# Verification

The theoretical equation derived earlier relating extension and temperature was

\[\sigma = \frac{F}{{{A_0}}} = \frac{{kTN}}{{{V_0}}}\left( {\lambda - \frac{1}{{{\lambda ^2}}}} \right)\]

(The subscript has been dropped from λ since we are only considering one direction.)
In the demonstration, F, A_{0}, V_{0}, N and k are all constant
(once the load has been attached). Therefore, in order to verify this equation we need to show that

\[\frac{{{T_1}}}{{{T_2}}} = \left[ {{\lambda _2} - \frac{1}{{\lambda _2^2}}/{\lambda _1} - \frac{1}{{\lambda _1^2}}} \right]\]

where the subscripts 1 and 2 refer to before and after the contraction respectively, not different directions as in the earlier derivation. The following observations were made:

- Before the experiment began it was noted that the top of the rubber band was 0.96 m from the base of the metre rule.
- It was also noted that the length of the weights was 0.13 m.
- The length of the rubber strip when unloaded, L
_{0}, was 0.205 m.

These values will be used in our calculation to calculate the final length of the rubber strip. Using the recorded observations we have

\[\frac{{{T_1}}}{{{T_2}}} = \frac{{296}}{{338}} = 0.876\]

From the video of the readings before and after heating

\({\lambda _1}\) = \(\frac{{{L_1}}}{{{L_0}}}\) = \(\frac{{\left( {0.96 - 0.077 - 0.13} \right)}}{{\left( {0.205} \right)}}\) = 3.673, so \({\lambda _1} - \frac{1}{{\lambda _1^2}} = 3.599\)

\({\lambda _1}\) = \(\frac{{{L_1}}}{{{L_0}}}\) = \(\frac{{\left( {0.96 - 0.154 - 0.13} \right)}}{{\left( {0.205} \right)}}\) = 3.298, so \({\lambda _2} - \frac{1}{{\lambda _2^2}} = 3.206\)

Therefore

\[\left[ {{\lambda _2} - \frac{1}{{\lambda _2^2}}/{\lambda _1} - \frac{1}{{\lambda _1^2}}} \right] = \frac{{3.206}}{{3.599}} = 0.891\]

So, to a close approximation,

\[\left[ {{\lambda _2} - \frac{1}{{\lambda _2^2}}/{\lambda _1} - \frac{1}{{\lambda _1^2}}} \right] = \frac {{T_1}}{{T_2}} \]

and the theoretical explanation is verified. The small discrepancy is attributed to conventional thermal expansion; rubbers
have relatively high expansivities (~ 50 x 10^{-6} K^{-1}), so a rise in *T* of
about 50 K will increase the length of a strip which is initially 0.75 m long by 2 mm.