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The most important constraint placed on us when developing these variables is that they have to be a property of the object: any point on the object must have the same value for the variable. We therefore cannot use our old variables, such as velocity, because some parts of a rotating disk move at different speeds than others, and a single number for velocity would not describe the motion of the entire disk. So what is a property of every point on a rotating object? Since every point rotates in a circle about a common axis, we can say that the angular displacement is the same for any point on a rotating object. That is, the angle which each point sweeps out in rotating is the same at any given time for any point on the object: The figure shows point P, located a distance r from the axis of rotation, moving a distance s as it rotates. The angle swept out by the point, which is the same for any point on the object, is given by:
μ = |
1 revolution = 2Π radians = 360o |
Angular displacement is an equivalent quantity to linear displacement. Indeed, by taking the linear displacement of a given particle on an object and dividing by the radius of that point, we derive angular displacement. The equivalency between linear and angular displacement leads us to a further realization: just as we define linear velocity from linear displacement, we similarly define angular velocity from angular displacement. If an object is displaced by an angle of Δμ during a time period of Δt, we define the average angular velocity as:
= |
σ = |
The rotational corollary of linear acceleration is angular acceleration, the rate of change of angular velocity. In the same manner as we derived the equations for average and instantaneous velocity, we define angular acceleration:
= | |||
α | = |
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