To solve the problem of measuring the length of a steel rod using a brass scale when both are subjected to a temperature change, we need to consider how the materials expand with temperature. The coefficients of linear expansion for steel and brass will play a crucial role in determining the new measurements at 100 °C.

Understanding Linear Expansion

Linear expansion refers to the change in length of a material when its temperature changes. The formula for linear expansion is given by:

ΔL = L₀ * α * ΔT

• ΔL = change in length
• L₀ = original length
• α = coefficient of linear expansion
• ΔT = change in temperature

Calculating the Change in Length

In this scenario, we have:

• Original length of the steel rod (L₀, steel) = 2 m
• Original length of the brass scale (L₀, brass) = 1 m
• Coefficient of linear expansion for steel (α₁) = 1.2 x 10^-5 / °C
• Coefficient of linear expansion for brass (α₂) = 1.8 x 10^-5 / °C
• Temperature change (ΔT) = 100 °C - 0 °C = 100 °C

Step 1: Calculate the Change in Length for Steel

Using the formula for linear expansion:

ΔL₁ = L₀, steel * α₁ * ΔT

Substituting the values:

ΔL₁ = 2 m * (1.2 x 10^-5) * (100)

ΔL₁ = 2 m * 1.2 x 10^-3 = 0.0024 m

Thus, the new length of the steel rod at 100 °C is:

L₁ = L₀, steel + ΔL₁ = 2 m + 0.0024 m = 2.0024 m

Step 2: Calculate the Change in Length for Brass

Now, we apply the same formula for the brass scale:

ΔL₂ = L₀, brass * α₂ * ΔT

Substituting the values:

ΔL₂ = 1 m * (1.8 x 10^-5) * (100)

ΔL₂ = 1 m * 1.8 x 10^-3 = 0.0018 m

Thus, the new length of the brass scale at 100 °C is:

L₂ = L₀, brass + ΔL₂ = 1 m + 0.0018 m = 1.0018 m

Step 3: Finding the Length of the Steel Rod as Measured by the Brass Scale

Now, we need to find out how long the steel rod appears when measured with the brass scale at 100 °C. The length of the steel rod remains 2.0024 m, but we need to express this length in terms of the expanded brass scale:

Using the ratio of the lengths:

Measured Length = (Length of Steel Rod) / (Length of Brass Scale)

Measured Length = 2.0024 m / 1.0018 m

Calculating this gives:

Measured Length ≈ 1.997 m

Final Result

Therefore, the length of the steel rod as measured by the brass scale when both are at 100 °C is approximately 1.997 m.