To solve the problem, we’ll start with part (a) to find the required cross-sectional area of the cable, and then proceed to part (b) to determine how much it stretches under the load.

Part (a): Cross-sectional Area of the Cable

Given:

• Load required to support: $F = 2 \times 1.3 \times 10^6 \, \text{N} = 2.6 \times 10^6 \, \text{N}$
• Tensile strength of steel: $\sigma$ (approximately $4.0 \times 10^8 \, \text{N/m}^2$)

Formula: The tensile strength is defined as: $\sigma = \frac{F}{A}$ Where:

• $\sigma$ = tensile strength
• $F$ = force (load)
• $A$ = cross-sectional area

Rearranging the formula to solve for $A$: $A = \frac{F}{\sigma}$

Calculation: $A = \frac{2.6 \times 10^6 \, \text{N}}{4.0 \times 10^8 \, \text{N/m}^2} = 0.0065 \, \text{m}^2$

Part (b): Stretch of the Cable

Given:

• Length of the cable: $L = 6.0 \, \text{m}$
• Young's modulus for steel: $E$ (approximately $2.0 \times 10^{11} \, \text{N/m}^2$)

Formula: The elongation (stretch) of the cable can be calculated using: $\Delta L = \frac{F L}{A E}$ Where:

• $\Delta L$ = change in length (stretch)
• $F$ = force (load)
• $L$ = original length
• $A$ = cross-sectional area
• $E$ = Young's modulus

Calculation: $\Delta L = \frac{(2.6 \times 10^6 \, \text{N}) (6.0 \, \text{m})}{(0.0065 \, \text{m}^2)(2.0 \times 10^{11} \, \text{N/m}^2)}$ $\Delta L = \frac{15.6 \times 10^6}{1.3 \times 10^6} = 0.012 \, \text{m} = 12 \, \text{mm}$

Summary of Results

(a) The cross-sectional area of the cable should be approximately 0.0065 m².

(b) The stretch of the cable when subject to a 1.3 million N load is approximately 0.012 m or 12 mm.

Would you like any further details on these calculations or have any questions? Here are some related questions to consider:

1. What are the implications of using a cable with insufficient cross-sectional area?
2. How do different materials compare in terms of tensile strength and Young's modulus?
3. What factors should be considered when designing a system that utilizes such cables?
4. How does temperature affect the tensile strength and Young's modulus of steel?
5. What safety factors are typically applied in engineering designs involving cables?

Tip: Always consider including a safety factor in your designs to account for unexpected loads or material defects.