Problem Statement
A line was measured with a steel tape which was exactly 30 metres long at 20°C under a pull of 100 N. The measured length was 1650.00 metres. The temperature during measurement was 30°C and the pull applied was 150 N. Find the true length of the line, given the following tape properties: cross-sectional area (A) = 0.025 cm², coefficient of thermal expansion (α) = 3.5 × 10⁻⁶ per °C, and modulus of elasticity (E) = 2.1 × 10⁵ N/mm².
Step-by-Step Solution
Key Information & Unit Conversion
- Nominal Tape Length (Lnom) = 30 m
- Standard Temperature (To) = 20°C
- Standard Pull (Po) = 100 N
- Measured Line Length (M) = 1650.00 m
- Measurement Temperature (Tm) = 30°C
- Measurement Pull (Pm) = 150 N
- Cross-sectional Area (A) = 0.025 cm² = 0.025 × (10 mm/cm)² = 2.5 mm²
- Coefficient of Thermal Expansion (α) = 3.5 × 10⁻⁶ /°C
- Modulus of Elasticity (E) = 2.1 × 10⁵ N/mm²
- Goal: Find the True Line Length (Ltrue).
Step 1: Calculate Temperature Correction per Tape Length (Ct)
This correction accounts for the expansion or contraction due to the difference between measurement temperature and standard temperature.
Ct = α (Tm – To) Lnom
Ct = (3.5 × 10⁻⁶ /°C) × (30°C – 20°C) × 30 m
Ct = (3.5 × 10⁻⁶) × (10) × 30 m
Ct = +0.00105 m
(Positive sign indicates expansion as Tm > To)
Step 2: Calculate Pull Correction per Tape Length (Cp)
This correction accounts for the stretch or relaxation due to the difference between measurement pull and standard pull. Ensure units are consistent (N, m, mm², N/mm²).
Cp = (Pm – Po) Lnom / (A × E)
Cp = (150 N – 100 N) × 30 m / (2.5 mm² × 2.1 × 10⁵ N/mm²)
Cp = (50 N) × 30 m / (5.25 × 10⁵ N)
Cp = 1500 m / (5.25 × 10⁵)
Cp ≈ +0.00286 m
(Positive sign indicates stretching as Pm > Po)
Step 3: Calculate Combined Correction & Actual Tape Length (Lact)
Sum the individual corrections to find the total change in the tape’s length under measurement conditions.
Combined Correction (Ctotal) = Ct + Cp
Ctotal = 0.00105 m + 0.00286 m = 0.00391 m
Calculate the actual length of the tape during the measurement:
Actual Tape Length (Lact) = Lnom + Ctotal
Lact = 30 m + 0.00391 m
Lact ≈ 30.0039 m
Step 4: Calculate True Line Length (Ltrue)
Use the ratio of the actual tape length to the nominal tape length to correct the total measured distance.
Ltrue = ( Lact / Lnom ) × Measured Length (M)
Ltrue = ( 30.0039 m / 30 m ) × 1650.00 m
Ltrue ≈ ( 1.00013 ) × 1650.00 m
Ltrue ≈ 1650.21 m
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Tape Standardization: Measuring tapes (especially steel) are calibrated to be a specific length (Lnom) only under defined standard conditions of temperature (To) and tension/pull (Po).
- Thermal Expansion (α): Physical materials expand when heated and contract when cooled. The change in length is proportional to the temperature change (Tm – To), the original length (Lnom), and the material’s coefficient of thermal expansion (α).
- Elasticity (E) & Tension Effects: Applying tension stretches elastic materials like steel tapes. The amount of stretch (or relaxation if tension is reduced) depends on the change in pull (Pm – Po), the tape’s original length (Lnom), its cross-sectional area (A), and its modulus of elasticity (E), often expressed by the formula ΔL = ΔP·L / (A·E).
- Combined Corrections: When multiple conditions deviate from standard (e.g., both temperature and pull), the total change in tape length is the algebraic sum of the individual corrections (Ctotal = Ct + Cp + …).
- Error Propagation in Measurement: The calculated difference (Ctotal) between the tape’s actual length during measurement (Lact = Lnom + Ctotal) and its nominal length causes a proportional error in the total measured distance. The true length is found by scaling the measured length: Ltrue = M × (Lact / Lnom).
Real-World Applications:
- Precise Engineering Surveys: Essential for setting out alignments for roads, railways, tunnels, and bridges where accuracy over long distances is critical.
- Geodetic Control Surveys: Establishing high-accuracy reference points requires accounting for all environmental effects on measuring equipment.
- Cadastral (Boundary) Surveys: Ensuring legal boundary measurements are accurate and reproducible often requires applying these corrections, especially when high precision is needed or conditions vary significantly.
- Structural Monitoring: Measuring small changes in distances on structures like dams, buildings, or bridges to detect deformation requires correcting measurements for temperature and potentially tension.
- Manufacturing and Calibration: Used in calibrating measurement instruments and ensuring precision in manufacturing processes requiring accurate length determination.
Why It Works:
The fundamental principle is that the steel tape’s physical length changes based on its environment and the force applied to it. The tape is only exactly 30m long under specific standard conditions (20°C, 100N pull).
First, we calculate the effect of temperature. Since the measurement temperature (30°C) was higher than standard (20°C), the tape expanded. This expansion adds a small amount (+0.00105 m) to its length.
Second, we calculate the effect of tension. The applied pull (150 N) was greater than standard (100 N), causing the tape to stretch elastically. This stretch adds another small amount (+0.00286 m) to its length.
Third, we combine these effects. The total correction per 30m tape length is the sum (+0.00391 m). This means that during the measurement, the tape was actually slightly longer (Lact = 30.00391 m, rounded to 30.0039 m) than its nominal 30m length.
Finally, we apply this understanding to the overall measurement. Since each ‘tape length’ laid down actually covered 30.0039 m instead of 30 m, the total distance measured (1650.00 m) corresponds to a slightly longer true distance. The correction factor is the ratio of the actual tape length to the nominal length (30.0039 / 30 ≈ 1.00013). Multiplying the measured length by this factor gives the true length of the line (1650.00 m × 1.00013 ≈ 1650.21 m).
