# Long answer type question. Derive an expression for strain energy per unit volume of the material of a wire. - Physics

Sum

Long answer type question.

Derive an expression for strain energy per unit volume of the material of a wire.

#### Solution

The expression for strain energy per unit volume:

1. Consider a wire of original length L and cross-sectional area A stretched by a force F acting along its length. The wire gets stretched and elongation l is produced in it.
2. If the wire is perfectly elastic then,
Longitudinal stress ="F"/"A"
Longitudinal strain = l/"L"
"Young’s modulus (Y)" = "longitudinal stress"/"longitudinal strain"
Y = ("F"//"A")/(l//"L") = "F"/"A" xx "L"/l
∴ F = "YAl"/"L"      ....(1)
3. The magnitude of stretching force increases from zero to F during the elongation of wire. Let ‘f’ be the restoring force and ‘x’ be its corresponding extension at certain instant during the process of extension.
∴ f = "YAx"/"L"     .....(2)
4. Let 'dW’ be the work done for the further small extension ‘dx’.
Work = force × displacement
∴ dW = fdx
∴ dW = "YAx"/"L"dx      .....(3) [From (2)]
5. The total amount of work done in stretching the wire from x = 0 to x = l can be found out by integrating equation (3).
W = $\int\limits_{0}^{l} dW = \int\limits_{0}^{l}\frac{YAx}{L} dx = \frac{YA}{L} \int\limits_{0}^{l} x dx$
∴ W = "YA"/"L" ["x"^2/2]_0^l
∴ W = "YA"/"L" [l^2/2 - 0^2/2]
∴ W = ("YA"l)/"L" xx l/2
But, ("YA"l)/"L" = "F"       .....[From (1)]
W = 1/2 xx "F" xx l
∴ Work done in stretching a wire,
W = 1/2 xx "load" xx "extension"
6. Work done by stretching force is equal to strain energy gained by the wire.
∴ Strain energy = 1/2 xx "load" xx "extension"
7. "Work done per unit volume" = "work done instretching wire"/"volume of wire"
= 1/2 xx ("F" xx l)/"V"
= 1/2 xx ("F" xx l)/("A" xx "L")
= 1/2 xx "F"/"A" xx l/"L"
= 1/2 xx "stress" xx "strain"
∴ "Strain energy per unit volume" = 1/2 xx "stress" xx "strain"
8. Other forms:
Since, Y = "stress"/"strain"
• Strain energy per unit volume
= 1/2 xx "stress" xx "stress"/"Y" = 1/2 xx ("stress")^2/"Y"
• Strain energy per unit volume
= 1/2 xx "Y" xx "strain" xx "strain" = 1/2 xx "Y" xx ("strain")^2
Concept: Strain Energy
Is there an error in this question or solution?

#### APPEARS IN

Balbharati Physics 11th Standard Maharashtra State Board
Chapter 6 Mechanical Properties of Solids
Exercises | Q 4. (iv) | Page 112