To solve this problem of finding the centroid of the plan area, we need to follow the process of integration.

Plan Breakdown:

• The given figure consists of two parts:

  • A rectangular section (2 m by 0.5 m) on the left.
  • A curve described by the equation $xy = 1$ on the right.

  The total width of the shape is 2 m, and the height is 2 m at the left and tapers down along the curve on the right.

Steps for Centroid Calculation:

1. Determine the boundaries of the region:

   • The left rectangle: $x = 0$ to $x = 0.5$.
   • The right curve: The equation $xy = 1$ implies $y = \frac{1}{x}$. This governs the curved part from $x = 0.5$ to $x = 2$.

2. Find the Area: The area of the region can be split into two:

   • Rectangular area: $A_1 = 2 \times 0.5 = 1 \text{ m}^2$.
   • Area under the curve from $x = 0.5$ to $x = 2$: $A_2 = \int_{0.5}^{2} \frac{1}{x} \, dx = [\ln x]_{0.5}^{2} = \ln(2) - \ln(0.5) = \ln(4)$ Therefore, the total area $A = A_1 + A_2$.

3. Calculate $\bar{x}$ (Centroid along x-axis): The x-coordinate of the centroid is given by: $\bar{x} = \frac{1}{A} \left( \int x dA \right)$ This will involve integrating separately for the rectangular and the curved parts.

4. Calculate $\bar{y}$ (Centroid along y-axis): The y-coordinate of the centroid is given by: $\bar{y} = \frac{1}{A} \left( \int y dA \right)$ This again involves breaking up the integration into two parts (rectangular and curved).

Once we perform these calculations, we'll have the coordinates of the centroid $(\bar{x}, \bar{y})$.

Would you like a step-by-step integration breakdown for $\bar{x}$ and $\bar{y}$?

Related Questions:

1. How do you apply integration to find centroids for different types of shapes?
2. Why is the centroid important in structural or mechanical applications?
3. What is the significance of the equation $xy = 1$ in this particular geometry?
4. Can you find the centroid for irregular shapes using approximation techniques?
5. How does the centroid change if the shape is rotated or scaled?

Tip:

When calculating centroids for composite areas, always break the shape into simpler geometric components and sum their contributions based on their areas and centroids.