THE PRACTICE STUDIO
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PRACTICE / Thermal expansion
The ruler changes along with the specimen
A freely expanding specimen and its ruler are initially at the calibration temperature. Their linear expansion coefficients are 30 × 10⁻⁶ K⁻¹ and 10 × 10⁻⁶ K⁻¹ respectively. Both are warmed uniformly by 100 K.
What is the fractional increase in the numerical length reading to first order?
Organize the givens & assumptions
- Specimen α = 30 × 10⁻⁶ K⁻¹
- Ruler α = 10 × 10⁻⁶ K⁻¹
- Common rise = 100 K
Uniform temperatures and the stated expansion coefficients; small thermal strains and first-order expansion unless a retained ratio is explicitly requested. Free expansion except for the named axial constraints. Constrained rods remain straight and linearly elastic; no buckling, yielding or moving supports.
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Problem: The ruler changes along with the specimen A freely expanding specimen and its ruler are initially at the calibration temperature. Their linear expansion coefficients are 30 × 10⁻⁶ K⁻¹ and 10 × 10⁻⁶ K⁻¹ respectively. Both are warmed uniformly by 100 K. Question: What is the fractional increase in the numerical length reading to first order? Givens: - Specimen α = 30 × 10⁻⁶ K⁻¹ - Ruler α = 10 × 10⁻⁶ K⁻¹ - Common rise = 100 K Assumptions: Uniform temperatures and the stated expansion coefficients; small thermal strains and first-order expansion unless a retained ratio is explicitly requested. Free expansion except for the named axial constraints. Constrained rods remain straight and linearly elastic; no buckling, yielding or moving supports. Choices (LaTeX): A. 0.002 B. 0.004 C. 0.003 D. -0.002 Ask for my approach first. Help me distinguish a physical misconception from an algebra error without revealing which choice is correct. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Start with the smallest hint.
Work through it with me.
One decision at a time. Pause, predict, then unpack the algebra.
A measurement compares two physical lengths, and both can change.
Compare scale factors
Both bodies expand, so count their changes in a ratio.
Why is this step valid?
Longer ruler intervals partly cancel the specimen extension.
Keep first-order terms
Expand the reciprocal denominator to the same accuracy.
Why is this step valid?
Products of two small fractional changes are beyond the requested order.
State the signed reading change
The positive difference means a larger numerical reading.
Why is this step valid?
The actual length increase and reading increase are different quantities.
The retained ratio gives 0.001998…, consistent to first order.
TRY THE IDEA AGAIN
If the ruler instead expanded faster than the specimen, the reading would…
YOUR LEARNING NOTE
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