Find the centre of gravity of 100 mm × 150 mm x 30 mm T-section

To find the **center of gravity (CG)** of a T-section, we follow these steps. A T-section consists of two rectangular parts: the **flange** and the **web**.

Given:
- Width of the flange = 100 mm
- Depth (height) of the flange = 30 mm
- Height of the web = 150 mm (including the flange height)
- Thickness of the web = 30 mm

### Step 1: Break down the T-section into two rectangles
1. **Flange dimensions**: 100 mm (width) × 30 mm (height)
2. **Web dimensions**: 30 mm (width) × (150 - 30) mm = 30 mm (height of the web excluding the flange).

### Step 2: Find the area of both parts
1. **Area of the flange (A₁)** = 100 mm × 30 mm = 3000 mm²
2. **Area of the web (A₂)** = 30 mm × 120 mm = 3600 mm²

### Step 3: Find the distances from a reference point (usually the bottom of the web) to the centroids of each part
1. **Distance from the bottom to the centroid of the flange (y₁)** = Total height - Half of flange height = 150 mm - (30 mm / 2) = 135 mm
2. **Distance from the bottom to the centroid of the web (y₂)** = Half of the web height = (120 mm / 2) = 60 mm

### Step 4: Apply the center of gravity formula
The vertical location of the center of gravity (**Yₐ**) is calculated using the formula:

\[
Yₐ = \frac{(A₁ \cdot y₁) + (A₂ \cdot y₂)}{A₁ + A₂}
\]

Substitute the values:

\[
Yₐ = \frac{(3000 \cdot 135) + (3600 \cdot 60)}{3000 + 3600}
\]
\[
Yₐ = \frac{405000 + 216000}{6600}
\]
\[
Yₐ = \frac{621000}{6600} = 94.09 \, \text{mm}
\]

Thus, the center of gravity of the T-section is approximately **94.09 mm** from the bottom.

### Horizontal center of gravity:
Since the flange and the web are symmetrically positioned, the horizontal center of gravity is at the midpoint of the width (50 mm).

### Final result:
The center of gravity of the T-section is at:
- **50 mm** horizontally (from the side),
- **94.09 mm** vertically (from the bottom).

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Would you like further details on this process or any clarifications? Here are some related questions:

1. How does changing the flange or web dimensions affect the center of gravity?
2. How is the moment of inertia of a T-section calculated?
3. How does the center of gravity relate to the stability of structures?
4. Can the center of gravity be outside the actual material?
5. How do we calculate the center of gravity for non-uniform shapes?

**Tip:** Knowing the center of gravity is crucial for understanding how a structure will balance and bear loads.

Learn how to calculate the center of gravity for a 100 mm × 150 mm × 30 mm T-section with step-by-step instructions. This tutorial includes key statics concepts, formulas for composite sections, and the application of the center of gravity formula. Suitable for Grades 11-12 and college-level engineering students.

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