C_3 - Chapter 3 Centroids and Center of Mass 3.1 First...

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Chapter 3 Centroids and Center of Mass 3.1 First Moment and Centroid of a Set of Points The position vector of a point P relative to a point O is r P and a scalar associated with P is s , e.g., the mass m of a particle situated at P . The first moment of a point P with respect to a point O is the vector M = s r P . The scalar s is called the strength of P . The set of n points P i , i = 1 , 2 ,..., n , is { S } , Fig. 3.1(a) { S } = { P 1 , P 2 P n } = { P i } i = 1 , 2 ,..., n . The strengths of the points P i are s i , i = 1 , 2 n , i.e., n scalars, all having the same dimensions, and each associated with one of the points of { S } . The centroid of the set { S } is the point C with respect to which the sum of the first moments of the points of { S } is equal to zero. The centroid is the point defining the geometric center of system or of an object. The position vector of C relative to an arbitrarily selected reference point O is r C , Fig. 3.1(b). The position vector of P i relative to O is r i . The position vector of P i relative to C is r i - r C . The sum of the first moments of the points P i with respect to C is n i = 1 s i ( r i - r C ) . If C is to be centroid of { S } , this sum is equal to zero { S } (a) (b) { S } P 1 () s 1 P i s i P n s n P i ( s i ) O r i r C C P 2 s 2 Fig. 3.1 (a) Set of points and (b) centroid of a set of points 1
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2 3 Centroids and Center of Mass n i = 1 s i ( r i - r C ) = n i = 1 s i r i - r C n i = 1 s i = 0 . The position vector r C of the centroid C , relative to an arbitrarily selected reference point O , is given by r C = n i = 1 s i r i n i = 1 s i . If n i = 1 s i = 0 the centroid is not defined. The centroid C of a set of points of given strength is a unique point, its location being independent of the choice of reference point O . The cartesian coordinates of the centroid C ( x C , y C , z C ) of a set of points P i , i = 1 ,..., n , of strengths s i , i = 1 n , are given by the expressions x C = n i = 1 s i x i n i = 1 s i , y C = n i = 1 s i y i n i = 1 s i , z C = n i = 1 s i z i n i = 1 s i . The plane of symmetry of a set is the plane where the centroid of the set lies, the points of the set being arranged in such a way that corresponding to every point on one side of the plane of symmetry there exists a point of equal strength on the other side, the two points being equidistant from the plane. A set { S 0 } of points is called a subset of a set { S } if every point of { S 0 } is a point of { S } . The centroid of a set { S } may be located using the method of decomposition : divide the system { S } into subsets; find the centroid of each subset; assign to each centroid of a subset a strength proportional to the sum of the strengths of the points of the corresponding subset; determine the centroid of this set of centroids. 3.2 Centroid of a Curve, Surface, or Solid The position vector of the centroid C of a curve, surface, or solid relative to a point O is
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3.2 Centroid of a Curve, Surface, or Solid 3 r C = Z τ r d τ Z τ d τ , (3.1) where, τ is a curve, surface, or solid, r denotes the position vector of a typical point of τ , relative to O , and d τ is the length, area, or volume of a differential element of τ . Each of the two limits in this expression is called an “integral over the domain
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This note was uploaded on 08/29/2011 for the course MECH 2110 taught by Professor Clark,b during the Spring '08 term at Auburn University.

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C_3 - Chapter 3 Centroids and Center of Mass 3.1 First...

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