To calculate the $y$-coordinate of the centroid ($\bar{y}$) for the composite area in the given figure, we use the centroid formula for composite areas:

$\bar{y} = \frac{\sum (A_i \cdot y_i)}{\sum A_i}$

Where:

• $A_i$ is the area of the $i$-th section.
• $y_i$ is the $y$-coordinate of the centroid of the $i$-th section.

We will calculate the area and $y_i$ for each segment (1, 2, 3, and 4) based on the given dimensions, then substitute into the formula. Let's proceed step-by-step.

Step 1: Areas and Centroids of Individual Sections

Section 1: Rectangle at the bottom-left

• Dimensions: $W_1 = 10 \, \text{mm}, H_1 = 23 \, \text{mm}$
• Area: $A_1 = W_1 \cdot H_1 = 10 \cdot 23 = 230 \, \text{mm}^2$
• Centroid $y_1$: Half of its height from the bottom, $y_1 = \frac{H_1}{2} = \frac{23}{2} = 11.5 \, \text{mm}$

Section 2: Rectangle at the bottom-right

• Dimensions: $W_2 = 24 \, \text{mm}, H_1/2 = 11.5 \, \text{mm}$
• Area: $A_2 = W_2 \cdot \frac{H_1}{2} = 24 \cdot 11.5 = 276 \, \text{mm}^2$
• Centroid $y_2$: Half of its height from the bottom, $y_2 = \frac{H_1}{4} = \frac{23}{4} = 5.75 \, \text{mm}$

Section 3: Triangle at the top-left

• Base: $W_1 = 10 \, \text{mm},$ Height: $H_2 = 11 \, \text{mm}$
• Area: $A_3 = \frac{1}{2} \cdot W_1 \cdot H_2 = \frac{1}{2} \cdot 10 \cdot 11 = 55 \, \text{mm}^2$
• Centroid $y_3$: Located at $\frac{H_2}{3}$ from the base (top of section 1), so $y_3 = H_1 + \frac{H_2}{3} = 23 + \frac{11}{3} \approx 26.67 \, \text{mm}$

Section 4: Circular hole

• Radius: $R = 6 \, \text{mm},$ Area (negative since it's a hole): $A_4 = -\pi R^2 = -\pi (6^2) = -113.1 \, \text{mm}^2$
• Centroid $y_4$: Centered vertically at $y_4 = H_1/2 = 11.5 \, \text{mm}$

Step 2: Weighted Sum of $A_i \cdot y_i$

$\sum (A_i \cdot y_i) = A_1 \cdot y_1 + A_2 \cdot y_2 + A_3 \cdot y_3 + A_4 \cdot y_4$ $= (230 \cdot 11.5) + (276 \cdot 5.75) + (55 \cdot 26.67) + (-113.1 \cdot 11.5)$ $= 2645 + 1587 + 1466.85 - 1300.65 = 4398.2 \, \text{mm}^3$

Step 3: Total Area

$\sum A_i = A_1 + A_2 + A_3 + A_4$ $= 230 + 276 + 55 - 113.1 = 447.9 \, \text{mm}^2$

Step 4: Centroid Calculation

$\bar{y} = \frac{\sum (A_i \cdot y_i)}{\sum A_i} = \frac{4398.2}{447.9} \approx 9.82 \, \text{mm}$

Final Answer

The $y$-coordinate of the centroid is approximately: $\boxed{9.82 \, \text{mm}}$

Would you like further clarification or assistance with other parts of this problem?

Additional Questions for Exploration:

1. How would the calculation change if the dimensions of one of the sections were altered?
2. What is the physical significance of the centroid in structural analysis?
3. How can you calculate the $x$-coordinate of the centroid for this figure?
4. What role does the negative area of the circular hole play in determining the centroid?
5. How would the calculation differ if the hole were a different shape (e.g., a square)?

Tip:

Always double-check your calculations for negative areas when dealing with composite areas involving holes or cutouts.