Question

Two rods of equal cross-sections, one of copper
and the other of steel are joined to form a
composite rod of length 2.0 m at 20°C , the
length of the copper rod is 0.5m. When the
temperature is raised to 120°C , the length of
composite rod increases to 2.002m. If the
composite rod is fixed between two rigid walls
and thus not allowed to expand, it is foundthat
the length of the component rod also do not
change with increase in temperature.
Calculate the Young's modulus of steel. Given
Young's modulus of copper
=1.3x10'N / m² ,the coefficient of linear
expansion of copper a, = 1.6x10-5 I°c.​

Answer 1

Answer:

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Answer 2

Young's modulus of steel is 26 N/m²

Given,

length of the composite rod at 20°C  (L₁) = 2.0 m

length of the copper rod at 20°C (l₁) = 0.5 m

length of the steel rod at 20°C (l₂) = (2.0 - 0.5) = 1.5 m

length of the composite rod at 120°C  (L₂) = 2.002 m

change in length of the composite rod (ΔL) = (L₂ - L₁) = (2.002 - 2.0) = 0.002 = 2 ×10⁻³ m

change in temperature (Δt) = 120 - 20 = 100°C

coefficient of linear expansion of copper (α₁) = 1.6 ×10⁻⁵/°C

ΔL = l₁α₁Δt + l₂α₂Δt

ΔL = (l₁α₁ + l₂α₂)Δt

2 ×10⁻³ = {(0.5) (1.6×10⁻⁵) + (1.5) (α₂)} (100)

$\frac{2 \times 10^{-3}}{100}$ = 0.8×10⁻⁵ + (1.5)(α₂)

2×10⁻⁵ - 0.8×10⁻⁵ = 1.5 × α₂

1.2×10⁻⁵ = 1.5×α₂

α₂ =  $\frac{1.2 \times 10^ {-5}}{1.5}$

α₂ = 0.8 ×10⁻⁵/°C

coefficient of linear expansion (α₂) = 0.8 ×10⁻⁵/°C

Young's modulus of copper rod (Y₁) = 1.3×10 N/m² = 13 N/m²

Young's modulus of steel (Y₂) =

Formula: $\frac{Y_2}{Y_1}$ =  $\frac{\alpha_1 }{\alpha_2}$

$\frac{Y_2}{13}$ =  $\frac{1.6 \times 10^{-5}}{0.8 \times 10^{-5}}$

$\frac{Y_2}{13}$ = 2

Y₂ = 2 × 13

Y₂ = 26 N/m²

young's modulus of steel is 26 N/m²