- 2074 Asoj · 9+3 marks
Centroidal MOI of plane figure; define centroid
Short answer
Divide area into rectangles and a triangle, locate centroid at x̄=38.97 cm, ȳ=50.36 cm. Applying parallel axis theorem gives I_xx = 11.95×10⁶ cm⁴, I_yy = 7.16×10⁶ cm⁴. Centroid: geometric centre; centre of gravity: weight balance point; axis of symmetry: mirror line.
Determine the moment of inertia about centroidal XX and YY axes of the plane figure shown in the figure below. Define centroid, centre of gravity and axes of symmetry. [Figure: L-shaped/stepped plane figure with dimensions 14 cm, 20 cm, 15 cm, 13 cm, 1.5 m and 1.2 m as printed.]
Reading of the figure: the printed dimensions are 14 cm (top), 20 cm, 15 cm, 13 cm, 1.5 m (left side) and 1.2 m (bottom). I take them all in cm: left side 150 cm, bottom 120 cm, top 14 cm, the 15 cm and 13 cm steps at the lower right, and a sloping edge that closes the figure. (The 20 cm label is the short vertical piece below the top, which is part of the 14 cm wide column and does not change the area.) The figure is not to scale.
Origin at the bottom-left corner. Vertices: (0,0), (120,0), (120,13), (105,13), (14,130), (14,150), (0,150).
Division into simple shapes
- R1: rectangle 14 × 150 (x from 0 to 14)
- R2: rectangle 106 × 13 at the bottom (x from 14 to 120)
- T: right triangle, base 91 and height 117, with its right angle at (14, 13)
Centroid
| Part | A (cm²) | x (cm) | y (cm) | A·x | A·y |
|---|---|---|---|---|---|
| R1: 14 × 150 | 2,100.00 | 7.00 | 75.00 | 14,700.0 | 157,500.0 |
| R2: 106 × 13 | 1,378.00 | 67.00 | 6.50 | 92,326.0 | 8,957.0 |
| T: 91 × 117 | 5,323.50 | 44.33 | 52.00 | 236,008.5 | 276,822.0 |
ΣA = 8,801.50 cm² x̄ = ΣAx/ΣA = 38.97 cm, ȳ = ΣAy/ΣA = 50.36 cm
Moments of inertia about the centroidal axes (parallel axis theorem)
Own values: rectangle bh³/12 and hb³/12, triangle bh³/36 and hb³/36.
| Part | I_x own (cm⁴) | A·dy² | I_y own | A·dx² |
|---|---|---|---|---|
| R1: 14 × 150 | 3,937,500.0 | 1,274,555.3 | 34,300.0 | 2,146,980.9 |
| R2: 106 × 13 | 19,406.8 | 2,651,345.1 | 1,290,267.3 | 1,082,316.8 |
| T: 91 × 117 | 4,048,521.8 | 14,247.8 | 2,449,105.8 | 152,872.6 |
I_xx = 11,945,577 cm⁴ ≈ 11.95 × 10⁶ cm⁴ I_yy = 7,155,843 cm⁴ ≈ 7.16 × 10⁶ cm⁴
Definitions
- Centroid: the point at which the whole area of a plane figure can be assumed to be concentrated. It is the geometric centre of the area and depends only on shape: x̄ = ∫x dA / A, ȳ = ∫y dA / A.
- Centre of gravity: the point through which the whole weight of a body acts, whatever its orientation. For a body of uniform density and in a uniform gravity field it coincides with the centroid of its volume (or area for a thin plate).
- Axis of symmetry: a line that divides a figure into two parts that are mirror images. The centroid always lies on an axis of symmetry, and the product of inertia about it is zero.
- Radius of gyration: the distance k from an axis at which the whole area could be concentrated to give the same moment of inertia, I = A k², so k = √(I/A).
✓ Answer checked by Claude Opus 5.5 (high effort)
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