Rotational Kinematics

Introduction to Rotational Kinematics

Overview

In week four, we define the relationship between angular position, angular velocity, and angular acceleration of point masses. We will analyze physics problems using any two of the variables mentioned above and predict the motion of an object at any given time using our three forms of modeling in rotational terms. We will introduce the rotational equivalence of the kinematic variables discussed thus far this semester including angular position, angular velocity, and angular acceleration.

Rotational Motion Variables

Angular Position (\theta)

  • Definition: The angle an object creates when moving through a circular path.
  • Analogy: Similar to linear position.
  • Units: Radians.

Angular Velocity (\omega)

  • Definition: The rate at which an object rotates through some angle theta.
  • Analogy: Similar to linear velocity.
  • Units: Radians per second (rad/s).
  • Vector Quantity: Has both magnitude and direction.

Angular Acceleration (\alpha)

  • Definition: The rate of change of angular velocity as an object moves through some angle theta.
  • Analogy: Similar to linear acceleration.
  • Units: Radians per second squared (rad/s^2).
  • Vector Quantity: Has both magnitude and direction.

Examples of Rotation

  • Twiddling thumbs (rotation about the thumb joint).
  • Throwing a baseball pitch (rotation about the shoulder joint).
  • A bike wheel (rotation about the center).
  • Planets revolving around the Sun.

Angular Displacement

  • Angular displacement is a scalar, not a vector.

Angular Velocity as a Vector

  • Determining Direction: Use the right-hand rule. Point the fingers of your right hand in the direction of the rotation, and your thumb indicates the direction of the angular velocity vector.
  • Alternative Visualization: When viewing the rotation face-on, counterclockwise rotation indicates a positive (towards you) angular velocity vector.

Angular Acceleration as a Vector

  • Direction: Relate it to angular velocity.
    • If angular acceleration points in the same direction as angular velocity, the rotation speeds up.
    • If angular acceleration points opposite to angular velocity, the rotation slows down.
  • Visualization: When viewing the rotation face-on, speeding up in the counterclockwise direction indicates a positive (towards you) angular acceleration.

Right-Hand Rule for Rotational Vectors

  • A fail-safe method for determining the direction of angular velocity and angular acceleration vectors.