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Difficulty: Very hardFundamental and Derived Quantities

Match each composite physical quantity or ratio on the left with its correct fundamental (SI base) unit decomposition on the right.

  • Electric potential gradientkg⋅m⋅s−3⋅A−1\text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{A}^{-1}
  • Coefficient of dynamic viscositykg⋅m−1⋅s−1\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}
  • Ratio of Planck's constant to moment of inertias−1\text{s}^{-1}
  • Specific latent heat divided by spatial temperature gradientm3⋅s−2⋅K−1\text{m}^3\cdot\text{s}^{-2}\cdot\text{K}^{-1}

Answer

Electric potential gradient matches kg⋅m⋅s−3⋅A−1\text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{A}^{-1}; Coefficient of dynamic viscosity matches kg⋅m−1⋅s−1\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}; Ratio of Planck's constant to moment of inertia matches s−1\text{s}^{-1}; Specific latent heat divided by spatial temperature gradient matches m3⋅s−2⋅K−1\text{m}^3\cdot\text{s}^{-2}\cdot\text{K}^{-1}.
Each physical quantity or ratio is systematically reduced to its SI base quantities (mass in kg, length in m, time in s, electric current in A, thermodynamic temperature in K) by substituting fundamental definitions of derived units.

Step-by-Step Solution

1
Decompose electric potential gradient into fundamental SI base units
Electric potential gradient=Electric PotentialDistance=WorkCharge×Distance=kg⋅m2⋅s−2A⋅s×m=kg⋅m⋅s−3⋅A−1\text{Electric potential gradient} = \frac{\text{Electric Potential}}{\text{Distance}} = \frac{\text{Work}}{\text{Charge} \times \text{Distance}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s} \times \text{m}} = \text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{A}^{-1}.
Electric potential is defined as energy per unit charge, and potential gradient is its spatial rate of change.
2
Decompose coefficient of dynamic viscosity into fundamental SI base units
η=Force×DistanceArea×Velocity=(kg⋅m⋅s−2)×mm2×(m⋅s−1)=kg⋅m−1⋅s−1\eta = \frac{\text{Force} \times \text{Distance}}{\text{Area} \times \text{Velocity}} = \frac{(\text{kg}\cdot\text{m}\cdot\text{s}^{-2}) \times \text{m}}{\text{m}^2 \times (\text{m}\cdot\text{s}^{-1})} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}.
Newton's law of viscosity relates shear force to surface area and velocity gradient.
3
Determine the base unit ratio of Planck's constant to moment of inertia
\frac{h}{I} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-1}}{\text{kg}\cdot\text{m}^2} = \text{s}^{-1}.
Planck's constant carries dimensions of angular momentum, while moment of inertia is mass multiplied by distance squared.
4
Decompose specific latent heat divided by spatial temperature gradient
\frac{L}{\frac{\Delta T}{\Delta x}} = \frac{\text{J}\cdot\text{kg}^{-1}}{\text{K}\cdot\text{m}^{-1}} = \frac{\text{m}^2\cdot\text{s}^{-2}}{\text{K}\cdot\text{m}^{-1}} = \text{m}^3\cdot\text{s}^{-2}\cdot\text{K}^{-1}.
Specific heat quantities represent thermal energy per unit mass, whereas temperature gradient represents thermal variation per unit displacement.

Key Concept

Fundamental SI base unit decomposition of derived physical quantities
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Topics

Fundamental and Derived Quantities

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