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Zorluk: OrtaPhysical Quantities, Units and Dimensions

The tensile stress σ\sigma on a solid wire subjected to a stretching force FF is defined as force per unit cross-sectional area, while the fractional change in length is the tensile strain ϵ\epsilon. If Young's modulus YY of the material is given by Y=σϵY = \frac{\sigma}{\epsilon}, which of the following is the SI unit of Young's modulus expressed in fundamental SI base units?

  1. kg⋅m−1⋅s−2\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}Cevap
  2. B
    kg⋅m⋅s−2\text{kg}\cdot\text{m}\cdot\text{s}^{-2}
  3. C
    kg⋅m−2⋅s−2\text{kg}\cdot\text{m}^{-2}\cdot\text{s}^{-2}
  4. D
    kg⋅m2⋅s−2\text{kg}\cdot\text{m}^{2}\cdot\text{s}^{-2}

Cevap

kg⋅m−1⋅s−2\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}
Young's modulus is calculated as tensile stress divided by tensile strain. Since tensile strain is the ratio of change in length to original length, it has no units. Therefore, the SI unit of Young's modulus is identical to that of stress. Stress is force divided by area: kg⋅m⋅s−2m2=kg⋅m−1⋅s−2\frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}.

Adım Adım Çözüm

1
Determine the dimensions of the force component
Force F=ma  ⟹  [F]=kg⋅m⋅s−2F = ma \implies [F] = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}
Mass has fundamental unit kg\text{kg} and acceleration has derived unit m⋅s−2\text{m}\cdot\text{s}^{-2}.
2
Determine the SI base units of tensile stress σ\sigma
[\sigma] = \frac{[F]}{\text{Area}} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}$
Stress is defined as force divided by cross-sectional area (A=m2A = \text{m}^2).
3
Evaluate the unit of Young's modulus YY
[Y] = \frac{[\sigma]}{[\epsilon]} = \frac{\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}}{1} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}$
Strain ϵ=ΔLL\epsilon = \frac{\Delta L}{L} is the ratio of two lengths and is dimensionless.

Anahtar Kavram

Derivation of SI base units for mechanical moduli from physical definitions
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Physical Quantities, Units and Dimensions

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