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ME 221 Statics Lecture #7 Sections 4.1 – 4.3

Learn about distributed loads, centroids, and center of gravity in statics. Understand how to find balance points and analyze geometric centers. Integration methods are crucial for solving problems in this lecture.

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ME 221 Statics Lecture #7 Sections 4.1 – 4.3

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  1. ME 221 Statics Lecture #7Sections 4.1 – 4.3 Lecture 7

  2. 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 Lecture 7

  3. Quiz #4 • Monday, June 7 Lecture 7

  4. 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 Lecture 7

  5. 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” Lecture 7

  6. Contact Distributed Load • Snow on roof, tire on road, bearing on race, liquid on container wall, ... Lecture 7

  7. z z x y z y y ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ w1(x1,y1,z1) w2(x2,y2,z2) w5(x5,y5,z5) w4(x4,y4,z4) w3(x3,y3,z3) x x ~ ~ ~ ~ ~ x w=x1w1+x2w2+x3w3+x4w4 +x5w5 Center of Gravity 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: Lecture 7

  8. 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. Or Lecture 7

  9. 20 10 ?? ?? ?? CG in Discrete Sense Where do we hold the bar to balance it? Find the point where the system’s weight may be balanced without the use of a moment. Lecture 7

  10. y r dw z x Discrete Equations Define a reference frame Lecture 7

  11. Center of Mass The total mass is given by M Mass center is defined by Lecture 7

  12. Continuous Equations Take our volume, dV, to be infinitesimal. Summing over all volumes becomes an integral. Note that dm = rdV . Center of gravity deals with forces and gdm is used in the integrals. Lecture 7

  13. If r 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 Lecture 7

  14. If the geometry of an object takes the form of a line (thin rod or wire), then the centroid may be defined as: Lecture 7

  15. 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. Lecture 7

  16. 2- 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. Lecture 7

  17. Symmetry Conditions y x • The centroid of some objects may be partially or completely specified by using the 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 Lecture 7

  18. In cases of more than one axis of symmetry, the centroid will be located at the intersection of these axes. Lecture 7

  19. Centroid of an Area • Geometric center of the area • Average of the first moment over the entire area • Where: Lecture 7

  20. 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. Lecture 7

  21. Centroid of a Volume • Geometric center of the volume • Average of the first moment over entire volume • In vector notation: Lecture 7

  22. Examples Lecture 7

  23. 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. Lecture 7

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