Engineering Mechanics: Statics Chapter 9: Center of Gravity and Centroid Engineering Mechanics: Statics

Chapter Objectives To discuss the concept of the center of gravity, center of mass, and the centroid. To show how to determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape. To use the theorems of Pappus and Guldinus for finding the area and volume for a surface of revolution. To present a method for finding the resultant of a general distributed loading and show how it applies to finding the resultant of a fluid.

Chapter Outline Center of Gravity and Center of Mass for a System of Particles Center of Gravity and Center of Mass and Centroid for a Body Composite Bodies Theorems of Pappus and Guldinus Resultant of a General Distributed Loading Fluid Pressure

9.1 Center of Gravity and Center of Mass for a System of Particles Locates the resultant weight of a system of particles Consider system of n particles fixed within a region of space The weights of the particles comprise a system of parallel forces which can be replaced by a single (equivalent) resultant weight having defined point G of application

9.1 Center of Gravity and Center of Mass for a System of Particles Resultant weight = total weight of n particles Sum of moments of weights of all the particles about x, y, z axes = moment of resultant weight about these axes Summing moments about the x axis, Summing moments about y axis,

9.1 Center of Gravity and Center of Mass for a System of Particles Although the weights do not produce a moment about z axis, by rotating the coordinate system 90° about x or y axis with the particles fixed in it and summing moments about the x axis, Generally,

9.1 Center of Gravity and Center of Mass for a System of Particles Where represent the coordinates of the center of gravity G of the system of particles, represent the coordinates of each particle in the system and represent the resultant sum of the weights of all the particles in the system. These equations represent a balance between the sum of the moments of the weights of each particle and the moment of resultant weight for the system.

9.1 Center of Gravity and Center of Mass for a System of Particles Center Mass Provided acceleration due to gravity g for every particle is constant, then W = mg By comparison, the location of the center of gravity coincides with that of center of mass Particles have weight only when under the influence of gravitational attraction, whereas center of mass is independent of gravity

9.2 Center of Gravity and Center of Mass and Centroid for a Body Center Mass A rigid body is composed of an infinite number of particles Consider arbitrary particle having a weight of dW

9.2 Center of Gravity and Center of Mass and Centroid for a Body Center Mass γ represents the specific weight and dW = γdV

9.2 Center of Gravity and Center of Mass and Centroid for a Body Density ρ, or mass per unit volume, is related to γ by γ = ρg, where g = acceleration due to gravity Substitute this relationship into this equation to determine the body’s center of mass

9.2 Center of Gravity and Center of Mass and Centroid for a Body Defines the geometric center of object Its location can be determined from equations used to determine the body’s center of gravity or center of mass If the material composing a body is uniform or homogenous, the density or specific weight will be constant throughout the body The following formulas are independent of the body’s weight and depend on the body’s geometry

9.2 Center of Gravity and Center of Mass and Centroid for a Body Volume Consider an object subdivided into volume elements dV, for location of the centroid,

9.2 Center of Gravity and Center of Mass and Centroid for a Body Area For centroid for surface area of an object, such as plate and shell, subdivide the area into differential elements dA

9.2 Center of Gravity and Center of Mass and Centroid for a Body Line If the geometry of the object such as a thin rod or wire, takes the form of a line, the balance of moments of differential elements dL about each of the coordinate system yields

9.2 Center of Gravity and Center of Mass and Centroid for a Body Line Choose a coordinate system that simplifies as much as possible the equation used to describe the object’s boundary Example Polar coordinates are appropriate for area with circular boundaries

9.2 Center of Gravity and Center of Mass and Centroid for a Body Symmetry The centroids of some shapes may be partially or completely specified by using conditions of symmetry In cases where the shape has an axis of symmetry, the centroid of the shape must lie along the line Example Centroid C must lie along the y axis since for every element length dL, it lies in the middle

9.2 Center of Gravity and Center of Mass and Centroid for a Body Symmetry For total moment of all the elements about the axis of symmetry will therefore be cancelled In cases where a shape has 2 or 3 axes of symmetry, the centroid lies at the intersection of these axes

9.2 Center of Gravity and Center of Mass and Centroid for a Body Procedure for Analysis Differential Element Select an appropriate coordinate system, specify the coordinate axes, and choose an differential element for integration For lines, the element dL is represented as a differential line segment For areas, the element dA is generally a rectangular having a finite length and differential width

9.2 Center of Gravity and Center of Mass and Centroid for a Body Procedure for Analysis Differential Element For volumes, the element dV is either a circular disk having a finite radius and differential thickness, or a shell having a finite length and radius and a differential thickness Locate the element at an arbitrary point (x, y, z) on the curve that defines the shape

9.2 Center of Gravity and Center of Mass and Centroid for a Body Procedure for Analysis Size and Moment Arms Express the length dL, area dA or volume dV of the element in terms of the curve used to define the geometric shape Determine the coordinates or moment arms for the centroid of the center of gravity of the element

9.2 Center of Gravity and Center of Mass and Centroid for a Body Procedure for Analysis Integrations Substitute the formations and dL, dA and dV into the appropriate equations and perform integrations Express the function in the integrand and in terms of the same variable as the differential thickness of the element The limits of integrals are defined from the two extreme locations of the element’s differential thickness so that entire area is covered during integration

9.2 Center of Gravity and Center of Mass and Centroid for a Body Example 9.1 Locate the centroid of the rod bent into the shape of a parabolic arc.

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Differential element Located on the curve at the arbitrary point (x, y) Area and Moment Arms For differential length of the element dL Since x = y2 and then dx/dy = 2y The centroid is located at

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Example 9.2 Locate the centroid of the circular wire segment.

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Differential element A differential circular arc is selected This element intersects the curve at (R, θ) Length and Moment Arms For differential length of the element For centroid,

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Example 9.3 Determine the distance from the x axis to the centroid of the area of the triangle

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Differential element Consider a rectangular element having thickness dy which intersects the boundary at (x, y) Length and Moment Arms For area of the element Centroid is located y distance from the x axis

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Example 9.4 Locate the centroid for the area of a quarter circle.

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Method 1 Differential element Use polar coordinates for circular boundary Triangular element intersects at point (R,θ) Length and Moment Arms For area of the element Centroid is located y distance from the x axis

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Method 2 Differential element Circular arc element having thickness of dr Element intersects the axes at point (r,0) and (r, π/2)

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Area and Moment Arms For area of the element Centroid is located y distance from the x axis

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Example 9.5 Locate the centroid of the area.

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Method 1 Differential element Differential element of thickness dx Element intersects curve at point (x, y), height y Area and Moment Arms For area of the element For centroid

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Method 2 Differential element Differential element of thickness dy Element intersects curve at point (x, y) Length = (1 – x)

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Area and Moment Arms For area of the element Centroid is located y distance from the x axis

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Example 9.6 Locate the centroid of the shaded are bounded by the two curves y = x and y = x2.

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Method 1 Differential element Differential element of thickness dx Intersects curve at point (x1, y1) and (x2, y2), height y Area and Moment Arms For area of the element For centroid

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Method 2 Differential element Differential element of thickness dy Element intersects curve at point (x1, y1) and (x2, y2) Length = (x1 – x2)

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Area and Moment Arms For area of the element Centroid is located y distance from the x axis

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Example 9.7 Locate the centroid for the paraboloid of revolution, which is generated by revolving the shaded area about the y axis.

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Method 1 Differential element Element in the shape of a thin disk, thickness dy, radius z dA is always perpendicular to the axis of revolution Intersects at point (0, y, z) Area and Moment Arms For volume of the element

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution For centroid Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Method 2 Differential element Volume element in the form of thin cylindrical shell, thickness of dz dA is taken parallel to the axis of revolution Element intersects the axes at point (0, y, z) and radius r = z

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Area and Moment Arms For area of the element Centroid is located y distance from the x axis

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Example 9.8 Determine the location of the center of mass of the cylinder if its density varies directly with its distance from the base ρ = 200z kg/m3.

9.2 Center of Gravity and Center of Mass and Centroid for a Body View Free Body Diagram Solution For reasons of material symmetry Differential element Disk element of radius 0.5m and thickness dz since density is constant for given value of z Located along z axis at point (0, 0, z) Area and Moment Arms For volume of the element

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution For centroid Integrations

9.2 Center of Gravity and Center of Mass and Centroid for a Body Solution Not possible to use a shell element for integration since the density of the material composing the shell would vary along the shell’s height and hence the location of the element cannot be specified