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Difficulty: HardFundamental and Derived Quantities

A physical quantity ZZ is defined by the expression Z=P⋅VI⋅tZ = \frac{P \cdot V}{I \cdot t}, where PP represents pressure, VV represents volume, II represents electric current, and tt represents time. Which of the following statements correctly classifies quantity ZZ and expresses it in fundamental SI base units?

  1. Quantity ZZ is a derived quantity with fundamental SI base units of kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.Answer
  2. B
    Quantity ZZ is a fundamental quantity with fundamental SI base units of kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.
  3. C
    Quantity ZZ is a derived quantity with fundamental SI base units of kg⋅m2⋅s−2⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}\cdot\text{A}^{-1}.
  4. D
    Quantity ZZ is a fundamental quantity with fundamental SI base units of kg⋅m⋅s−2⋅A−1\text{kg}\cdot\text{m}\cdot\text{s}^{-2}\cdot\text{A}^{-1}.

Answer

Quantity ZZ is a derived quantity with fundamental SI base units of kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.
The quantity ZZ represents electric potential (voltage), which is defined as work done per unit electric charge (W/QW/Q). Since electric potential is derived from mass, length, time, and electric current, it is a derived physical quantity. Expanding P⋅VP \cdot V yields energy units (kg⋅m2⋅s−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}), and dividing by I⋅tI \cdot t (A⋅s\text{A}\cdot\text{s}) gives the correct base units of kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.

Step-by-Step Solution

1
Determine the SI base unit decomposition for pressure (PP) and volume (VV).
Pressure P=ForceArea=kg⋅m⋅s−2m2=kg⋅m−1⋅s−2P = \frac{\text{Force}}{\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}. Volume V=m3V = \text{m}^3. Therefore, P⋅V=(kg⋅m−1⋅s−2)⋅m3=kg⋅m2⋅s−2P \cdot V = (\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}) \cdot \text{m}^3 = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}.
Decomposing composite physical quantities into base units of mass, length, and time.
2
Determine the SI base unit decomposition for the denominator I⋅tI \cdot t.
I⋅t=Electric current×Time=A⋅sI \cdot t = \text{Electric current} \times \text{Time} = \text{A}\cdot\text{s}.
Electric current (amperes, A) and time (seconds, s) are fundamental SI quantities.
3
Evaluate the full expression for Z=P⋅VI⋅tZ = \frac{P \cdot V}{I \cdot t} in SI base units.
Base units of Z=kg⋅m2⋅s−2A⋅s=kg⋅m2⋅s−3⋅A−1Z = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.
Apply laws of indices to simplify base unit ratios.
4
Classify physical quantity ZZ.
Because ZZ (electric potential) is defined in terms of fundamental quantities (kg, m, s, A) rather than existing independently, it is a derived quantity.
The seven fundamental SI quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.

Key Concept

Classification of physical quantities into fundamental and derived types and resolving complex derived quantities into base SI units.
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Topics

Fundamental and Derived Quantities

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