一定角加速度による回転 - II

The third equation of rotational motion is independent of time and expresses the final angular velocity in terms of initial angular velocity, angular acceleration, and angular displacement. It is derived by rearranging the first kinematic equation to solve for time, then substituting that expression into the second equation. This equation is useful when time is unknown but displacement and velocities are known.

The fourth equation is obtained by substituting the first kinematic equation for rotational motion into the second equation. This equation relates the final angular position to the initial angular position, displacement under constant angular velocity, and displacement due to changing angular velocity. It describes how an object's total angular position changes with constant angular acceleration.

The third equation is derived by eliminating time from the first two kinematic equations. By rearranging the first equation to express time and substituting it into the second equation, time cancels out. This makes the equation useful for solving problems where time is not given or needed, focusing instead on velocities and displacements.

All rotational kinematic equations are valid only when an object rotates about a fixed axis with constant angular acceleration. These conditions ensure the relationships between rotation angle, angular velocity, angular acceleration, and time remain consistent. If angular acceleration varies or the axis changes, different approaches are required.

The choice depends on which variables are present in the problem. If time is unknown, use the third equation. If you need final angular position, use the fourth equation. Sometimes multiple equations must be combined to solve a problem. Identify known and unknown variables first, then select the equation that contains all relevant variables.

Rotational kinematics describes motion using relationships between rotation angle, angular velocity, angular acceleration, and time without considering causes. Rotational dynamics examines the forces and torques causing motion. Kinematics answers how an object moves; dynamics explains why it moves that way.

Angular displacement represents the total change in rotation angle between initial and final positions. In the third kinematic equation, the difference between final and initial angular displacements appears as a key variable relating angular velocities and acceleration. This displacement term allows solving for velocity without knowing the time interval.