10.4:
Rotation with Constant Angular Acceleration – II
The first and second equations of rotational motion with constant angular acceleration both have time as a variable.
However, the third equation is independent of time. To derive the third equation, begin by rearranging the first equation of rotational motion to obtain an expression for time. Then, substitute the value of time in the second equation of rotational motion.
Now, rearrange the θ0 term and multiply both sides by 2αz. Simplifying it further gives an expression for the final angular velocity in terms of initial angular velocity, angular acceleration, and the difference between the final and initial angular displacements. This is the third equation of rotational motion.
On the other hand, the fourth equation can be obtained by substituting the first equation for rotational motion into the second equation.
This equation gives the relationship between the final angular position of an object with respect to the initial angular position, the rotation with constant angular velocity, and the rotation due to a change in angular velocity.
10.4:
Rotation with Constant Angular Acceleration – II
Kinematics is the description of motion. The kinematics of rotational motion discusses the relationships between rotation angle, angular velocity, angular acceleration, and time. One can describe many things with great precision using kinematics, but kinematics does not consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity without any consideration of its cause. Thus, rotational kinematics does not represent the laws of nature.
The first two kinematic equations for rotational motion involve time as a variable, but the third is independent of time. This equation expresses the angular velocity of an object with respect to its initial angular velocity and total angular displacement due to constant angular acceleration. The fourth kinematic equation represents the final angular displacement of an object as a function of the initial angular displacement, the displacement under constant angular velocity, and displacement under the constant angular acceleration.
These kinematics equations enable us to solve many problems in physics and engineering. The choice of kinematic equations in solving a problem depends on the variables present in the problem. Sometimes, it is necessary to use multiple equations to solve a problem. It is important to note that all the rotational kinematics equations are valid only when an object is rotating about a fixed axis with constant angular acceleration.
This text is adapted from Openstax, University Physics Volume 1, Section 10.2: Rotational with Constant Angular Acceleration.