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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 αiron = 1.2 × 10−5/K and αbrass = 1.8 × 10−5/K, what should we take as length of each strip?
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Solution
According to the problem, L1 – Lb = 10 cm where,
L1 = length of iron scale
Lb = 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
⇒ LB = 20 cm.
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