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By the end of this section, you will be able to:
  • Describe the physical meaning of rotational variables as applied to fixed-axis rotation
  • Explain how angular velocity is related to tangential speed
  • Calculate the instantaneous angular velocity given the angular position function
  • Find the angular velocity and angular acceleration in a rotating system
  • Calculate the average angular acceleration when the angular velocity is changing
  • Calculate the instantaneous angular acceleration given the angular velocity function

So far in this text, we have mainly studied translational motion, including the variables that describe it: displacement, velocity, and acceleration. Now we expand our description of motion to rotation—specifically, rotational motion about a fixed axis. We will find that rotational motion is described by a set of related variables similar to those we used in translational motion.

Angular velocity

Uniform circular motion (discussed previously in Motion in Two and Three Dimensions ) is motion in a circle at constant speed. Although this is the simplest case of rotational motion, it is very useful for many situations, and we use it here to introduce rotational variables.

In [link] , we show a particle moving in a circle. The coordinate system is fixed and serves as a frame of reference to define the particle’s position. Its position vector from the origin of the circle to the particle sweeps out the angle θ , which increases in the counterclockwise direction as the particle moves along its circular path. The angle θ is called the angular position    of the particle. As the particle moves in its circular path, it also traces an arc length s .

Figure is a graph that shows a particle moving counterclockwise. Vector r from the origin of the co-ordinate system to the point s on the pass of a particle forms an angle theta with the X axis.
A particle follows a circular path. As it moves counterclockwise, it sweeps out a positive angle θ with respect to the x- axis and traces out an arc length s .

The angle is related to the radius of the circle and the arc length by

θ = s r .

The angle θ , the angular position of the particle along its path, has units of radians (rad). There are 2 π radians in 360 ° . Note that the radian measure is a ratio of length measurements, and therefore is a dimensionless quantity. As the particle moves along its circular path, its angular position changes and it undergoes angular displacements Δ θ .

We can assign vectors to the quantities in [link] . The angle θ is a vector out of the page in [link] . The angular position vector r and the arc length s both lie in the plane of the page. These three vectors are related to each other by

s = θ × r . .

That is, the arc length is the cross product of the angle vector and the position vector, as shown in [link] .

Figure is an XYZ coordinate system that shows three vectors. Vector Theta points in the positive Z direction. Vector s is in the XY plane. Vector r is directed from the origin of the coordinate system to the beginning of the vector s.
The angle vector points along the z- axis and the position vector and arc length vector both lie in the xy -plane. We see that s = θ × r . All three vectors are perpendicular to each other.

The magnitude of the angular velocity    , denoted by ω , is the time rate of change of the angle θ as the particle moves in its circular path. The instantaneous angular velocity    is defined as the limit in which Δ t 0 in the average angular velocity ω = Δ θ Δ t :

ω = lim Δ t 0 Δ θ Δ t = d θ d t ,

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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