A wire of length L0 is supplied heat to raise its temperature by T. If γ 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

Question

A wire of length L0 is supplied heat to raise its temperature by T. If &gamma; 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 (A) (1/2) &gamma;2 T2 Y (B) (1/3) &gamma;2 T2 Y3 (C) (1/18) (&gamma;2 T2/Y) (D) (1/18) &gamma;2 T2 Y

Tardigrade Admin · Accepted Answer

Due to heating the length of the wire increases.

∴

Longitudinal strain is produced

\Rightarrow \frac{Δ L}{L} = α \times Δ T

Elastic potential energy per unit volume

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{Δ L}{L})^{2} = \frac{1}{2} \times Y \times α^{2} \times Δ T^{2}

or

E = \frac{1}{2} \times Y \times (\frac{γ}{3})^{2} \times T^{2} = \frac{1}{18} γ^{2} Y T^{2}

[As

γ = 3 α

and

Δ T = T

(given)]

Q. A wire of length $L_{0}$ is supplied heat to raise its temperature by $T$ . If $γ$ 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

$\frac{1}{2} γ^{2} T^{2} Y$

$\frac{1}{3} γ^{2} T^{2} Y^{3}$

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

$\frac{1}{18} γ^{2} T^{2} Y$

Q. A wire of length L0​ is supplied heat to raise its temperature by T. If γ 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

21​γ2T2Y

31​γ2T2Y3

181​Yγ2T2​

181​γ2T2Y

Q. A wire of length $L_{0}$ is supplied heat to raise its temperature by $T$ . If $γ$ 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

$\frac{1}{2} γ^{2} T^{2} Y$

$\frac{1}{3} γ^{2} T^{2} Y^{3}$

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

$\frac{1}{18} γ^{2} T^{2} Y$