ME 221 Statics

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ME 221 Statics Lecture 13Sections 4.1 4.2
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Homework 5

• Chapter 3 problems
• 48, 51, 55, 57, 61, 62, 65, 71, 72 75
• May use MathCAD, etc. to solve
• Due Friday, October 3

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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

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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

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Contact Distributed Load

• Snow on roof, tire on road, bearing on race,
  liquid on container wall, ...

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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
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Or
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CG in Discrete Sense

• Where do we hold the bar to balance it?

Find the point where the systems weight may be
balanced without the use of a moment.
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Discrete Equations
Define a reference frame
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Center of Mass
The total mass is given by M
Mass center is defined by
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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.
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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

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

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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.
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Symmetry conditions

• 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 centoid is located along the
y-axis
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In cases of more than one axis of symmetry, the
centroid will be located at the intersection of
these axes.