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True length of 1650 m line

Short answer

Only temperature and pull corrections apply: temperature +0.0578 m, pull +0.1571 m. True length of the line = 1650.215 m.

A line was measured with a steel tape which was exactly 30 metres at 20°C at a pull of 100 N, the measured length being 1650.00 metres. The temperature during measurement was 30°C and the pull applied was 150 N. Find the length of the line, if the cross-sectional area of the tape was 0.025 sq.cm. The coefficient of expansion of the material of tape per 1°C = 3.5×10−63.5\times10^{-6} and the modulus of elasticity of the material of the tape = 2.1×1052.1\times10^{5} N/mm².

Assumption. No weight or support condition is given, so the tape is taken as fully supported and no sag correction is applied.

Data

  • L=1650.00L = 1650.00 m; T0=20∘T_0 = 20^\circC, T=30∘T = 30^\circC; P0=100P_0 = 100 N, P=150P = 150 N
  • A=0.025A = 0.025 cm2=2.5^2 = 2.5 mm2^2; E=2.1×105E = 2.1\times 10^{5} N/mm2^2, so AE=2.5×2.1×105=525,000AE = 2.5\times 2.1\times 10^{5} = 525{,}000 N
  • α=3.5×10−6\alpha = 3.5\times 10^{-6}/∘^\circC

1. Temperature

Ct=3.5×10−6×(30−20)×1650=+0.0578 mC_t = 3.5\times 10^{-6}\times(30 - 20)\times 1650 = +0.0578\ \text{m}

2. Pull

Cp=(150−100)×1650525,000=+0.1571 mC_p = \frac{(150 - 100)\times 1650}{525{,}000} = +0.1571\ \text{m}

3. True length

Ltrue=1650.00+0.0578+0.1571=1650.2149 mL_{true} = 1650.00 + 0.0578 + 0.1571 = 1650.2149\ \text{m}

Answer: length of the line =1650.215= 1650.215 m.

✓ Answer checked by Claude Opus 5.5 (high effort)

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