To determine the centroid of the built-up section in the given figure, we need to consider the individual rectangles that make up the shape and then calculate the centroid of the entire structure by taking the weighted average of the centroids of these individual parts. The steps are as follows:

1. Identify individual sections: The figure consists of three rectangles.

   • Rectangle 1: Bottom horizontal rectangle (width $120$ mm, height $20$ mm).
   • Rectangle 2: Vertical rectangle (width $20$ mm, height $80$ mm).
   • Rectangle 3: Top horizontal rectangle (width $100$ mm, height $10$ mm).

2. Calculate the area and centroid of each rectangle:

   • Rectangle 1: $A_1 = 120 \times 20 = 2400 \, \text{mm}^2$ Centroid (from the origin $O$): $\bar{x}_1 = \frac{120}{2} = 60 \, \text{mm}$ $\bar{y}_1 = \frac{20}{2} = 10 \, \text{mm}$

   • Rectangle 2: $A_2 = 20 \times 80 = 1600 \, \text{mm}^2$ Centroid (from the origin $O$): $\bar{x}_2 = 40 + \frac{20}{2} = 50 \, \text{mm}$ $\bar{y}_2 = 20 + \frac{80}{2} = 60 \, \text{mm}$

   • Rectangle 3: $A_3 = 100 \times 10 = 1000 \, \text{mm}^2$ Centroid (from the origin $O$): $\bar{x}_3 = 10 + \frac{100}{2} = 60 \, \text{mm}$ $\bar{y}_3 = 20 + 80 + \frac{10}{2} = 105 \, \text{mm}$

3. Calculate the combined centroid of the structure:

   • Total area: $A = A_1 + A_2 + A_3 = 2400 + 1600 + 1000 = 5000 \, \text{mm}^2$

   • $\bar{x}$-coordinate of the centroid: $\bar{x} = \frac{A_1 \bar{x}_1 + A_2 \bar{x}_2 + A_3 \bar{x}_3}{A} = \frac{2400 \times 60 + 1600 \times 50 + 1000 \times 60}{5000}$ $\bar{x} = \frac{144000 + 80000 + 60000}{5000} = \frac{284000}{5000} = 56.8 \, \text{mm}$

   • $\bar{y}$-coordinate of the centroid: $\bar{y} = \frac{A_1 \bar{y}_1 + A_2 \bar{y}_2 + A_3 \bar{y}_3}{A} = \frac{2400 \times 10 + 1600 \times 60 + 1000 \times 105}{5000}$ $\bar{y} = \frac{24000 + 96000 + 105000}{5000} = \frac{225000}{5000} = 45 \, \text{mm}$

The coordinates of the centroid are: $\bar{x} = 56.8 \, \text{mm}, \quad \bar{y} = 45 \, \text{mm}$

If you have any questions or need more details, feel free to ask!

Here are 5 questions you might find interesting:

1. How do you find the centroid of more complex shapes?
2. What is the significance of the centroid in engineering applications?
3. How does the method change for finding the centroid of a three-dimensional object?
4. Can you explain the concept of the center of mass and how it relates to the centroid?
5. How do you calculate the moment of inertia for composite areas?

Tip: Always break down complex shapes into simpler components to make calculations easier and more accurate.