Spinning in a Circle—Kinematics and Vectors, Part II

  1. Introduction

    We continue the discussion on kinematics by building on the previous lesson. Here we emphasize the fact that rotational motion and linear motion have identical kinematic expresseions. We also introduce vectors, vector operations, and the ways vectors are used in rotational motion.

  2. Objectives

    The student will see that we describe rotational kinematics in exactly the same way as linear kinematics. The student will gain experience solving problems in both paradigms. Vectors should be familiar to the students but we will review the meaning of vectors, vector notation, and operations with vectors. The student will understand how vectors are used to describe both rotational and linear motion.

  3. Materials

    Notebooks and paper are required. Some students may want to use calculators but they are not needed.

  4. Vocabulary
  5. Presentation

    Again, this lesson tends to be primarily a lecture format. Make every effort to get the students involved. Encourage them to recognize the similarities between linear and rotational motion and they might engage more readily. Repeat much of the previous lesson to emphasize the formal correspondence between rotational and linear motion. Also emphasize the the fact that all motion can be decomposed into a linear part and a rotational part. Spend some time reiterating rotational vocabulary and point out the differences in linear and rotational motion vocabulary. The conversion between rotational motion and linear motion needs an explanation. Below are the definitions of angular velocity and angular acceleration along with their linear counterparts.

    definition of linear velocity
    definition of linear acceleration
    definition of angular velocity
    definition of angular acceleration

    The basic kinematic equaitons are similar too.


    linear position vs time (no acceleration)
    linear velocity vs time (constant acceleraton)
    linear position vs time (constant acceleration)
    linear velocity vs position (constant acceleration)
    angle vs time (no acceleration)
    angular velocity vs time (constant acceleration
    angular position vs time (constant acceleration
    angular velocity vs angle (constant acceleration

    We also need to convert between rotational motion and linear motion. In the equations below, s represents the length of arc at a distance of r from the center of a circle. We always use radians for angular measure, θ. Note that the first equation below can serve as the definition of a radian.

    conversions between linear and angular quantities

    We can think of the first equation above as describing either the distance around a circle or the distance that a wheel would roll on the ground.

    Remind the students that we often use angles larger than 2π and smaller than 0.

    peridicity of angular measure

    Provide a short review of vectors and vector addition and vector products.

    vector addition
    scalar product
    vector product

    The right hand rule for angular position, velocity and acceleration.

    vector product
  6. Evaluation

    Secondary texts on vector algebra will have plenty of problems that can be used to evaluate students and give them practice. Also, the instructor can divise plenty of simple problems about vector kinematics.