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V_-_Center_of_Gravity_and_Centroid

# V_-_Center_of_Gravity_and_Centroid - CENTER OF GRAVITY AND...

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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

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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

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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

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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 γ is the specific weight of the body, then dW = γ dV = = = V V V V V V dV dV z z dV dV y y dV dV x x γ γ γ γ γ γ ~ ; ~ ; ~
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 ρ is the density of the body, then dM = ρ dV, (remember that γ = ρg and hence, ρ = γ/g ) = = = V V V V V V dV dV z z dV dV y y dV dV x x ρ ρ ρ ρ ρ ρ ~ ; ~ ; ~ CENTRE OF MASS OF A BODY

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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. = = = L L L L L L dL dL z z dL dL y y dL dL x x ~ ; ~ ; ~ For a line

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CALCULATION STEPS FOR CG, CM AND CENTROID OF A BODY 2. Set up the integrand for numerator and denominator 1. Set up the appropriate boundary of integration, which is the same for both numerator and denominator Triple integration for a volume Double integration for an area Line integration for a line For CG, the integrands for both the numerator and the denominator contain g For CM, the integrands for both the numerator and the
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