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Uncertainty analysis for a simple thermal expansion experiment
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Image of Fig. 1.
Fig. 1.

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. .

Image of Fig. 2.
Fig. 2.

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.

Image of Fig. 3.
Fig. 3.

Effects of the wire's elasticity and mass. When the wire is at room temperature (upper position) the displacement of the load is nonzero due to elastic stretching of the wire. The displacement 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).

Image of Fig. 4.
Fig. 4.

Empirical determination of the massless wire regime. Shown are measurements of the quantity as a function of , where and are the room-temperature displacement and mass of the load, respectively. We use the dependence of on as an indicator of the validity of the massless wire assumption. When is constant with respect to 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.

Image of Fig. 5.
Fig. 5.

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 m/m⋅K.

Image of Fig. 6.
Fig. 6.

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


Generic image for table
Table I.

Partial uncertainties in determination of α.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Uncertainty analysis for a simple thermal expansion experiment