In
an experiment 400kg mass is Suspended at one end of the  wire of length 3m
and area
of
cross-section 20.2 cm2
. If the increase in length is 0.4cm, find stress, strain and young’smodulus.

To solve this problem, we'll calculate the **stress**, **strain**, and **Young's modulus** using the given dataTo solve this problem, we'll calculate the **stress**, **strain**, and **Young's modulus** using the given 

### 1. **Stress** (\(\sigma\)):
Stress is defined as the force per unit area. The force in this context is the weight of the mass, which is given by \( F = mg \), where:
- \( m = 400 \, \text{kg} \) (mass)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

The cross-sectional area \( A \) of the wire is given as \( 20.2 \, \text{cm}^2 \). We need to convert this to \( \text{m}^2 \).

#### Steps:
1. **Calculate the force**:
   \[
   F = 400 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 3920 \, \text{N}
   \]

2. **Convert the area to \( \text{m}^2 \)**:
   \[
   A = 20.2 \, \text{cm}^2 = 20.2 \times 10^{-4} \, \text{m}^2 = 0.00202 \, \text{m}^2
   \]

3. **Calculate the stress**:
   \[
   \sigma = \frac{F}{A} = \frac{3920 \, \text{N}}{0.00202 \, \text{m}^2} \approx 1.94 \times 10^6 \, \text{N/m}^2
   \]

### 2. **Strain** (\(\epsilon\)):
Strain is defined as the ratio of the change in length to the original length. 

Given:
- Original length \( L = 3 \, \text{m} \)
- Increase in length \( \Delta L = 0.4 \, \text{cm} = 0.004 \, \text{m} \)

#### Calculate the strain:
\[
\epsilon = \frac{\Delta L}{L} = \frac{0.004 \, \text{m}}{3 \, \text{m}} \approx 1.33 \times 10^{-3}
\]

### 3. **Young's Modulus** (\( E \)):
Young's modulus is defined as the ratio of stress to strain.

#### Calculate Young's modulus:
\[
E = \frac{\sigma}{\epsilon} = \frac{1.94 \times 10^6 \, \text{N/m}^2}{1.33 \times 10^{-3}} \approx 1.46 \times 10^9 \, \text{N/m}^2
\]

### Final Answers:
- **Stress**: \( \sigma \approx 1.94 \times 10^6 \, \text{N/m}^2 \)
- **Strain**: \( \epsilon \approx 1.33 \times 10^{-3} \)
- **Young's Modulus**: \( E \approx 1.46 \times 10^9 \, \text{N/m}^2 \)

Would you like further details on any of these calculations, or do you have any questions?

### Relative Questions:
1. How would the stress change if the mass were doubled?
2. What would be the effect on Young's modulus if the wire material were different?
3. How does the length of the wire affect the strain?
4. Can stress and strain be directly measured in this experiment?
5. What assumptions are made in this calculation of Young's modulus?

### Tip:
Young's modulus is a material property, so it will remain constant for a given material regardless of the dimensions of the wire.

Learn how to calculate stress, strain, and Young's modulus using mechanics principles. This problem involves a 400kg mass suspended on a wire with specific dimensions and cross-sectional area. Explore the formulas and detailed solutions for stress, strain, and Young's modulus calculation.

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