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Section 13.4 Notes - If is a continuous density function on...

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Summary of Section 6.6 Mass is a measure of a body’s resistance to changes in motion. The center of mass is the point at which a fulcrum could be located to attain equilibrium. The moment about the origin is a measure of the tendency of a system to rotate about the origin. Similarly, the moment about the x-axis is a measure of the tendency of a system to rotate about the x-axis. A planar lamina is a thin, flat plate of material of constant density. Density is a measure of mass per unit of volume. For a planar lamina density is considered to be a measure of mass per unit of area. If the density is constant , Mass = ∫ ∫ ∫ ∫ = = R R dA dA A ρ ρ ρ In section 13.4 we will extend the formulas from section 6.6 to the case where the density is variable
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Unformatted text preview: If is a continuous density function on the lamina corresponding to a plane region R, then the mass m of the lamina is given by ∫ ∫ = R dA y x m ) , ( . Moments and Center of Mass of a Variable Density Planar Lamina Let be a continuous density function on the planar lamina R. The moments of mass with respect to the x- and y- axes are ∫ ∫ = R x dA y x y M ) , ( and ∫ ∫ = R y dA y x x M ) , ( . If m is the mass of the lamina, then the center of mass is ( 29 . , , = m M m M y x x y If R represents a simple plane region rather than a lamina, the point ( 29 y x , is called the centroid of the region....
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