Question

Calculate the length of a brass rod which shows the same expansion as an iron rod of length 4m, when both rods are heated through the same range of temperature.
Coefficient of linear expansion of brass is 0.000018 per K and iron is 0.000012 per K.

Answer 1

$a_b$ - coefficient of linear expansion of brass

$a_i$ - coefficient of linear expansion of iron

$l_b$ - initial length of brass rod

$l_i$ - initial lenghth of iron rod

$t$ - growth of temperature

After expansion, length of brass rod is equal to $l_b(1+a_b*t)$. Thus, its expansion is $l_b(1+a_b*t)-l_b=l_b(1+a_b*t-1)=l_b*a_b*t$

In analogy, expansion of iron rod is $l_i*a_i*t$

$l_b*a_b*t=l_i*a_i*t$

$l_b*a_b=l_i*a_i$

$l_b=\frac{a_i}{a_b}*l_i=\frac{0.000012}{0.000018}*4=\frac{8}{3} m$

Answer: $\frac{8}{3} m$