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Difficulty: MediumFormulating and Modifying Hypotheses

To study thermal expansion in solids, a researcher proposed that the coefficient of linear expansion (α\alpha) of a metal rod depends on its starting length. Specifically, the researcher hypothesized that longer rods of the same metal would exhibit a larger value of α\alpha when subjected to the same temperature change.

To test this hypothesis, the researcher measured the physical expansion of three copper rods under identical heating conditions. The results of the measurements and the calculated values of α\alpha are recorded in the table below.

Copper RodInitial Length (L0L_0, m\text{m})Temperature Increase (ΔT\Delta T, ∘C^\circ\text{C})Expansion (ΔL\Delta L, mm\text{mm})Coefficient of Linear Expansion (α\alpha, 10−6 ∘C−110^{-6}\text{ }^\circ\text{C}^{-1})
Rod A1.0500.8517
Rod B2.0501.7017
Rod C3.0502.5517

Based on the results of the experiment, does the data support the researcher's hypothesis, and how should the hypothesis be modified?

  1. A
    Yes; the data show that the total expansion of the rod increases from 0.85 mm0.85\text{ mm} to 2.55 mm2.55\text{ mm} as the initial length increases.
  2. B
    Yes; the data show that the coefficient of linear expansion increases proportionally with the initial length of the copper rod.
  3. C
    No; the data show that the coefficient of linear expansion decreases as the initial length of the rod increases.
  4. No; the data show that the coefficient of linear expansion remains constant at 17×10−6 ∘C−117 \times 10^{-6}\text{ }^\circ\text{C}^{-1} regardless of the rod's initial length.Answer

Answer

The experimental data does not support the hypothesis because the coefficient of linear expansion remains constant at 17×10−6 ∘C−117 \times 10^{-6}\text{ }^\circ\text{C}^{-1} for all initial lengths, meaning the hypothesis should be modified to state that the coefficient of linear expansion is independent of the initial length of the rod.
The correct option accurately states that the hypothesis is not supported because the coefficient of linear expansion (α\alpha) remains constant at 17×10−6 ∘C−117 \times 10^{-6}\text{ }^\circ\text{C}^{-1} for all three rods, regardless of their initial length. A scientific hypothesis must be rejected or modified when the measured property does not change in the predicted direction.

Step-by-Step Solution

1
Identify the core assertion of the researcher's hypothesis.
The hypothesis predicts that as the initial length (L0L_0) of a metal rod increases, the coefficient of linear expansion (α\alpha) of that metal will also increase.
Understanding the hypothesis defines the independent variable (initial length) and the dependent variable (coefficient of linear expansion) that must be compared using the data.
2
Analyze the data table to observe how the coefficient of linear expansion (α\alpha) behaves as the initial length increases.
As the initial length increases from 1.0 m1.0\text{ m} to 2.0 m2.0\text{ m} and then 3.0 m3.0\text{ m}, the calculated coefficient of linear expansion (α\alpha) remains constant at 17×10−6 ∘C−117 \times 10^{-6}\text{ }^\circ\text{C}^{-1}.
Evaluating the trend of the target variable (α\alpha) against the independent variable (L0L_0) determines whether the hypothesis is supported or refuted.
3
Determine the proper modification of the hypothesis based on the constant values of α\alpha.
Since α\alpha did not increase, the hypothesis is refuted. The modified hypothesis must reflect that the coefficient of linear expansion remains constant and is independent of the initial length of the rod.
Comparing the empirical data directly to the proposed hypothesis allows the researcher to correctly modify it to align with the experimental results.

Key Concept

Formulating and Modifying Hypotheses
Estimated Time:1m 15s
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Topics

Formulating and Modifying Hypotheses

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