Centroid

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Centroid ??????????

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• ??????? x ??? y ????? ???????????? centroid
  ??????????????????????
• ???????????? centroid ?????????????
• ?????????????????????? centroid ???????????????

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9.3 Composite Bodies

• Procedure for Analysis
• Summations
• Determine the coordinates of the center of
  gravity by applying the center of gravity
  equations
• If an object is symmetrical about an axis, the
  centroid of the objects lies on the axis

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• Example 9.10
• Locate the centroid of the plate area.

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• Solution
• Composite Parts
• Plate divided into 3 segments
• Area of small rectangle considered negative

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• Solution
• Moment Arm
• Location of the centroid for each piece is
  determined and indicated in the diagram

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• Solution
• Summations

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???????? Pappus and Guldinus

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• ????AB ?????????????AO
• ???????????????????????

o
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• ????????
• ????????????????? ABC
• ????????????? AC
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  ?/????????????

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• ???????
• ????????????????? ??????????????????????????????
  ????? centroid ??????????

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• ???????
• ????????????????? ??????????????????????????????
  ?????? centroid ??????????

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9.4 Theorems of Pappus and Guldinus

• Composite Shapes
• The above two mentioned theorems can be applied
  to lines or areas that may be composed of a
  series of composite parts
• Total surface area or volume generated is the
  addition of the surface areas or volumes
  generated by each of the composite parts

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9.4 Theorems of Pappus and Guldinus

• Example 9.12
• Show that the surface area of a sphere is
• A 4pR2
• and its volume
• V 4/3 pR3

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• Solution
• Surface Area
• Generated by rotating semi-arc about the x axis
• For centroid,
• For surface area,

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• Solution
• Volume
• Generated by rotating semicircular area about the
  x axis
• For centroid,
• For volume,