A steel measuring tape which is correct at 20 °C is used to measure a 2000 m cross-country skiing course while the temperature is -0.8 °C. What is the correct length of the course? [table]
Coefficients of thermal expansion Material Coefficient of thermal expansion K-1 Concrete 13 ´ 10-6 Steel 12 ´ 10-6 Brass 19 ´ 10-6 Iron 12 ´ 10-6 Copper 17 ´ 10-6
[/table] Give your answer to the nearest half metre.

Question

BlackTom AI · Accepted Answer

Linear thermal expansion: ΔL = α L0 ΔT, so the tape’s markings change length when temperature changes.

1. Relevant concepts
- Coefficient of linear expansion α for steel = 12 × 10^-6 K^-1 (given).
- If a tape is correct at 20.0 °C but used at a different temperature T, each marked length changes by factor (1 + α ΔT) where ΔT = T_used − T_reference.
- A contracted tape unit means an indicated length is larger than the true length unless corrected.

2. Step-by-step reasoning
- Reference temperature: 20.0 °C. Measurement temperature: −0.8 °C.
- Temperature change: ΔT = T_used − T_ref = −0.8 − 20.0 = −20.8 °C.
- Fractional change in length of the tape: α ΔT = (12 × 10^-6) × (−20.8) = −249.6 × 10^-6 = −2.496 × 10^-4.
- Multiplicative factor for one tape-marked length: 1 + α ΔT = 1 − 2.496 × 10^-4 = 0.9997504.

3. Relating indicated reading to true length
- The tape shows 2000.0 m. Each “1 m” marked on the tape is actually 0.9997504 m long at the colder temperature, so the true length L_true = indicated_length × (1 + α ΔT).
- Compute: L_true = 2000.0 × 0.9997504 = 1999.5008 m.

4. Why this is correct
- Because the steel tape has contracted at −0.8 °C relative to 20 °C, each marked metre is slightly shorter; multiplying the indicated reading by the contraction factor gives the actual (shorter) true length.

5. Common incorrect approaches
- Using ΔT as +20.8 instead of −20.8 would give the wrong sign (predicting expansion rather than contraction).
- Forgetting to multiply the indicated reading by (1 + α ΔT) and instead subtracting an absolute amount would be incorrect.

6. Conclusion
- Rounding to the nearest half metre gives 1999.5 m, which matches the provided answer.

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