First slide

Thermal expansion

Question

A wire of length L₀ is supplied heat to raise its temperature by T. if g is the coefficient of volume expansion of the wire and Y is Young’s modulus of the wire then the energy density stored in the wire is

Difficult

A

$\frac{1}{2} γ^{2} T^{2} Y$
B

$\frac{1}{3} γ^{2} T^{2} Y^{3}$
C

$\frac{1}{18} \frac{γ^{2} T^{2}}{Y}$
D

$\frac{1}{18} γ^{2} T^{2} Y$

Solution

Elastic potential energy per unit volume
$\begin{array}{l} E = \frac{1}{2} \times Stress \times Strain = \frac{1}{2} \times Y \times (Strain)^{2} \\ \Rightarrow E = \frac{1}{2} \times Y \times {(\frac{ΔL}{L})}^{2} = \frac{1}{2} \times Y \times a^{2} \times {ΔT}^{2} \end{array}$

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