Engineering Mechanics Statics Chapter 5 Distributed Forces Introduction

Engineering Mechanics: Statics Chapter 5: Distributed Forces

Introduction ¡ ‘Concentrated’ force does not exist, since every force applied mechanically is distributed over a finite contact area. ¡ When the applied region is not negligible, we must account for the actual manner the force is distributed ¡ Three categories: l Line distribution (N/m) l Area distribution (N/m 2, Pa) l Volume distribution or body force (N/m 3)

Centers of Mass ¡ Body force due to gravitational attraction of the earth (weight) is the most commonly encountered distributed force ¡ This section treats the determination of the point in a body through which the resultant gravitational force acts. ¡ Center of gravity l ¡ Apply the principle of moments (The moment of the resultant force W equal the sum of the moments about the same axis of the force d. W acting on all particles) Center of mass

Centers of Mass and Centroids ¡ In vector form where and ¡ ¡ The density r of a body is its mass per unit volume, thus , etc. When the density r of a body is uniform throughout, the expressions define purely geometrical property of the body and the term centroid is used.

Centroids ¡ Centroid of lines ¡ Centroid of areas ¡ Centroid of volumes ¡ In a homogeneous body, center of mass always lies on a plane of symmetry! ¡ Since the moments due to symmetrically located elements will always cancel.

Choice of Element for Integration ¡ Order of element ¡ Choice of coordinate system ¡ Continuity ¡ Centroidal coordinate of element ¡ Discard higher-order term (integration of ydx instead of ydx+1/2 dydx)

Sample Problem 5/2 Determine the distance centroid of its area from the base of a triangle of altitude h to the

Sample Problem 5/5 Locate the centroid of the volume of a hemisphere of radius r with respect to its base.

Composite Bodies ¡ When a body can be divided into several parts whose mass centers are easily determined, we use the principle of moments and treat each part as a finite element of the whole

Sample Problem 5/6 Locate the centroid of the shaded area. 120 40 40 30 30 20 20 20 50 Dimensions in mm

Problem 5/57 The rigidly connected unit consists of a 2 -kg circular disk, a 1. 5 -kg round shaft and a 1 -kg square plate. Determine the z-coordinate of the mass center of the unit.