Find the Youngs modulus of the rod:
A composite rod is made by joining a copper rod end to end with a second rod of different material but of same cross-section. At 25oC, the composite rod is 1 m in length of which the length of copper rod is 30 cm. At 125oC, the length of the composite rod increases by 1.91 mm. While the composite rod is not permitted to expand through holding it among two rigid walls, it is found that the length of constituents does not change with rise in temperature. Find the Young's modulus and coefficient of linear expansion of the second rod. For copper, α = 1.7 × 10- 5 oC-1, and Y = 1.3 × 1011 Nm- 2.
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 25oC, and α2 and Y2 be coefficient of linear expansion and Young's modulus for the material of this rod. Then, at initial temperature θ1 = 25oC, l1 + l2 = 1 m,
l2 = 1 - l1 = 1 - 0.3 = 0.7 m
Let the final temperature θ2 = 125oC, the increases in the length of cooper rod and that of other material be Δl1 and Δl2, respectively.
Δl1 + Δl2 = 1.91 mm = 1.91 × 10-3 m . . . (A)
Δl1 = l1 α1 (θ2 - θ1) = 0.3 × 1.7 × 10- 5 × (125 - 25) = 0.51 × 10-3 m
Δl2 = l2 α2 (θ2 - θ1) = 0.7 × α2 × (125 - 25) = 70 α2
Substituting for Δl1 and Δl2 in Eq. (A), we get
0.51 × 10- 3 + 70 α2 = 1.91 × 10-3
or, α2 = 2.0 × 10- 5 oC -1
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, thus, tension produced in copper rod F1 = Y1AΔl1 /l1, and tension generates in the rod of other material, F2 = Y2 A Δl2/ l2. The tension developed in the two rods will be same, i.e. F1= F2.
(Y1 A Δl1)/l1 = (Y2 A Δl2)/l2
or Y2 = Y1 l2Δl1/l1Δl2 ⇒ Y2 = 1.3 × 1011 × 0.7×0.51×10-3/0.3 × 1.40 × 10-3
or Y2 = 1.105 × 1011 Nm-2