297Chap5

# 297Chap5 - Center of gravity of a two-dimensional body...

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Center of gravity of a two-dimensional body Gravitational attraction of the earth on a rigid body, a distributed force, can be represented by a single equivalent force W applied at the center of gravity.

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CE 297 2 Center of gravity of a two-dimensional body (cont.) For a flat horizontal plate Σ F z : Σ M y : Σ M x : W = ∆ W 1 + ∆ W 2 + . . . xW = x 1 W 1 + x 2 W 2 + . . . yW = y 1 W 1 + y 2 W 2 + . . . W = dW xW = x dW yW = y dW
CE 297 3 Center of gravity of a two-dimensional body (cont.) For a wire lying in the xy plane Σ F z : Σ M y : Σ M x : W = ∆ W 1 + ∆ W 2 + . . . xW = x 1 W 1 + x 2 W 2 + . . . yW = y 1 W 1 + y 2 W 2 + . . . W = dW xW = x dW yW = y dW

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CE 297 4 Center of gravity of a two-dimensional body (cont.) Σ F z : Σ M y : Σ M x : W = ∆ W 1 + ∆ W 2 + . . . xW = x 1 W 1 + x 2 W 2 + . . . yW = y 1 W 1 + y 2 W 2 + . . . W = dW xW = x dW yW = y dW The center of gravity is located at x = x dW W y = y dW W Note: For a wire lying in the xy plane, the center of gravity may not be located on the wire.
CE 297 5 Centroids of areas and lines For a homogeneous flat horizontal plate of uniform thickness W = γ t A W = γ tA where γ = specific weight t = thickess of the plate Σ F z : Σ M y : Σ M x : W = ∆ W 1 + ∆ W 2 + . . . xW = x 1 W 1 + x 2 W 2 + . . . yW = y 1 W 1 + y 2 W 2 + . . . W = dW xW = x dW yW = y dW

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CE 297 6 Centroids of areas and lines (cont.) Σ F z : Σ M y : Σ M x : A = ∆ A 1 + ∆ A 2 + . . . x A = x 1 A 1 + x 2 A 2 + . . . yA = y 1 A 1 + y 2 A 2 + . . . A = dA x A = x dA yA = y dA The centroid of an area is located at x = x dA A y = y dA A The centroid is a geometric prop- erty.
CE 297 7 Centroids of areas and lines For a homogeneous wire of uniform cross section W = γ a L W = γ aL where γ = specific weight a = cross sectional area of the wire Σ F z : Σ M y : Σ M x : W = ∆ W 1 + ∆ W 2 + . . . xW = x 1 W 1 + x 2 W 2 + . . . yW = y 1 W 1 + y 2 W 2 + . . . W = dW xW = x dW yW = y dW

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CE 297 8 Centroids of areas and lines (cont.) Σ F z : Σ M y : Σ M x : L = ∆ L 1 + ∆ L 2 + . . . xL = x 1 L 1 + x 2 L 2 + . . . yL = y 1 L 1 + y 2 L 2 + . . . L = dL xL = x dL yL = y dL The centroid of the line L is located at x = x dL L y = y dL L The centroid is a geometric prop- erty.
CE 297 9 First moments of areas and lines For areas Q y = x dA = x A First moment of A with respect to y -axis Q x =

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