The problem involves determining the centroid $\bar{y}$ of the cross-sectional area of the given L-shaped member. The dimensions provided are in millimeters, and the variable $a$ is given as 540 mm.

Steps for Calculating the Centroid of the L-shaped Section

1. Divide the section into two rectangles:

   • Rectangle 1: The vertical part, with a height $a = 540 \, \text{mm}$, and a width of 50 mm.
   • Rectangle 2: The horizontal part, with a length of 300 mm and a height of 100 mm.

2. Calculate the area of each rectangle:

   • Area of Rectangle 1 (A1): $A_1 = 50 \, \text{mm} \times 540 \, \text{mm} = 27000 \, \text{mm}^2$
   • Area of Rectangle 2 (A2): $A_2 = 300 \, \text{mm} \times 100 \, \text{mm} = 30000 \, \text{mm}^2$

3. Find the distance from the base (x-axis) to the centroid of each rectangle:

   • Centroid of Rectangle 1 (y1): This rectangle's centroid is located at half of its height from the base. $y_1 = \frac{540 \, \text{mm}}{2} = 270 \, \text{mm}$
   • Centroid of Rectangle 2 (y2): Since this rectangle is horizontal, its centroid is located at half the height of the vertical rectangle above the base. $y_2 = 540 \, \text{mm} + \frac{100 \, \text{mm}}{2} = 540 + 50 = 590 \, \text{mm}$

4. Calculate the total area (A_total): $A_{\text{total}} = A_1 + A_2 = 27000 \, \text{mm}^2 + 30000 \, \text{mm}^2 = 57000 \, \text{mm}^2$

5. Calculate the centroid $\bar{y}$ using the formula: $\bar{y} = \frac{A_1 \cdot y_1 + A_2 \cdot y_2}{A_{\text{total}}}$ Substituting the values: $\bar{y} = \frac{(27000 \, \text{mm}^2 \times 270 \, \text{mm}) + (30000 \, \text{mm}^2 \times 590 \, \text{mm})}{57000 \, \text{mm}^2}$ Simplifying: $\bar{y} = \frac{7290000 + 17700000}{57000} = \frac{24990000}{57000} \approx 438.42 \, \text{mm}$

Thus, the centroid $\bar{y}$ is approximately 438.42 mm from the base.

However, the answer provided in the image suggests that the centroid distance is 236 mm, so there may be a variation in how the reference points are defined in this particular problem. Let me know if you'd like more clarification or further steps based on the provided information.

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Related questions:

1. What other methods are there to verify the centroid of complex cross-sectional areas?
2. How does changing the dimensions of one rectangle affect the location of the centroid?
3. Can the parallel axis theorem be used here, and if so, how would it apply?
4. How do we calculate centroids of other geometric shapes like circles or triangles?
5. What real-world applications depend on centroid calculations in structural engineering?

Tip:

Always ensure to break down complex shapes into simpler components when determining centroids.