10.6:
Relating Angular And Linear Quantities – II
When an object is moving in a non-uniform circular motion, the linear acceleration is represented as a centripetal and tangential component.
The centripetal or the radial component is associated with a change in the direction of velocity, expressed in terms of velocity and radius of the circle. Replacing velocity by omega times radius, a relationship between centripetal acceleration and angular velocity is obtained.
The tangential acceleration component is parallel to instantaneous velocity and is associated with the change in magnitude of velocity. Replacing velocity by omega times radius, a relationship between tangential acceleration and angular acceleration is obtained.
Since tangential acceleration is associated only with speed and not the direction of motion, the angular acceleration is positive when the angular velocity is increasing and negative when the angular velocity is decreasing.
10.6:
Relating Angular And Linear Quantities – II
In the case of circular motion, the linear tangential speed of a particle at a radius from the axis of rotation is related to the angular velocity by the relation:
This could also apply to points on a rigid body rotating about a fixed axis. In a circular motion, both uniform and nonuniform, there exists a centripetal acceleration. The centripetal acceleration vector points inward from the particle executing circular motion toward the axis of rotation. In uniform circular motion, when the angular velocity is constant and the angular acceleration is zero, we observe a linear acceleration—that is, centripetal acceleration—since the tangential speed is constant. If the circular motion is nonuniform, then the rotating system has an angular acceleration, and we have both a linear centripetal acceleration and linear tangential acceleration.
The centripetal acceleration is due to a change in the direction of tangential velocity, whereas the tangential acceleration is due to any change in the magnitude of the tangential velocity. The tangential and centripetal acceleration vectors are always perpendicular to each other. To complete this description, a total linear acceleration vector is assigned to a point on a rotating rigid body or a particle executing circular motion at a radius r from a fixed axis. The total linear acceleration vector is the vector sum of the centripetal and tangential accelerations. The total linear acceleration vector in the case of nonuniform circular motion points at an angle between the centripetal and tangential acceleration vectors.
This text is adapted from Openstax, University Physics Volume 1, Section 10.3: Relating Angular and Translational Quantities.