angular acceleration and angular velocity are defined in the z axis, and for the purposes of basic mechanics, remains stationary.

There are also analogs to the basic equations of motion. Assuming constant angular acceleration we can get:
ωz0z + αzt
θ=θ+ ω0zt + 1/2αzt2

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