Objectives To discuss the concept of the center of gravity, center of mass, and centroids (centers of area). To show how to determine the location of the.

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Presentation transcript:

Objectives To discuss the concept of the center of gravity, center of mass, and centroids (centers of area). To show how to determine the location of the center of gravity and centroid for a system of particles and a body of arbitrary shape. 12/1/2018 Chapter 9 - Centroids

Center of Gravity The center of gravity G is a point which locates the resultant weight of a system of particles. The weights of the particles is considered to be a parallel force system. The system of weights can be replaced by a single weight acting at the Center of Gravity. 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

Total Weight x location: y location: z location: 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

Center of Mass 12/1/2018 Chapter 9 - Centroids

Center of Gravity and Centroid for a Body Consider a body to be a system of an infinite number of particles 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

Center of Gravity of a Body 12/1/2018 Chapter 9 - Centroids

CENTROID The centroid C is a point which defines the geometric center of an object. Its location can be determined by formulas similar to those used for center of gravity or center of mass. 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

Centroid of a Volume 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

Centroid of an Area 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

Centroid of a Line 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

PROBLEM 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

PROBLEM 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

y y = (h/b) (b - x) h C x b 12/1/2018 Chapter 9 - Centroids

Strip Method 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

PROBLEM 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

Composite Bodies 12/1/2018 Chapter 9 - Centroids

Composite Bodies If a body is made up of several simpler bodies then a special technique can be used. 12/1/2018 Chapter 9 - Centroids

Procedure Divide body into several subparts. If the body has a hole or cutout, treat that as negative area. Centroid will lie on line of symmetry. Create Table and calculate centroid. 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

xc yc xc A yc A Body Area 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

Locate Centroid of the Composite Area 12/1/2018 Chapter 9 - Centroids

1 ft 3 ft 2 1 3 1 ft 2 ft 3 ft 12/1/2018 Chapter 9 - Centroids

Segment A (ft2) x y xA yA 1 4.5 1 1 4.5 4.5 2 6 -1 1.5 -6 9 1 4.5 1 1 4.5 4.5 2 6 -1 1.5 -6 9 3 1 -2.5 0.5 -2.5 0.5  A = 11.5  xA = -4  xA = 14 12/1/2018 Chapter 9 - Centroids

3 3 ft 2 1 1 ft 1 ft 2 ft 3 ft 12/1/2018 Chapter 9 - Centroids

Segement A (ft2) x y xA yA 1 4.5 1 1 4.5 4.5 2 9 -1.5 1.5 -13.5 13.5 1 4.5 1 1 4.5 4.5 2 9 -1.5 1.5 -13.5 13.5 3 -2.5 -2.5 2 5 -4  A = 11.5  xA = -4  xA = 14 12/1/2018 Chapter 9 - Centroids

9.55 1 in 1 in 1 in 1 in 3 in 12/1/2018 Chapter 9 - Centroids

9.55 1 in 1 in 2 3 1 5 4 1 in 1 in 3 in Break into sub-areas 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

9.55 1 in 3 1 in 1 1 in 1 in 2 3 in 12/1/2018 Chapter 9 - Centroids

9.55 1 in 3 1 in 1 1 in 1 in 2 3 in 12/1/2018 Chapter 9 - Centroids

9.55 1 in 3 1 in 1 1 in 1 in 2 3 in 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

h y b 12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

12/1/2018 Chapter 9 - Centroids

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