Problem Statement
Steel railway tracks each 12.0 m long are laid at 20°C. What gap should be left between consecutive tracks so that there is no compression when temperature reaches 50°C? ($\alpha = 1.2 \times 10^{-5}$ °C$^{-1}$)
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
Steel railway tracks each 12.0 m long are laid at 20°C. What gap should be left between consecutive tracks so that there is no compression when temperature reaches 50°C? ($\alpha = 1.2 \times 10^{-5}$ °C$^{-1}$)
Given Information
- $L_0 = 12.0$ m
- $\Delta T = 50 – 20 = 30°C$
- $\alpha = 1.2 \times 10^{-5}$ °C$^{-1}$
Physical Concepts & Formulas
The expansion must fit within the gap. Required minimum gap = $\Delta L$ for each rail:
$$\Delta L = L_0 \alpha \Delta T$$
Step-by-Step Solution
Step 1: Calculate expansion.
$$\Delta L = 12.0 \times 1.2 \times 10^{-5} \times 30$$
Step 2: Compute.
$$\Delta L = 12.0 \times 3.6 \times 10^{-4} = 4.32 \times 10^{-3} \text{ m}$$
Step 3: Convert.
$$\Delta L = 4.32 \text{ mm}$$
Worked Calculation
$$\Delta L = 12.0 \times 1.2 \times 10^{-5} \times 30 = 4.32 \times 10^{-3} \text{ m} = 4.32 \text{ mm}$$
Answer
Gap required $= 4.32$ mm
Physical Interpretation
Each 12-meter rail expands by about 4.3 mm over a 30°C temperature rise. If no gap is left, adjacent rails push against each other creating compressive thermal stress that can cause tracks to buckle sideways — a dangerous phenomenon called “sun kink” that has derailed trains. Modern welded tracks use prestressing instead of gaps.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\Delta L = 12.0 \times 1.2 \times 10^{-5} \times 30 = 4.32 \times 10^{-3} \text{ m} = 4.32 \text{ mm}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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