m23 Center of Mass (Centroid) (1)

m23 Center of Mass (Centroid) (1) - MATH023 Center of...

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MATH023 Center of Mass ( Centroid )
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Objectives At the end of the period, you should be able to: Locate the centers of mass of simple regions and regions bounded by curves . Locate the centers of mass of solids of revolution .
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See-Saws We all remember the fun see-saw of our youth. But what happens if . . .
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Balancing Unequal Masses Moral Both the masses and their positions affect whether or not the “see saw” balances.
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Balancing Unequal Masses Need: M 1 d 1 = M 2 d 2 M 1 M 2 d 1 d 2
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Changing our Point of View The great Greek mathematician Archimedes said, “give me a place to stand and I will move the Earth,” meaning that if he had a lever long enough he could lift the Earth by his own effort.
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In other words. . . We can think of leaving the masses in place and moving the fulcrum. It would have to be a pretty long see-saw in order to balance the school bus and the race car, though!
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In other words. . . (We still) need: M 1 d 1 = M 2 d 2 M 2 d 1 d 2 M 1
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What happens if there are many things trying to balance on the see-saw ? Where do we place the fulcrum? Mathematical Setting First we fix an origin and a coordinate system. . . 0 1 -1 -2 2
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Mathematical Setting And place the objects in the coordinate system. . . 0 M 2 M 1 M 3 M 4 d 2 d 1 d 3 d 4 Except that now d 1 , d 2 , d 3 , d 4 , . . . denote the placement of the objects in the coordinate system, rather than relative to the fulcrum. (Because we don’t, as yet, know where the fulcrum will be!)
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Mathematical Setting And place the objects in the coordinate system. . . 0 M 2 M 1 M 3 M 4 d 2 d 1 d 3 d 4 Place the fulcrum at some coordinate . is called the center of mass of the system. x x x
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Mathematical Setting And place the objects in the coordinate system. . . 0 M 2 M 1 M 3 M 4 d 2 d 1 d 3 d 4 In order to balance 2 objects, we needed: M 1 d 1 = M 2 d 2 OR M 1 d 1 - M 2 d 2 =0 For a system with n objects we need: x 1 1 2 2 3 3 ( ) ( ) ( ) ( ) 0 n n M d x M d x M d x M d x
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Finding the Center of Mass of the System 1 1 2 2 3 3 ( ) ( ) ( ) ( ) 0 leads to the following set of calculations n n M d x M d x M d x M d x x 1 1 1 2 2 2 3 3 3 0 n n n M d M x M d M x M d M x M d M x Now we solve for . 1 1 2 2 3 3 1 2 3 n n n M d M d M d M d M x M x M x M x 1 1 2 2 3 3 1 2 3 n n n M d M d M d M d M M M M x 1 1 2 2 3 3 1 2 3 And finally . . . n n n M d M d M d M d x M M M M
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The Center of Mass of the System 1 1 2 2 3 3 1 2 3 n n n M d M d M d M d x M M M M In the expression The numerator is called the first moment of the system The denominator is the total mass of the system
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Center of Mass Our main objective here is to find the point P on which a thin plate of any given shape balances horizontally as shown. This point is called the center of mass (or center of gravity) of the plate.
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Center of Mass The center of mass of the system is located at where m = Σ m i is the total mass of the system.
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