ch9_1_3

IDE 50 - StaticsFall Semester 2008

Centroids

•Remember:

•Mass is a property of a physical object thatquantifies the amount of matter it contains.

•Weight is a force that results from the actionof gravity on matter.

Chapter 9: CG, CM, &Centroids

§9.1-2

•Definitions:

•Center of gravity (CG) - Point which locates theresultant weight of a physical object or system of particles.

•Center of mass (CM) - Point where the mass of aphysical object or system of particles can be assumed to beconcentrated.

•Centroid - Point which defines the geometric centerof a physical object or system of particles.

§9.1-2


•Note:

•The center of gravity and the center of mass coincidewhen the gravitational field the object is subjected to has thesame magnitude and direction everywhere.

•On Earth, in general engineering applications, theassumption of a uniform gravity field is appropriate touse.

•The centroid will coincide with the CM and CG only ifthe material composing the physical object ishomogeneous.

§9.1-2


•When do we need to calculate the Centerof Mass/Center of Gravity?

•When we want to show the weight of a body asa concentrated force.

•You can represent the weight of an object by asingle equivalent force acting at its center of mass.

•We have already been doing this on a simplifiedlevel...

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200 lb

“...a 200 lb block...”


•However, for more complicated shapes, we will need touse the CG/CM equations...

•Center of Gravity:

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•• ϒϒ is the is the specific weightspecific weight (weight per unit volume) of the body (weight per unit volume) of the body•• are the coordinates of the Center of Gravity are the coordinates of the Center of Gravity•• are the coordinates of the center of the differential element used to are the coordinates of the center of the differential element used to

analyze the bodyanalyze the body


x =˜ x !d V"! d V"

y =˜ y ! d V"! d V"

z =˜ z !d V"! d V"

x =˜ x ! d V"!d V"

y =˜ y ! d V"!d V"

z =˜ z !d V"!d V"

•However, for more complicated shapes, we will need touse the CG/CM equations...

•Center of Mass:

§9.1-2

•• ρρ is the is the density density (mass per unit volume) of the body(mass per unit volume) of the body•• are the coordinates of the Center of Mass are the coordinates of the Center of Mass•• are the coordinates of the center of the differential element used to are the coordinates of the center of the differential element used to

analyze the bodyanalyze the body


•Centroid:•The centroid represents the geometric center of an object.

§9.1-2Note that in some cases the centroid is located at a point that is not on the object.Note that in some cases the centroid is located at a point that is not on the object.


volume centroid

area centroid

line centroid

x =˜ x dV!d V!

y =˜ y dV!d V!

z =˜ z d V!d V!

x =%x dA!dA!

y =%y dA!dA!

z =%z dA!dA!

x =%x dL!dL!

y =%y dL!dL!

z =%z dL!dL!

•Procedure for integration...

1.Set up coordinate axes and choose a differential element• Lines: dL (line segment)

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1.Set up coordinate axes and choose a differential element• Lines: dL (line segment)• Area: dA (rectangle with finite length, differential width)

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1.Set up coordinate axes and choose a differential element• Lines: dL (line segment)• Area: dA (rectangle with finite length, differential width)• Volume: dV (circular disk or rectangular slab with differential

thickness)

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2.Express dL, dA, dV in coordinates that define the boundary of thebody

• Express x, y, z for centroid or CG case as equations• x, y, z are CG or centroid locations for the differential

element3.Integrate

• Establish limits and integrate

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~ ~ ~~ ~ ~

•Simplifying the analysis:

•Symmetry - The centroid will lie on any axis of symmetry for thebody.

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volume example



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area example



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line example

• Examples:• Problem 9-1

• Determine the distance x to the center of mass of thehomogeneous rod bent into the shape shown. If the rod has amass per unit length of 0.5 kg/m, determine the reactions at thefixed support O.


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• Locate the centroid x and y of the shaded area.


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• Locate the centroid of the solid.


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•Composite Bodies:

•Often a complicated body can be broken into simpler parts, eachof which has an easily computed centroid (or CG).

•The overall centroid is found by:


x =x i Ai!Ai!

y =y i Ai!

Ai!z =

z i Ai!Ai!

§9.3

•Composite Bodies:

•Centroids are tabulated in your text (back leaf).


§9.3

•Procedure for composite bodies...

1.Break the body into simpler parts2.Establish coordinate axes and determine xi, yi, zi foreach part i3.Complete summations according to the centroidequations

§9.3



• Locate the center of mass (x, y) of the four particles.


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• Determine the location y of the centroidal axis x x of the beam’scross-sectional area. Neglect the size of the corner welds at Aand B for the calculation.


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• Locate the centroid (x, y) of the member’s cross-sectional area.


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