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Unformatted text preview: Section 6.7 Moments and Centers of Mass Goal: Some objects behave as if all their mass is concentrated at a single point called the center of mass. We will develop a way to find centers of mass for one and two dimensional systems. Point Masses: Look at a system of point masses along a line: The center of mass is the point at which we could place a fulcrum and have the line balance. mi xi is called the _____________________ of mi The moment about the origin of the system is: Moment measures the tendency of a system to twist. (if you multiply moment by gravity, you get torque) The system’s total mass is: The center of mass is: Example 1: Given two point masses, mA = 60 lb, mB = 100 lb on a line where A is located 3 feet from the fulcrum, where should B be located to have the two masses balanced? Point masses in 2 dimensions In a two dimensional system of point masses m1 , m2 , ..., mn at positions ( x1 , y1 ), ( x2 , y2 ), ..., ( xn , yn ) : n Total Mass: m = ∑ mi i =1 n Moment about yaxis: M y = ∑ mi xi i =1 n Moment about xaxis: M x = ∑ mi yi i =1 ⎛ My Mx ⎞
Center of Mass: ( x , y ) = ⎜ ,
⎠ Example 2: m1 = 4, m2 = 8, m3 = 3, m4 = 2
P1 = (−2, 3), P2 = (2, −6 ), P3 = ( 7, −3), P4 = (5,1) Find the center of mass ( x , y ) . Center of Mass of Solids: The center of mass of a rectangle is at its center: Mass=(density)(area) in 2 dimensions, so we can use area to determine center of mass. Density is constant in our problems. Example 3: Find the center of mass of the object below. The density is 5. Center of mass of a triangle: x A + x B + xC
yA + yB + yC x=
3 Example 4: Find the center of mass of the following region. Density = 1. Example 5: Find the center of mass. δ = 3 . Thin Plates Divide the region into n strips parallel to the y – axis. Assume the center of mass of each sub‐rectangle is at its geometric center. Then if δ ( xi ) is the density of the ith rectangle and ΔAi is the area of the ith rectangle, Δmi = In this course we will assume density is constant and write δ ( xi ) = δ 1b
δ ∫ f ( x )2 − g( x )2 dx
x = ab y = 2 ba
δ ∫ ( f ( x ) − g( x )) dx
δ ∫ ( f ( x ) − g( x ))dx
b δ ∫ x( f ( x ) − g( x )) dx
a a Suppose we integrate over the y‐axis. d
δ ∫ f ( y)2 − g( y)2 dy
δ ∫ y( f ( y) − g( y))dy
x=2 d y = d
δ ∫ f ( y) − g( y)dy
δ ∫ f ( y) − g( y)dy
c c Example 6: Let R be the lamina with density 3 in the region in the xy plane bounded by x = e y , x = 1, and y = 2. Find the mass of the lamina. Example 7: Find the center of mass of the region bounded by y = 2 x and y = x 2 . Let δ = 1 . Using Symmetry Example 8: Find the centroid of the region bounded by y = 1 and y = x 2 . Let δ = 1 . Example 9: Find the center of mass of the region bounded by x = y 2 − y and y = x with constant density δ = 1 . ...
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This note was uploaded on 10/02/2011 for the course MATH 2214 at Virginia Tech.