17 - COM

# 17 - COM - y M x m i i 1 n Ex m 1 = 4 m 2 = 8 m 3 = 3 m 4 =...

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Center of Mass M = moment = (mass)(directed distance from point P) Moment measures the tendency of a mass to rotate around point P. (If we multiply moment by gravity we get torque.) Ex. m A = 60 lb. m B = 100 lb. d A = 3 feet What should d B equal so that A and B are in balance?

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The total moment of n masses with respect to the origin is x 1 m 1 x 2 m 2 ... x n m n x i m i i 1 n Suppose we have n masses along a horizontal line Somewhere a fulcrum can be placed where these masses will balance. Let x be the coordinate where the masses balance. x is the center of mass – the balance point.
x x i m i i 1 n m i i 1 n total moment wrt origin total mass Two dimensions M x = Moment with respect to the x-axis. M x measures the tendency to rotate around the x-axis. M y = Moment with respect to the y-axis. M y measures the tendency to rotate around the y-axis. M x = m i y i i 1 n M

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Unformatted text preview: y M x m i i 1 n Ex. m 1 = 4 m 2 = 8 m 3 = 3 m 4 = 2 P 1 (-2, 3) P 2 (2, -6) P 3 (7, -3) P 4 (5, 1) Find the center of mass x , y . Solids The center of mass of a rectangle is in the center Since mass = (density)(area) in 2D, we can use area to find the COM. Density in all of our problems will be constant. Ex. Find the center of mass of the following region. Suppose the density is 5. The center of mass of a triangle ABC can be written x x A x B x C 3 ±±±±±±±±±±± y y A y B y C 3 Ex. Find the center of mass of the following region. δ 1 Ex. Suppose there is a hole in the region. Find the center of mass. 1 Do: Find the center of mass. 1. m 1 = 10 P 1 (-1, 2) m 2 = 6 P 2 (5, 0) 2. Let δ 3...
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