II. Workout the following problems
8. A vertical, solid steel post 25 cm in diameter and 2.50 m long is required to support a load of 8000 kg .
You can ignore the weight of the post. What are (a) the stress in the post; (b) the strain in the post; a
© the change in the post’s length when the load is applied?

Question

UpStudy AI · Accepted Answer

Sure, let's work through each part of the problem step by step.

### **Given:**
- **Diameter of the post (D):** 25 cm = 0.25 m
- **Radius of the post (r):** D/2 = 0.125 m
- **Length of the post (L):** 2.50 m
- **Load (mass, m):** 8000 kg
- **Acceleration due to gravity (g):** 9.81 m/s² (standard value)
- **Young's Modulus for steel (E):** $$2.00 \times 10^{11}$$ Pa

### **(a) Calculating the Stress in the Post**

**Stress (σ)** is defined as the force applied per unit area.

1. **Calculate the Force (F):**
   $$
   F = m \times g = 8000\, \text{kg} \times 9.81\, \text{m/s}² = 78,480\, \text{N}
   $$

2. **Calculate the Cross-Sectional Area (A):**
   $$
   A = \pi r² = \pi \times (0.125\, \text{m})² \approx 3.1416 \times 0.015625\, \text{m}² \approx 0.0491\, \text{m}²
   $$

3. **Calculate the Stress (σ):**
   $$
   \sigma = \frac{F}{A} = \frac{78,480\, \text{N}}{0.0491\, \text{m}²} \approx 1.60 \times 10^6\, \text{Pa} \, \text{or} \, 1.60\, \text{MPa}
   $$

### **(b) Calculating the Strain in the Post**

**Strain (ε)** is the deformation experienced by the material in the direction of the applied force, defined as:
$$
\epsilon = \frac{\sigma}{E}
$$

1. **Calculate Strain (ε):**
   $$
   \epsilon = \frac{1.60 \times 10^6\, \text{Pa}}{2.00 \times 10^{11}\, \text{Pa}} = 8.00 \times 10^{-6}
   $$

### **(c) Calculating the Change in Length of the Post**

**Change in Length (ΔL)** is given by:
$$
\Delta L = \epsilon \times L
$$

1. **Calculate Change in Length (ΔL):**
   $$
   \Delta L = 8.00 \times 10^{-6} \times 2.50\, \text{m} = 2.00 \times 10^{-5}\, \text{m} = 0.02\, \text{mm}
   $$

### **Final Answers:**
- **(a)** The stress in the post is **1.60 × 10⁶ Pa** (or **1.60 MPa**).
- **(b)** The strain in the post is **8.00 × 10⁻⁶**.
- **(c)** The change in length of the post is **0.02 mm**.
- **(a)** Stress in the post: 1.60 × 10⁶ Pa (1.60 MPa) - **(b)** Strain in the post: 8.00 × 10⁻⁶ - **(c)** Change in length: 0.02 mm

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Solución

Given:

(a) Calculating the Stress in the Post

(b) Calculating the Strain in the Post

© Calculating the Change in Length of the Post

Final Answers:

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