Correction per tape length (kg data, 58)

Reading. The tape was standardised with the specified pull (fully supported) and is used supported at the two ends only, so the sag correction applies. Corrections are temperature, pull and sag.

Data

• $l = 30$ m; $T_0 = 15^\circ$C, $T = 20^\circ$C; $P_0 = 10$ kg, $P = 16$ kg
• $W = 0.8$ kg; density $= 0.00786$ kg/cm$^3$; $E = 2.1\times 10^{6}$ kg/cm$^2$; $\alpha = 7.1\times 10^{-7}$/$^\circ$C

Area of the tape

$$A = \frac{0.8}{0.00786\times 3000} = 0.03393\ \text{cm}^2,\qquad AE = 0.03393\times 2.1\times 10^{6} = 71{,}247\ \text{kg}$$

1. Temperature

$$C_t = 7.1\times 10^{-7}\times 5\times 30 = +0.000107\ \text{m}$$

2. Pull

$$C_p = \frac{(16 - 10)\times 30}{71{,}247} = +0.002526\ \text{m}$$

3. Sag

$$C_s = -\frac{W^{2}\,l}{24\,P^{2}} = -\frac{0.8^{2}\times 30}{24\times 16^{2}} = -0.003125\ \text{m}$$

4. Correction per tape length

$$C = 0.000107 + 0.002526 - 0.003125 = -0.000492\ \text{m}$$

Answer: correction per tape length $= -0.0005$ m (about $-0.5$ mm), so the true length of one tape length is 29.9995 m.