ME221Lecture 71 ME 221 Statics Lecture #7 Sections 4.1 – 4.3

ME221Lecture 72 Homework #3 Chapter 3 problems: –48, 55, 57, 61, 62, 65 & 72 Chapter 4 problems: –2, 4, 10, 11, 18, 24, 39 & 43 –Must use integration methods to solve Due Monday, June 7

ME221Lecture 73 Quiz #4 Monday, June 7

ME221Lecture 74 Distributed Forces (Loads); Centroids & Center of Gravity The concept of distributed loads will be introduced Center of mass will be discussed as an important application of distributed loading –mass, (hence, weight), is distributed throughout a body; we want to find the “balance” point

ME221Lecture 75 Distributed Loads Two types of distributed loads exist: –forces that exist throughout the body e. g., gravity acting on mass these are called “body forces” –forces arising from contact between two bodies these are called “contact forces”

ME221Lecture 76 Contact Distributed Load Snow on roof, tire on road, bearing on race, liquid on container wall,...

ME221Lecture 77 Center of Gravity x y z w 5 (x 5,y 5,z 5 ) ˜ ˜ ˜ x y z w 3 (x 3,y 3,z 3 ) ˜ ˜ ˜ w 1 (x 1,y 1,z 1 ) ˜ ˜ ˜ w 2 (x 2,y 2,z 2 ) ˜ ˜ ˜ w 4 (x 4,y 4,z 4 ) ˜ ˜ ˜ The weights of the n particles comprise a system of parallel forces. We can replace them with an equivalent force w located at G(x,y,z), such that: x w=x 1 w 1 +x 2 w 2 +x 3 w 3 +x 4 w 4 +x 5 w 5 ~ ~ ~ ~ ~ y z x

ME221Lecture 78 Or Where are the coordinates of each point. Point G is called the center of gravity which is defined as the point in the space where all the weight is concentrated.

ME221Lecture 79 CG in Discrete Sense Where do we hold the bar to balance it? 2010 ?? Find the point where the system’s weight may be balanced without the use of a moment.

ME221Lecture 710 Discrete Equations Define a reference frame x y z dw r

ME221Lecture 711 Mass center is defined by The total mass is given by M Center of Mass

ME221Lecture 712 Continuous Equations Take our volume, dV, to be infinitesimal. Summing over all volumes becomes an integral. Note that dm =  dV. Center of gravity deals with forces and gdm is used in the integrals.

ME221Lecture 713 If  is constant These coordinates define the geometric center of an object (the centroid) In case of 2-D, the geometric center can be defined using a differential element dA

ME221Lecture 714 If the geometry of an object takes the form of a line (thin rod or wire), then the centroid may be defined as:

ME221Lecture 715 Procedure for Analysis 1-Differential element Specify the coordinate axes and choose an appropriate differential element of integration. For a line, the differential element is dl For an area, the differential element dA is generally a rectangle having a finite height and differential width. For a volume, 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 differential thickness.

ME221Lecture Size Express the length dl, dA, or dv of the element in terms of the coordinate used to define the object. 3-Moment Arm Determine the perpendicular distance from the coordinate axes to the centroid of the differential element. 4- Equation Substitute the data computed above in the appropriate equation.

ME221Lecture 717 x y Symmetry Conditions In the case where the shape of the object has an axis of symmetry, then the centroid will be located along that line of symmetry. In this case, the centroid is located along the y-axis The centroid of some objects may be partially or completely specified by using the symmetry conditions

ME221Lecture 718 In cases of more than one axis of symmetry, the centroid will be located at the intersection of these axes.

ME221Lecture 719 Centroid of an Area Geometric center of the area –Average of the first moment over the entire area –Where:

ME221Lecture 720 Centroid of an Area Is then defined as an integral over the area. Integration of areas may be accomplished by the use of either single integrals or double integrals.

ME221Lecture 721 Centroid of a Volume Geometric center of the volume –Average of the first moment over entire volume –In vector notation:

ME221Lecture 722 Examples

ME221Lecture 723 Homework Assignments 4, 5 & 6 Combination of hand-calculated and computer-generated solutions (MatLab). 15 points possible for each. Must register for 1 of 2 MatLab sessions on Wednesdays June 9, 16 & 23 (12:40-2:30pm or 5:00-6:50pm). ME221 Wednesday lectures will be 10:20am to 11:10am. Will be assigned to a group with 2 ME221 & 2 CSE131 students. Group members will receive the same grade for MatLab part.