Advertisement Remove all ads

Advertisement Remove all ads

Sum

**Long answer type question.**

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

Advertisement Remove all ads

#### Solution

**The expression for strain energy per unit volume:**

- 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.
- 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) - 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) - Let 'dW’ be the work done for the further small extension ‘dx’.

Work = force × displacement

∴ dW = fdx

∴ dW = `"YAx"/"L"`dx .....(3) [From (2)] - 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"` - Work done by stretching force is equal to strain energy gained by the wire.

∴ Strain energy = `1/2 xx "load" xx "extension"` - `"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"` **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?

Advertisement Remove all ads

#### APPEARS IN

Advertisement Remove all ads