Chapter Objectives Chapter Outline Concept of the center of gravity, center of mass, and the centroid Determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape Chapter Outline Center of Gravity and Center of Mass for a System of Particles Composite Bodies

9.1 Center of Gravity and Center of Mass for a System of Particles An object is composed of infinitesimal bodies and let each body has the weight dW Let the body is in the 3D in the x, y, z frame The total weight of all the bodies can be represented by a single force W at point G

9.1 Center of Gravity and Center of Mass for a System of Particles Center of Gravity (use equivalent force and moment) Resultant weight of an object = the sum of all infinitestimal weights ofall bodies Resultant moment about any axis =The sum of moments of all bodies about the axis Let G is the location of the equivalent force, which is equal to

9.1 Center of Gravity and Center of Mass for a System of Particles In summary, for discrete quantity For continuous quantity

9.1 Center of Gravity and Center of Mass for a System of Particles Center Mass Using the relationship W = mg, consider the centre of mass with contant g In general, the centre of mass is the centre of gravity The centre of mass is usually used in Dynamics problems

9.1 Center of Gravity and Center of Mass for a System of Particles Centroid of a volume For an object with uniformly distributed mass, hence or Hence the centre of volume is

9.1 Center of Gravity and Center of Mass for a System of Particles Centroid of an Area For an are with the same thickness thoughout the area The area can be subdivide into dA

9.1 Center of Gravity and Center of Mass for a System of Particles Centroid of a Line For an object with a line shape, consider a small length dL From Pythagorus or

Example 9.1 Locate the centroid of the rod bent into the shape of a parabolic arc.

Example 9.1 Differential element dL is 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 of a small element is located at

Example 9.1 Integrations

Example 9.2 Locate the centroid of the circular wire segment shown.

Example 9.2 Polar coordinate is more suitable since the arc is circular Differential element A circular arc is selected, it intersects the curve at Area and Moment Arms The differential length of the element The centroid of a small element is located at Integration

Example 9.3 Determine the distance measured from the x axis to the centroid of the area of the triangle shown.

Example 9.3 Differential element A rectangular element with thickness dy, intersect at (x,y) Area and Moment Arms The area of the element The centroid of a small element is located at Integration

Example 9.4 Locate the centroid for the area of a quarter circle shown.

Example 9.4 Polar coordinate is used. Differential element A triangle element with angle intersect at Area and Moment Arms The area of the element The centroid of the triangle element is located at (Ex. 9.3) Integration

Example 9.5 Locate the centroid for the area shown below.

Example 9.5 Solution I (Fig. a) Differential element A rectangular element with thickness dx, has a height y. Area and Moment Arms The area of the element The centroid is located at Integration

Example 9.5 Solution II (Fig. b) Differential element A rectangular element with thickness dy, has a length (1-x). Area and Moment Arms The area of the element The centroid is located at Integration

9.2 Composite Bodies is composed of objects with simple geometries such as triangles, rectangles, circles Consider individual geometries to find the centroid

9.2 Composite Bodies Analysis method Composite Parts Devide a big object in to a series of simple geometries For the void or spaces, the geometries have negative values Moment Arms Locate the frame and then find the centroid for each simple geometry Summations Find the centroid the previous centroid equaiton

Example 9.10 Locate the centroid of the plate area.

Solution Composite Parts Plate divided into 3 segments. Area of small rectangle considered “negative”.

Solution Moment Arm Location of the centroid for each piece is determined and indicated in the diagram. Summations