Examkin

Question

Difficulty: HardFundamental and Derived Quantities

The rate of change of linear momentum per unit cross-sectional area is a derived physical quantity. When resolved into fundamental SI base units, which of the following physical quantities has the exact same SI base unit decomposition?

  1. A
    Surface tension
  2. Energy densityAnswer
  3. C
    Power
  4. D
    Specific heat capacity

Answer

Energy density
The rate of change of momentum is force (FF), which has SI base units of kg⋅m⋅s−2\text{kg}\cdot\text{m}\cdot\text{s}^{-2}. Dividing by area (m2\text{m}^2) gives kg⋅m−1⋅s−2\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}. Energy density is defined as energy per unit volume, which decomposes to kg⋅m2⋅s−2m3=kg⋅m−1⋅s−2\frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{m}^3} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}. Both derived quantities possess identical fundamental SI base unit representations.

Step-by-Step Solution

1
Determine the physical definition of the rate of change of linear momentum per unit area.
By Newton's second law, rate of change of momentum equals force (F=ΔpΔtF = \frac{\Delta p}{\Delta t}). Dividing by cross-sectional area (AA) yields force per unit area (pressure or stress), FA\frac{F}{A}.
Force is defined as the time rate of change of linear momentum.
2
Express force per unit area in terms of fundamental SI base units.
Force=mass×acceleration=kg⋅m⋅s−2\text{Force} = \text{mass} \times \text{acceleration} = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}. Therefore, FA=kg⋅m⋅s−2m2=kg⋅m−1⋅s−2\frac{F}{A} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}.
Mass (kg\text{kg}), length (m\text{m}), and time (s\text{s}) are fundamental base quantities.
3
Analyze energy density in fundamental SI base units.
Energy density=EnergyVolume=kg⋅m2⋅s−2m3=kg⋅m−1⋅s−2\text{Energy density} = \frac{\text{Energy}}{\text{Volume}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{m}^3} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}.
Decomposing energy into work (force ×\times distance) and dividing by volume reveals its base unit equivalence.

Key Concept

Decomposition of derived quantities into fundamental SI base units
Estimated Time:1m 30s
Rate this question

Topics

Fundamental and Derived Quantities

More questions from this topic