Question

We would like to prepare a scale whose length does not change with temperature. It is proposed to prepare a unit scale of this type whose length remains, say 10 cm. We can use a bimetallic strip made of brass and iron each of different length whose length (both components) would change in such a way that difference between their lengths remain constant. If α_iron = 1.2 × 10⁻⁵/K and α_brass = 1.8 × 10⁻⁵/K, what should we take as length of each strip?

Answer 1

According to the problem, L₁ – L_b = 10 cm where,

L₁ = length of iron scale

L_b = Length of brass scale

This condition is possible if the change in length of both rods is remain the same at all temperatures.

Change in length of iron rod,

`ΔL = α_IL_IΔT`

Change in length of brass rod,

`ΔL = α_BL_BΔT`

As the change will equal in both the rods, so

`α_IL_IΔT = α_BL_BΔT`

⇒ `α_IL_I = α_BL_B`

⇒ `L_I/L_B = α_B/α_I`

Here, `α_B = 1.8 xx 10^-5 K^-1, α_I = 1.2 xx 10^-5 K^-1`

∴ `L_I/L_B = (1.8 xx 10^-5)/(1.2 xx 10^-5) = 3/2`

`L_I = 3/2 L_B`

As, `L_I - L_B` = 10 cm

∴ `3/2 L_B - L_B` = 10

⇒ `1/2 L_B` = 10

⇒ L_B = 20 cm.