FIGURE EX4.22 shows the angular-velocity-versus-time graph...

FIGURE EX4.22 shows the angular-velocity-versus-time graph for a particle moving in a circle, starting from $\theta_{0}=0$ rad at $t=0$ s. Draw the angular-position-versus-time graph. Include an appropriate scale on both axes.
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Key Concepts

Graphical Representation and Scale Selection

Choosing an appropriate scale for both axes is essential when plotting kinematic graphs. For angular position versus time graphs, the time axis must be labeled with time intervals and the angular position axis with angular displacement (typically in radians). Proper scaling ensures that the key features of the motion, such as periods of acceleration or constant velocity, are clearly visible and accurately represented.

Integration of Angular Velocity

This concept involves recognizing that the angular position function is obtained by integrating the angular velocity function over time. Mathematically, the angular position theta(t) is given by the initial angular position plus the integral of the angular velocity omega(t) with respect to time. This process accumulates the angular displacement over the interval, translating instantaneous rates of rotation into a total angular displacement.

Relationship Between Angular Position and Angular Velocity Graphs

Understanding this relationship is crucial in rotational kinematics. The instantaneous angular velocity at any point in time corresponds to the slope of the angular position versus time graph. A constant angular velocity results in a linear angular position graph, while a variable angular velocity produces a graph with changing slopes, indicating nonuniform rotation.

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