Answer:
a) N = 145,833,674.52174 = 1.458 × 10⁸ strands
b) diameter of single rope with the same effect = 2.415 cm
Explanation:
Hooke's law explains that stress is directly proportional to strain.
Stress ∝ Strain.
Stress = E × Strain
E = constant of proportionality = Young's Modulus = 4.0 ✕ 10⁹ N/m².
Stress = (Load/Total Cross sectional Area)
Load = a pair of 85 kg mountain climbers = 85 × 2 × 9.8 = 1666 N
Total Cross sectional Area = (Number of strands) × (Area of one strand) = A
Strain = (ΔL/L)
ΔL = 1.00 cm = 0.01 m
L = 11 m
Strain = (0.01/11) = 0.0009091
Stress = (Young's Modulus) × (Strain) = (4.0 ✕ 10⁹) × (0.0009091) = 3,636,363.64 N/m²
(Load/ total Area) = 3,636,363.64
Total area = (Load/3,636,363.64) = (1666/3,636,363.64) = 0.00045815 m²
Recall,
Total Cross sectional Area = (Number of strands) × (Area of one strand)
Area of one strand = (πd²/4)
diameter of one strand = 2 μm = (2×10⁻⁶) m
Area of one strand = (πd²/4)
= π × (2×10⁻⁶)² ÷ 4 = (3.142 × 10⁻¹²) m²
Total Cross sectional Area = (Number of strands) × (Area of one strand)
0.00045815 = N × (3.142 × 10⁻¹²)
N = 145,833,674.52174 = 1.458 × 10⁸ strands
b) If it was a single rope, the cross sectional Area would just be equal to the total cross sectional Area obtained in (a)
A = 0.00045815 m²
A = (πD²/4)
where D = diameter of the single rope
0.00045815 = (πD²/4)
D² = (4×0.00045815) ÷ π = 0.0005833347
D = 0.02415 m = 2.415 cm
Hope this Helps!!!
