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  • Print publication year: 2018
  • Online publication date: April 2019

15 - Kinematics of Rigid Bodies

Summary

Introduction

So far, the motion of particles has been analyzed under kinematics and kinetics. Earlier, the rectilinear and curvilinear motion of a particle was studied in the chapter 10 and 11, respectively, where a body was idealized as a particle and does not have any rotational motion. However, in some cases we cannot make such assumptions where a body consists considerable rotary motion. In this chapter, we will analyse motion of rigid bodies having rotary and general plane motion for parameters like displacement, time, velocity and acceleration without cause of motion (force).

Rotational Motion

In this motion all the particles of a rigid body continuously rotate about a fixed axis. Here all the particles have parallel path in the plane of rotation. The examples are moving fan, pump, rotating shaft in a lathe machine and drilling operation, etc.

Angular Displacement, Angular Velocity and Angular Acceleration

Angular displacement is the displacement of a body on the plane of rotation about a fixed axis. It is a vector quantity and expressed in radians. For example, consider a link rotates clockwise about a fixed axis as shown in Fig. 15.1. If the link turns by angle θ during time t from OX-axis, the angular displacement of all particles on the link will be θ in clockwise direction. In one revolution the angular displacement of link will be equal to 2π radians.

Angular Velocity is the rate by which angular displacement varies with respect to time. It is represented by ω. If a rotating body turns by angle dθ in time dt then angular velocity will be given by

It is a vector quantity. The unit of angular velocity is rad/sec, however in engineering devices; it is expressed in r.p.m. (revolutions per minute).

If the angular velocity is expressed in r.p.m. then it can be converted into rad/sec as follows:

Consider a shaft rotating with N r.p.m.

Then number of revolutions per second will be

The angular displacement in one revolution of shaft will be 2π radians

Thus total angular displacement per second will be

Angular Acceleration is the rate at which angular velocity varies with respect to time. It is represented by α.