CHAPTER 9 CENTER OF GRAVITY and CENTROID Objective

CHAPTER 9 CENTER OF GRAVITY and CENTROID

Objective • To discuss the concept of the center of gravity, center of mass, and the centroid. • To show to determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape.

Chapter Outline 1. Center of Gravity and Center of Mass for a System of Particles 2. Center of Gravity and Center of Mass and Centroid for a Body 3. Composite Bodies

APPLICATIONS To design the structure for supporting a water tank, we will need to know the weights of the tank and water as well as the locations where the resultant forces representing these distributed loads are acting. How can we determine these weights and their locations?

APPLICATIONS (continued) One concern about a sport utility vehicle (SUVs) is that it might tip over while taking a sharp turn. One of the important factors in determining its stability is the SUV’s center of mass. Should it be higher or lower for making a SUV more stable? How do you determine its location?

9. 1 Center of Gravity and Center of Mass for a System of Particles Center of Gravity • 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 Center of Gravity • 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 Center of Gravity • 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 Center of Gravity • 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.

Center of gravity

Centroid - a point which defines the geometric center of an object - coincides with the center of mass or center of gravity only if the material composing the body is uniform or homogeneous -consider 3 specific cases: 1. VOLUME 2. AREA 3. LINE

VOLUME if an object is subdivided into volume elements d. V, centroid for the volume of the object can be determined by computing the “moment” of the elements about each of the coordinate axes

AREA -plate, shell LINE -thin rod, wire

Symmetry in cases where the shape has an axis of symmetry, the centroid of the shape will lie along the axis if the shape has two or three axes of symmetry, the centroid lies at the intersection of these axes

formulas used to locate the center of gravity or the centroid simply represent a balance between the sum of the moments of all the system and the moment of the “resultant” for the system in some cases the centroid is located at a point that is not on the object. ring-the centroid is at its center

Ex 1

[solution]

9. 2 Composite Bodies -a series of connected “simpler” shaped bodies -the need for integration can be eliminated

Ex. 2 Locate the centroid of the plate area shown

Quiz 2 Given: The part shown. a c b Find: The centroid of the part. d Solution: 1. This body can be divided into the following pieces ; rectangle (a) + triangle (b) + quarter circular (c) – semicircular area (d)

Steps 2 & 3: Make up and fill the table using parts a, b, c, and d. a c b d Segment Rectangle Triangle Q. Circle Semi-Circle Area A(in 2) x (in) y (in) 18 3 1. 5 4. 5 7 1 9 / 4 – 4(3) / (3 ) – /2 0 4(1) / (3 ) 28. 0 A x ( in 3) A y ( in 3) 54 31. 5 – 9 0 27 4. 5 9 - 2/3 76. 5 39. 83

·C 4. Now use the table data and these formulas to find the coordinates of the centroid. x = ( x A) / ( A ) = 76. 5 in 3/ 28. 0 in 2 = 2. 73 in y = ( y A) / ( A ) = 39. 83 in 3 / 28. 0 in 2 = 1. 42 in

Tutorial 3 Q 2. Locate the center of gravity of the steel machine element. The diameter of each hole is 1 in.

SOLUTION: • Form the machine element from a rectangular parallelepiped and a quarter cylinder and then subtracting two 1 -in. diameter cylinders.

5 - 26

5 - 27

9. 3 Theorems of Pappus and Guldinus - used to find the surface area and volume of any object of revolution surface area of revolution is generated by revolving a plane curve about a nonintersecting fixed axis in the plane of the curve volume of revolution is generated by revolving a plane area about a nonintersecting fixed axis in the plane of the area

Surface Area Volume

the surface area the amount of water that can be stored in this water tank the surface area the amount of roofing material used on this storage building (surface area)

Ex. 3 Show that the surface area of sphere is A=4 R 2 and its volume is V = 4/3 R 3