give answer in 2 step with explanation at the end of each step and final answer at the end:| Problem 4. The cross section of a structural component is composed of three sections as shown |
below (num

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:| Problem 4. The cross section of a structural component is composed of three sections as shown |
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moment of inertia | ; of the cross section shown with respect to centroidal axis (xbar-xbar) axis. (60
pts.)
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VIPSolutions · Accepted Answer

## Step 1: Find the Area and Centroid ($\bar{y}$) of the Composite Section

### **Areas:**
- **Section 1 (Parabolic Area):**
  - Equation: $ y^3 = x^2 $
  - Limits: $ x =  $ to $ x = 1 $, $ y =  $ to $ y = .5 $
  - Area ($A_1$): Use integration.
- **Section 2 (Hole - Circle):**
  - Radius = .5 m
  - Area ($A_2$): $ A_2 = -\pi(.5)^2 = -.785 $ m² (subtract since it's a hole)
- **Section 3 (Rectangle):**
  - Dimensions: 1 m × 1 m
  - Area ($A_3$): $ 1 \times 1 = 1 $ m²

#### **Total Area ($A_{total}$):**
$$
A_{total} = A_1 + A_3 + A_2
$$

#### **Centroid ($\bar{y}$):**
$$
\bar{y} = \frac{\sum (A_i y_i)}{A_{total}}
$$
Where $y_i$ is the centroid of each section relative to the x-axis.

**Explanation:**  
In this step, we calculate the individual areas and their centroid locations to determine the centroid of the composite section, which is needed for the parallel axis theorem in the next step.

---

## Step 2: Calculate the Moment of Inertia about the Centroidal Axis ($I_{x'}$)

1. **Moment of Inertia about x-axis ($I_{x}$) for Each Section:**
   - **Section 1:** Use integration for the parabolic segment.
   - **Section 2:** $I_{x,2} = -\frac{\pi r^4}{4}$ (negative, since it's a hole)
   - **Section 3:** $I_{x,3} = \frac{1}{12} b h^3$
2. **Apply the Parallel Axis Theorem:**
   $$
   I_{x'} = \sum [I_{x,i} + A_i (y_i - \bar{y})^2]
   $$
   Where $I_{x,i}$ is the moment of inertia of each section about the x-axis, and $(y_i - \bar{y})$ is the distance from each section's centroid to the centroid of the composite section.

**Explanation:**  
This step involves calculating each section's moment of inertia about the x-axis, then shifting each to the centroidal axis using the parallel axis theorem, and summing up for the final answer.

---

## **Final Answer Format:**

$$
I_{x'} = \sum [I_{x,i} + A_i (y_i - \bar{y})^2]
$$

**Where:**
- $I_{x,i}$ = moment of inertia of each section about the x-axis
- $A_i$ = area of each section (negative for holes)
- $y_i$ = centroid location of each section relative to x-axis
- $\bar{y}$ = centroid of the whole section

**You need to perform the integrations and calculations to get the final numeric value.** If you need help with specific integrations or values, let me know!

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