15.3:
Kinematic Equations for Rotation
For the rigid body rotating with constant angular acceleration, similar to linear kinematics, kinematics equations for rotational motion can be established.
Considering point A on the rigid body executing circular motion, the translational velocity can be formulated by determining the time derivatives of the displacement equation.
Here, the translational velocity is continually tangential to the circular path. It can be represented using the vector product of angular velocity and the position vector. The expression rAsinθ corresponds to the radius of the circular path followed by point A.
Equally, the linear acceleration of point A can be described as the sum of the normal and tangential acceleration components.
The tangential component gives the time rate of change of magnitude of the velocity, whereas the normal component gives the time rate of change of direction of the velocity.
The acceleration can be expressed in the vector form by taking the time derivative of the vector equation of the translational velocity.
Here, the first term gives tangential acceleration, whereas the second term gives a normal component of acceleration.
15.3:
Kinematic Equations for Rotation
In mechanics, when one observes a rigid body in rotational motion with constant angular acceleration, it is possible to establish equations for its rotational kinematics. This process resembles how linear kinematics are dealt with in simpler motion studies.
For instance, imagine a point A on a rigid body engaged in circular motion. The translational velocity of this particular point can be calculated by taking the time derivatives of the displacement equation, which essentially measures the change in position of point A over time. This translational velocity is always tangential to the circular path traced by point A, implying that at any given moment, the direction of the velocity is tangent to the circle at that point. This relationship can be mathematically expressed using the vector product of the angular velocity and the position vector.
Further examination of point A's motion allows for the description of its linear acceleration. This is achieved by summing up the normal and tangential acceleration components.
These two distinct components provide different aspects of the acceleration. The tangential component reveals the rate of change of the velocity's magnitude over time, indicating how rapidly point A's speed (not the direction) changes. Conversely, the normal component provides information on the rate of change of the velocity's direction, showing how swiftly point A alters its course of motion.
Lastly, the acceleration can be represented in vector form, which is derived by taking the time derivative of the vector equation of the translational velocity.
In this representation, the initial term corresponds to the tangential acceleration, while the subsequent term provides the normal component of acceleration. Together, these terms offer a comprehensive description of the acceleration of point A on the rotating rigid body.