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Difficulty: HardFundamental and Derived Quantities

A physical quantity XX is defined by the expression X=P⋅Vm⋅tX = \frac{P \cdot V}{m \cdot t}, where PP represents pressure, VV represents volume, mm represents mass, and tt represents time. Which of the following statements correctly classifies XX and expresses its unit strictly in terms of fundamental SI base units?

  1. XX is a derived quantity, and its unit in SI base units is m2⋅s−3\text{m}^2 \cdot \text{s}^{-3}.Answer
  2. B
    XX is a fundamental quantity, and its unit in SI base units is m2⋅s−3\text{m}^2 \cdot \text{s}^{-3}.
  3. C
    XX is a derived quantity, and its unit in fundamental units is N⋅m⋅kg−1⋅s−1\text{N} \cdot \text{m} \cdot \text{kg}^{-1} \cdot \text{s}^{-1}.
  4. D
    XX is a derived quantity, and its unit in SI base units is kg⋅m2⋅s−3\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3}.

Answer

The physical quantity XX is a derived quantity, and its unit expressed strictly in fundamental SI base units is m2⋅s−3\text{m}^2 \cdot \text{s}^{-3}.
The quantity XX is defined via a mathematical formula involving pressure, volume, mass, and time, which classifies it as a derived quantity. Replacing each component with its fundamental SI base units gives pressure as kg⋅m−1⋅s−2\text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2} and volume as m3\text{m}^3, making the numerator kg⋅m2⋅s−2\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}. Dividing by the denominator (m⋅t=kg⋅sm \cdot t = \text{kg} \cdot \text{s}) cancels out kilograms and leaves m2⋅s−3\text{m}^2 \cdot \text{s}^{-3}, consisting purely of fundamental base units.

Step-by-Step Solution

1
Classify the physical quantity XX
XX is a derived physical quantity.
Fundamental physical quantities in the SI system are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Because XX is calculated from a combination of other quantities, it is derived.
2
Express pressure (PP) and volume (VV) in terms of fundamental SI base units
P=kg⋅m−1⋅s−2P = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2} and V=m3V = \text{m}^3.
Pressure is defined as force per unit area (kg⋅m⋅s−2m2)\left(\frac{\text{kg} \cdot \text{m} \cdot \text{s}^{-2}}{\text{m}^2}\right), and volume has the base unit m3\text{m}^3.
3
Calculate the base unit expression for the numerator P⋅VP \cdot V
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}.
Combining the powers of length (metres) gives −1+3=2-1 + 3 = 2.
4
Divide by the denominator m⋅tm \cdot t to obtain the base units of XX
X=kg⋅m2⋅s−2kg⋅s=m2⋅s−3X = \frac{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{kg} \cdot \text{s}} = \text{m}^2 \cdot \text{s}^{-3}.
The unit of mass (kg\text{kg}) cancels completely, and dividing by time (s\text{s}) reduces the exponent of seconds from −2-2 to −3-3.

Key Concept

Fundamental quantities are independent base quantities defined by the SI system, whereas derived quantities are defined algebraically from fundamental quantities. Reducing derived units to SI base units requires breaking down all non-base units into metres (m), kilograms (kg), seconds (s), amperes (A), kelvins (K), moles (mol), or candelas (cd).
Estimated Time:2m 0s
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Topics

Fundamental and Derived Quantities

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