W_-_Center_of_Gravity_and_Centroid2

W_-_Center_of_Gravity_and_Centroid2 - CENTER OF CENTER OF...

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Unformatted text preview: CENTER OF CENTER OF GRAVITY AND GRAVITY AND CENTROID CENTROID Center of Gravity, Center of Mass Center of Gravity, Center of Mass and Centroid and Centroid LEARNING OBJECTIVES LEARNING OBJECTIVES Be able to understand the concepts of Be able to understand the concepts of center of gravity, center of mass, and center of gravity, center of mass, and centroid centroid Be able to determine the locations of Be able to determine the locations of these points for a system of particles or a these points for a system of particles or a body body PRE-REQUISITE KNOWLEDGE PRE-REQUISITE KNOWLEDGE Units of measurement Units of measurement Integrations over lines, areas and Integrations over lines, areas and volumes volumes SYSTEM OF PARTICLES Center of gravity (CG) is a point which locates the resultant weight of a system of particles n n n W W W W x W x W x x + + + + + + = ... ... 2 1 2 2 1 1 n n n W W W W y W y W y y + + + + + + = ... ... 2 1 2 2 1 1 n n n W W W W z W z W z z + + + + + + = ... ... 2 1 2 2 1 1 Center of mass (CM) is a point which locates the resultant mass of a system of particles n n n m m m m x m x m x x + + + + + + = ... ... 2 1 2 2 1 1 n n n m m m m y m y m y y + + + + + + = ... ... 2 1 2 2 1 1 n n n m m m m z m z m z z + + + + + + = ... ... 2 1 2 2 1 1 m 1 m 2 m n CG and CM have the same location when the gravity, g is constant, which is not always true, e.g. planets. SYSTEM OF PARTICLES CENTRE OF GRAVITY OF A BODY Center of gravity (CG) is a point which locates the resultant weight of a body = = = V V V V V V dW dW z z dW dW y y dW dW x x ~ ; ~ ; ~ If is the specific weight of the body, then dW = dV = = = V V V V V V dV dV z z dV dV y y dV dV x x ~ ; ~ ; ~ Center of mass (CM) is a point which locates the resultant mass of a body = = = V V V V V V dM dM z z dM dM y y dM dM x x ~ ; ~ ; ~ if is the density of the body, then dM = dV, (remember that = g and hence, = /g ) = = = V V V V V V dV dV z z dV dV y y dV dV x x ~ ; ~ ; ~ CENTRE OF MASS OF A BODY The centroid of a body is a point which locates the geometric center of the body = = = V V V V V V dV dV z z dV dV y y dV dV x x ~ ; ~ ; ~ = = = A A A A A A dA dA z z dA dA y y dA dA x x ~ ; ~ ; ~ For an area CENTROID OF A BODY For a volume CENTROID OF A BODY-Continued Note: CM and centroid are the same if the body is homogeneous (constant density). If an object has an axis of symmetry, then the centroid of the object is located on that axis....
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This note was uploaded on 07/26/2009 for the course CE 221 taught by Professor Buch during the Spring '08 term at Michigan State University.

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W_-_Center_of_Gravity_and_Centroid2 - CENTER OF CENTER OF...

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