Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# 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, A0, V0, 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, L0, 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.