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Angular displacement is a measure of the angle through which an object moves on a circular path. It represents the angle between the initial and final positions of the object. Formally, if we denote angular displacement by the Greek letter theta \( \theta \), it is defined as the change in the angle as an object rotates around a point or axis. In our exercise, the angular position of a swinging door is given as a function of time \( t \) by \( \theta = 5.00 + 10.0 t + 2.00 t^{2} \).

To determine the angular displacement at a specific time, we simply substitute the time value into this equation. At \( t = 0 \) seconds, we calculated that the angular displacement was \( \theta = 5.00 \) radians. At \( t = 3.00 \) seconds, the displacement increased to \( \theta = 53 \) radians. This shows how angular displacement changes with time and can be calculated for any given moment during the door's swing.

Angular velocity tells us how fast an object is rotating, specifically how quickly the angular displacement changes with respect to time. It is commonly represented by the Greek letter omega \( \omega \). To find the angular velocity, we take the first derivative of the angular position \( \theta \) with respect to time \( t \).

In our example, \( \omega = d\theta/dt = 10.0 + 4.00t \), giving us a formula for the angular velocity at any time during the door's motion. At \( t = 0 \) seconds, angular velocity is \( \omega = 10.0 \) radians per second. This value increases as time progresses; for instance, at \( t = 3 \) seconds, it is \( \omega = 22.0 \) radians per second, indicating that the door swings open faster as time goes by.

Angular acceleration is the rate at which angular velocity changes with time, analogous to linear acceleration in translational motion. Represented by the Greek letter alpha \( \alpha \), it is calculated as the second derivative of angular displacement \( \theta \) or the first derivative of angular velocity \( \omega \) with respect to time \( t \).

From our exercise, we determine that the angular acceleration of the door is \( \alpha = d^{2}\theta/dt^{2} = 4.00 \) radians per second squared. This value remains constant regardless of the particular moment we choose (whether at \( t = 0 \) or at \( t = 3 \) seconds), suggesting that the door's angular velocity increases at a constant rate over time. Constant angular acceleration implies uniform changes in angular velocity, which is a crucial concept when analyzing rotational motion.