10.3:
Rotation with Constant Angular Acceleration – I
The equations derived for linear motion also hold true for rotational motion if the linear motion variables are replaced by their rotational motion counterparts.
Let ω0z be the angular velocity of the rotating object at any time t equal to zero and ωz be its final angular velocity at later time t. If the angular acceleration, αz, of the object is constant, it can be written as the difference between final and initial angular velocity over time t.
Rearranging this expression, the first kinematic equation for rotational motion is obtained. Thus, the angular velocity at any time equals the sum of initial angular velocity and the change in angular velocity.
To derive the second equation, two equations for average angular velocity are used. One, average angular velocity equals the total change in angular displacement over time t.
Two, for constant acceleration, average angular velocity equals the average of the initial and final velocity. Solving these two equations, the second equation of rotational motion is obtained.
Here, the angular position of an object at any time is represented as the sum of its initial angular position, the displacement moved under constant initial angular velocity, and the angular displacement traveled during the change in angular velocity.
10.3:
Rotation with Constant Angular Acceleration – I
If angular acceleration is constant, then we can simplify equations of rotational kinematics, similar to the equations of linear kinematics. This simplified set of equations can be used to describe many applications in physics and engineering where the angular acceleration of a system is constant.
Using our intuition, we can begin to see how rotational quantities such as angular displacement, angular velocity, angular acceleration, and time are related to one another. For example, if a flywheel has an angular acceleration in the same direction as its angular velocity vector, its angular velocity increases with time, as does its angular displacement. On the contrary, if the angular acceleration is opposite in direction to the angular velocity vector, its angular velocity decreases with time. These physical situations, along with many others, can be described with a consistent set of rotational kinematic equations under constant angular acceleration. The method to investigate rotational motion in this way is called kinematics of rotational motion.
To begin, note that if a system is rotating under constant acceleration, then the average angular velocity follows a simple relation, because the angular velocity is increasing linearly with time. The average angular velocity is simply half the sum of the initial and final values. From this, an equation relating the angular position, average angular velocity, and time can be obtained.
This text is adapted from Openstax, University Physics Volume 1, Section 10.2: Rotational with Constant Angular Acceleration.