Determine the Youngs modulus:

A composite rod is build by joining a copper rod end to end with a second rod of different material but of similar cross-section. At 25^oC, the composite rod is 1 m in length of that the length of copper rod is 30 cm. At 125^oC, the length of the composite rod enhance by 1.91 mm. While the composite rod is not permitted to expand by holding it among two rigid walls, this is found that the length of constituents does not change with rise in temperature. Determine the Young's modulus & coefficient of linear expansion of the second rod. For copper, α = 1.7 × 10- 5 ^oC^{- 1}, & Y = 1.3 × 10¹¹ Nm^{- 2}.

Solution

For copper rod α₁ = 1.7 × 10- 5 ^oC- 1, and Y₁ = 1.3 × 1011 Nm^{- 2}, l1 = 30 cm = 0.3 m. Let l₂ be initial length of the rod of other material at 25^oC, and α₂ & Y₂ be coefficient of linear expansion and Young's modulus for the material of this rod. After that, at initial temperature θ₁ = 25^oC, l₁ + l₂ = 1 m, l₂  = 1 - l₁  = 1 - 0.3 = 0.7 m

Assume the final temperature θ₂ = 125^oC, enhance in the length of cooper rod and that of other material be Δl₁ & Δl₂, respectively.

Then,

Δl₁ + Δl₂ = 1.91 mm = 1.91 × 10^{- 3}  m

Now,

Δl₁ = l₁ α₁  (θ₂  - θ₁ ) = 0.3 × 1.7 × 10^{- 5} × (125 - 25) = 0.51 × 10^{- 3}  m

Also,

Δl₂  = l₂ α₂  (θ₂  - θ₁ ) = 0.7 × α₂  × (125 - 25) = 70 α₂

By substituting for Δl₁ and Δl₂ in Eq. (A), we obtain

0.51 × 10^{- 3} + 70 α₂  = 1.91 × 10^{- 3}

or,

α₂  = 2.0 × 10- 5 ^oC ^-1

Now,

Y =   (F/A)/ (Δl/C) =  FL/ A Δl  ⇒ F = (Y A Δl)/l

Let A be the cross-section of each of the two rods, therefore, tension produced in copper rod F1 = (Y ₁A Δl)/ l₁, and tension produced in the rod of other material, F = Y₂ A Δl₂/ l₂. The tension developed in the two rods shall be same that means F₁ = F₂.

 Therefore,

Y₁ A Δl₁ / l1= Y₂ A Δl₂/ l₂

or         Y₂  = Y₁  (l₂ Δl₁ )/( l₁ Δl₂) ⇒ Y₂  = 1.3 × 10¹¹ ×(0.7 × 0.51 × 10^{- 3})/ (0.3 × 1.40 × 10^{- 3})

or

Y₂ = 1.105 × 10¹¹ Nm^{- 2}