Question

A wire of length L and area of cross-section A is held straight between two rigid supports.

It has zero stress at 0o

C. Show that the stress in the wire at 100o

C is 100 αY, where α is

coefficient of linear expansion and Y is young's modulus of elasticity of the wire.

Answer 1

we know, Young's modulus = stress/strain
strain = original length/change in length
Let original length of wire = L at 0°C

after 100°C, due to thermal expansion length of wire increases .
so, length or wire at 100°C , L' =L(1 + α∆T)
here, α is linear coefficient of expansion and ∆T is the change in temperature.
so, L' =L + Lα(100- 0) = L + 100αL
L' - L = ∆L = 100αL
hence, change in length , ∆L = 100αL

now, Young's modulus = stress/strain
Y = stress/{∆L/L}
stress = {∆L/L} × Y = (100αL/L) × Y
stress = 100αY

hence, stress = 100αY $\textbf{\underline{hence proved}}$

Answer 2

Thermal expansion or extension or strain due to the increase in temperature

= Delta L = L × a × (100° - 0°) = strain

= 100 a L

Thus strain and the stress produced in the material due to its ends being tightly held are related by Young's modulus Y.

Stress = Y × strain = 100 a L Y

That's the answer..