Question

A steel cable with a radius of 1.5 cm supports a chairlift at a ski area. If the maximum stress is not to exceed 108 N m^–2, what is the maximum load the cable can support ?

Answer 1

Given

• Radius of cable: \( r = 1.5 \,\text{cm} = 0.015 \,\text{m} \)
• Maximum allowable stress: \( \sigma_{\text{max}} = 10^{8} \,\text{N m}^{-2} \)

1. Area of cross-section of the cable

For a circular cross-section:

\( A = \pi r^{2} = \pi (0.015)^{2} = \pi \times 2.25 \times 10^{-4} \,\text{m}^{2} \approx 7.07 \times 10^{-4} \,\text{m}^{2} \)

2. Relation between stress and load

Stress is force per unit area:

\( \sigma_{\text{max}} = \dfrac{F_{\text{max}}}{A} \Rightarrow F_{\text{max}} = \sigma_{\text{max}} \, A \)

3. Maximum load

\( F_{\text{max}} = 10^{8} \times 7.07 \times 10^{-4} \,\text{N} = 7.07 \times 10^{4} \,\text{N} \)

Maximum load the cable can support: \( F_{\text{max}} \approx 7.1 \times 10^{4} \,\text{N} \) (about \( 71 \,\text{kN} \)).