A unit scale is to be prepared whose length does not change with temperature and remains 20 cm, using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their lengths remains constant. If length of brass is 40 cm and length of iron will be __________ cm. (αᵢᵣₒₙ=1.2 × 10⁻⁵ K⁻¹ and α_(brass)=1.8 × 10⁻⁵ K⁻¹).

Solution

Physics sir · 4 steps

The gap stays fixed only if both strips grow by the same amount, so set Lα equal for the two.

1. Equal expansions. For a temperature rise Δ T, each strip grows by Δ L = Lα Δ T. The difference in lengths stays the same only if brass and iron grow by exactly the same amount.

   L_(brass) α_(brass) Δ T = Lᵢᵣₒₙ αᵢᵣₒₙ Δ T

2. Put in numbers. Δ T cancels, which leaves 40 × 1.8 × 10⁻⁵ = Lᵢᵣₒₙ × 1.2 × 10⁻⁵. This gives Lᵢᵣₒₙ = (40 × 1.8)/1.2 = 60 cm.
3. Check. Iron has the smaller α, so it must be the longer strip. The difference is 60 - 40 = 20 cm, which is exactly the scale length the question asks for.
4. So. The length of the iron strip is 60 cm.

The common slip: 26.67. Swapping the two coefficients, as in 40 × 1.2 = L × 1.8, gives about 26.7 cm. The difference is then not 20 cm, so the answer fails the check.

The idea to keep. For a length difference that does not change with temperature, the two rods need equal Lα. The rod with the smaller α must be the longer one.

Answer from the NTA final answer key. Solution by PrepWiser.