A tape 100 m long, 6.35 mm wide, and 0.5 mm thick was used to measure a line, with the apparent measured length being 1986.96 m. The tape was standardised under a pull of 67.5 N. However, it was found that the pull actually used during the measurement was 77.5 N. What was the true length of the line, assuming the tape was standardized and used on the flat?

Key Information

• Nominal Tape Length (L_nom) = 100 m
• Tape Width (w) = 6.35 mm
• Tape Thickness (t) = 0.5 mm
• Standard Pull (P_o) = 67.5 N
• Young’s Modulus (E) = 200,000 N/mm²
• Measured Line Length (M) = 1986.96 m
• Measurement Pull (P_m) = 77.5 N
• Condition: Used ‘on the flat’ (no sag correction needed). Temperature assumed standard or constant.
• Goal: Find the True Line Length (L_true).

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

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

Step 3: Calculate Actual Tape Length (L_act)

Step 4: Calculate True Line Length (L_true)

Final Result

Core Concepts:

• Tape Standardization (Pull): Tapes are manufactured and calibrated to provide their nominal length (L_nom) under a specific standard tension or pull (P_o).
• Elasticity (E) & Tension Correction: Applying a pull different from the standard (P_m ≠ P_o) causes the tape to stretch or contract elastically. The change in length (C_p) is calculated using the formula C_p = (P_m – P_o)L_nom / (AE), where A is the cross-sectional area and E is Young’s Modulus.
• Cross-Sectional Area (A): For a tape with a rectangular cross-section, the area is simply the product of its width and thickness (A = w × t). This area resists the stretching force.
• Measurement “On the Flat”: This condition implies the tape is fully supported along its length, eliminating the need for a sag correction which would be necessary if the tape were suspended between points.
• Error Propagation: The correction per tape length (C_p) determines the actual length of the tape during measurement (L_act = L_nom + C_p). The true length of the measured line is found by applying the ratio (L_act / L_nom) to the total measured length (M).

Real-World Applications:

• Precise Distance Measurement: Critical in land surveying, civil engineering layout (roads, bridges, buildings), and construction where specified tolerances must be met.
• Baseline Measurement: Establishing high-accuracy baselines for triangulation or trilateration networks.
• Industrial Metrology: Aligning large machinery, verifying dimensions of large manufactured components (e.g., in aerospace or shipbuilding).
• Calibration Services: Understanding and applying pull corrections is essential when calibrating tapes or other length-measuring devices.
• Sports Measurements: While less critical, understanding tension can matter in precise measurements for athletic track events or field markings.

Why It Works:
The steel tape behaves as an elastic material. It has a defined length (100 m) only when subjected to its standard pull (67.5 N). During the measurement, a higher pull (77.5 N) was applied. This excess tension (77.5 N – 67.5 N = 10 N) caused the tape to stretch slightly. First, the tape’s cross-sectional area (A = 3.175 mm²) is calculated from its width and thickness, as this area resists the stretching force. Second, the amount of stretch per 100m tape length (C_p) is calculated using the formula incorporating the change in pull, the nominal length, the area, and the material’s stiffness (Young’s Modulus E). This results in a stretch C_p = +0.001575 m. Third, the actual length of the tape during measurement (L_act) is found by adding this stretch to the nominal length: L_act = 100 m + 0.001575 m = 100.001575 m. Finally, since each ‘100m’ segment measured actually covered 100.001575 m on the ground, the total recorded length (M = 1986.96 m) needs to be scaled up proportionally. The true length is calculated as L_true = M × (L_act / L_nom) = 1986.96 m × (100.001575 / 100) ≈ 1986.991 m. The true length is slightly longer than the measured length because the tape was stretched during measurement.