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Angular Displacement

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:

Figure %: Point P on an object sweeping out an angle μ
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 =    

Where s is the arc length shown in , r is the distance from the point to the axis of rotation, and μ is the measure of the angle. Note: Up to this point we have measured angles in degrees. We now introduce a new, more useful measurement called a radian. A radian is defined by the following relation:

1 revolution = 2Π radians = 360o    

90 degrees is equivalent to Π/2 radians, 180 degrees is equivalent to Π radians, etc. By convention, we define the positive direction of rotation to be counterclockwise.

Angular Velocity

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:

=    

And, using calculus, we define the instantaneous angular velocity as:

σ =    

Like angular displacement, angular velocity is identical for every point on a rotating object, and essentially describes the rate at which an object rotates.

Angular Acceleration

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=  

These equations for angular displacement, velocity, and acceleration bear striking resemblance to our definitions of translational variables. To see this, simply substitute x every time you see μ, v every time you see σ, and a every time you see α. The yield are the translational equations for displacement, velocity, and acceleration. This similarity will allow us to easily derive kinematic equations for rotational motion

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