Center of gravity – problems and solutions

1. Determine the coordinate of the center of gravity of the object as shown in the figure below.

Solution :

Divide the object into three parts.

Area of part 1 (A₁) = (2)(6) = 12 cm²

The center point lies on the x axis (x₁) = 1/2 (2) = 1 cm

The center point lies on the y axis (y₁) = 1/2 (6) = 3 cm

Area of part 2 (A₂) = (4)(2) = 8 cm²

The center point lies on the x axis (x₂) = 2 + (1/2)(4) = 2 + 2 = 4 cm

The center point lies on the y axis (y₂) = 2 + (1/2)(2) = 2 + 1 = 3 cm

Area of part 3 (A₃) = (2)(6) = 12

The center point lies on the x axis (x₃) = 2 + 4 + (1/2)(2) = 2 + 4 + 1 = 7 cm

The center point lies on the y axis (y₃) = 1/2 (6) = 3 cm

Coordinate of the center of gravity at x axis :

Coordinate of the center of gravity at y axis :

Coordinate of the center of gravity of the object is at x axis and y axis (x , y) = (4, 3)

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2. Determine the coordinate of the center of gravity of object , about the x axis.

Solutions

Divide the object into three parts, A, B, C, and D.

Calculate the area of each part :

A_A = ½ (base)(height) = ½ (1.5)(3) = (1.5)(1.5) = 2.25

A_B = (length)(width) = (4.5-1.5)(1) = (3)(1) = 3

A_C = ½ (base)(height) = ½ (6-4.5)(3) = (1.5)(1.5) = 2.25

A_D = ½ (base)(height) = ½ (4.5-1.5)(6-3) = ½ (3)(3) = (1.5)(3) = 4.5

y_A = 1/3 (3) = 1

y_B = 1/2 (1) = 0,5

y_C = 1/3 (3) = 1

y_D = 3 + (1/3)(6-3) = 3 + (1/3)(3) = 3 + 1 = 4

Coordinate of the center of gravity at y axis :

Coordinate of the center of gravity about the x axis is 2 cm.

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3. Determine coordinate of the center of gravity of the object, as shown in figure.

Solution :

Divide the object into four parts, A, B, C, and D.

Calculate the area of each part.

A_A = (length)(width) = (4)(3) = 12

A_B = ½ (base)(height) = ½ (6-4)(3) = ½ (2)(3) = (1)(3) = 3

A_C = ½ (base)(height) = ½ (8-6)(3) = ½ (2)(3) = (1)(3) = 3

A_D = (length)(width) = (8)(6-3) = (8)(3) = 24

y_A = 1/2 (3) = 1.5

y_B = 3 – (1/3)(3) = 3 – 1 = 2

y_C = 3 – (1/3)(3) = 3 – 1 = 2

y_D = 3 + (1/2)(6-3) = 3 + (1/2)(3) = 3 + 1.5 = 4.5

x_A = 1/2 (4) = 2

x_B = 4 + (1/2)(6-4) = 4 + (1/2)(2) = 4 + 1 = 5

x_C = 6 + (1/2)(8-6) = 6 + (1/2)(2) = 6 + 1 = 7

x_D = 1/2 (8) = 4

Coordinate of the center of gravity at the x axis :

Coordinate of the center of gravity at y axis :

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