^{1,a)}, Punit R. Gandhi

^{1,b)}and Geoffrey Z. Iwata

^{2,c)}

### Abstract

We describe a simple experiment for measuring the thermal expansion coefficient of a metal wire and discuss how the experiment can be used as a tool for exploring the interplay of measurement uncertainty and scientific models. In particular, we probe the regimes of applicability of three models of the wire: stiff and massless, elastic and massless, and elastic and massive. Using both analytical and empirical techniques, we present the conditions under which the wire's mass and elasticity can be neglected. By accounting for these effects, we measure Nichrome's thermal expansion coefficient to be 17.1(1.3) μm/m⋅K, which is consistent with the accepted value at the 8% level.

The authors acknowledge helpful discussions with Joel Corbo, Brian Estey, Nathan Leefer, Jenna Pinkham, and William Semel. This work was supported by the Berkeley Compass Project and the Associated Students of the University of California through the Educational Enhancement Fund. D.R.D.F. and G.Z.I. were supported by the National Science Foundation under Grant No. PHY-1068875.

I. INTRODUCTION

II. MODELS OF THE WIRE

A. Stiff, massless wire

B. Elastic, massless wire

C. Elastic, massive wire

III. RESULTS AND DISCUSSION

IV. FUTURE DIRECTIONS

### Key Topics

- Elasticity
- 39.0
- Thermal expansion
- 25.0
- Error analysis
- 9.0
- Experiment design
- 8.0
- Thermal models
- 6.0

## Figures

Experimental apparatus. The ends of a wire are attached to blocks of wood via small metal hooks. Both blocks are fastened to a level, rigid surface (not shown) via c-clamps. Insulated copper wires are attached to the hooks to facilitate electrical connections to an ac power supply. Our apparatus is an adaptation of similar setups used in Refs. 10–12 .

Experimental apparatus. The ends of a wire are attached to blocks of wood via small metal hooks. Both blocks are fastened to a level, rigid surface (not shown) via c-clamps. Insulated copper wires are attached to the hooks to facilitate electrical connections to an ac power supply. Our apparatus is an adaptation of similar setups used in Refs. 10–12 .

Schematic of experimental setup. A load (filled circle) is suspended from the midpoint of a taut wire (solid lines). The wire is connected in series with an ac power supply and an ammeter. Long-dashed lines indicate electrical connections and arrows represent the forces experienced by the load.

Schematic of experimental setup. A load (filled circle) is suspended from the midpoint of a taut wire (solid lines). The wire is connected in series with an ac power supply and an ammeter. Long-dashed lines indicate electrical connections and arrows represent the forces experienced by the load.

Effects of the wire's elasticity and mass. When the wire is at room temperature (upper position) the displacement H 0 of the load is nonzero due to elastic stretching of the wire. The displacement H of a hot wire (lower position) is due to a combination of elastic and thermal effects. In the massless wire approximation, the wire hangs in a triangle (dashed lines). A massive wire, on the other hand, hangs in the shape of a loaded catenary (solid lines).

Effects of the wire's elasticity and mass. When the wire is at room temperature (upper position) the displacement H 0 of the load is nonzero due to elastic stretching of the wire. The displacement H of a hot wire (lower position) is due to a combination of elastic and thermal effects. In the massless wire approximation, the wire hangs in a triangle (dashed lines). A massive wire, on the other hand, hangs in the shape of a loaded catenary (solid lines).

Empirical determination of the massless wire regime. Shown are measurements of the quantity as a function of m, where H 0 and m are the room-temperature displacement and mass of the load, respectively. We use the dependence of C on m as an indicator of the validity of the massless wire assumption. When C is constant with respect to m the wire's mass is negligible; otherwise it is not. For our apparatus, the massless wire regime is realized when . The dashed line represents the weighted average of the data collected in this regime (filled circles). Error bars in this and subsequent figures represent statistical uncertainties.

Empirical determination of the massless wire regime. Shown are measurements of the quantity as a function of m, where H 0 and m are the room-temperature displacement and mass of the load, respectively. We use the dependence of C on m as an indicator of the validity of the massless wire assumption. When C is constant with respect to m the wire's mass is negligible; otherwise it is not. For our apparatus, the massless wire regime is realized when . The dashed line represents the weighted average of the data collected in this regime (filled circles). Error bars in this and subsequent figures represent statistical uncertainties.

Experimental determination of Nichrome's thermal expansion coefficient α as a function of the temperature difference of the wire relative to room temperature. The data were analyzed using the massless, elastic wire model of Sec. II B . The solid line represents the weighted mean of the data, and the dashed lines represent the 65% confidence interval due to statistical uncertainties. When taking systematic uncertainties into account (Table I ), we find , which is consistent with the accepted value of Nichrome's thermal expansion coefficient of 17.3 μm/m⋅K. 18

Experimental determination of Nichrome's thermal expansion coefficient α as a function of the temperature difference of the wire relative to room temperature. The data were analyzed using the massless, elastic wire model of Sec. II B . The solid line represents the weighted mean of the data, and the dashed lines represent the 65% confidence interval due to statistical uncertainties. When taking systematic uncertainties into account (Table I ), we find , which is consistent with the accepted value of Nichrome's thermal expansion coefficient of 17.3 μm/m⋅K. 18

Empirical determination of the stiff wire regime. Shown are measurements of the dimensionless quantity as a function of temperature change . When X is negligibly small, the wire's elasticity can be neglected. For comparison, a dashed line corresponding to X = 0 has been included. Because our data yield nonzero measurements of X over a broadrange of temperatures, the stiff wire regime is not realized in our experiment.

Empirical determination of the stiff wire regime. Shown are measurements of the dimensionless quantity as a function of temperature change . When X is negligibly small, the wire's elasticity can be neglected. For comparison, a dashed line corresponding to X = 0 has been included. Because our data yield nonzero measurements of X over a broadrange of temperatures, the stiff wire regime is not realized in our experiment.

## Tables

Partial uncertainties in determination of α.

Partial uncertainties in determination of α.

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