Maharashtra State BoardHSC Science (General) 11th
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Long answer type question. Derive an expression for strain energy per unit volume of the material of a wire. - Physics

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Sum

Long answer type question.

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

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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
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APPEARS IN

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