To measure a baseline, a steel tape 30m long, standardized at 15°C with a pull of 100 N, was used. Find the correction per tape length if the temperature at the time of measurement was 20°C and the pull exerted was 160 N. The weight of 1 cubic cm of steel is 0.0786 N, the total weight of the tape is 8 N, the modulus of elasticity (E) is 2.1 × 10⁵ N/mm² and the coefficient of expansion (α) is 7.1 × 10⁻⁷ per °C.

Key Information

• Nominal Tape Length (L_nom) = 30 m = 3000 cm
• Standard Temperature (T_o) = 15°C
• Standard Pull (P_o) = 100 N
• Measurement Temperature (T_m) = 20°C
• Measurement Pull (P_m) = 160 N
• Density of Steel (ρ) = 0.0786 N/cm³
• Total Weight of Tape (W) = 8 N
• Modulus of Elasticity (E) = 2.1 × 10⁵ N/mm² (Assumed unit for consistency)
• Coefficient of Thermal Expansion (α) = 7.1 × 10⁻⁷ /°C
• Goal: Find the Total Correction per Tape Length (C_total).

Step 1: Calculate Tape Cross-Sectional Area (A)

Step 2: Calculate Pull Correction per Tape Length (C_p)

Step 3: Calculate Temperature Correction per Tape Length (C_t)

Step 4: Calculate Total Correction per Tape Length (C_total)

Final Result

Core Concepts:

• Deriving Physical Properties: When a direct property like cross-sectional area (A) isn’t given, it can sometimes be calculated from other knowns like total weight (W), length (L), and material density (ρ) using the relationship W = A × L × ρ.
• Tape Standardization: Tapes are calibrated for a nominal length (L_nom) under standard temperature (T_o) and pull (P_o).
• Thermal Expansion (α): Temperature differences (T_m – T_o) cause length changes proportional to α and L_nom.
• Elasticity (E) & Tension Effects: Pull differences (P_m – P_o) cause stretching/relaxation proportional to L_nom and inversely proportional to Area (A) and Modulus of Elasticity (E).
• Combined Corrections: The net effect on tape length is the algebraic sum of individual corrections due to different physical factors (temperature, pull, sag, etc.).

Real-World Applications:

• Precise Engineering Surveys: Calculating corrections is standard practice when high accuracy is needed for setting out structures, alignments, etc.
• Geodetic Control & Monitoring: Essential for high-precision measurements where environmental factors significantly impact results.
• Calibration & Metrology: Understanding how to derive and apply corrections is fundamental in calibrating measuring instruments.
• Forensic Engineering: Analyzing measurement data where original instrument specifications might be incomplete requires deriving properties if possible.
• Quality Control: Assessing the impact of environmental variations on measurements in various industrial and scientific settings.

Why It Works:
The goal is to find the total change in the tape’s length due to differences between field conditions (20°C, 160 N) and standard conditions (15°C, 100 N). First, the cross-sectional area (A) of the tape is calculated because it’s essential for the pull correction formula. Using the tape’s total weight, length, and the density of steel, A is found to be approximately 3.39 mm². Second, the temperature correction (C_t) is calculated. The field temperature is higher than standard (20°C > 15°C), causing the tape to expand by +0.0001065 m per 30m length. Third, the pull correction (C_p) is calculated. The pull applied in the field is greater than standard (160 N > 100 N), causing the tape to stretch elastically. Using the calculated area (A) and the modulus of elasticity (E), this stretch is found to be approximately +0.0025 m per 30m length. Finally, the total correction per tape length (C_total) is the sum of the individual corrections. Since both corrections are positive (lengthening), C_total = C_t + C_p ≈ +0.0001 m + +0.0025 m = +0.0026 m. This means that under the measurement conditions, the 30m tape was effectively 0.0026 m longer than its standard length.