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Difficulty: MediumFundamental and Derived Units

Which of the following correctly expresses the derived SI unit of power, the watt (W\text{W}), in terms of fundamental SI base units?

  1. A
    kg⋅m2⋅s−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}
  2. kg⋅m2⋅s−3\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}Answer
  3. C
    kg⋅m⋅s−3\text{kg}\cdot\text{m}\cdot\text{s}^{-3}
  4. D
    kg⋅m2⋅s3\text{kg}\cdot\text{m}^2\cdot\text{s}^{3}

Answer

The watt (W\text{W}) expressed in fundamental SI base units is kg⋅m2⋅s−3\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}.
Power is defined as energy transferred or work done per unit time (P=Wt\text{P} = \frac{W}{t}). Work is force times distance (N⋅m=kg⋅m2⋅s−2\text{N}\cdot\text{m} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}). Dividing work by time (s\text{s}) yields kg⋅m2⋅s−3\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}.

Step-by-Step Solution

1
Recall the defining formula for power
Power=WorkTime=Force×DisplacementTime\text{Power} = \frac{\text{Work}}{\text{Time}} = \frac{\text{Force} \times \text{Displacement}}{\text{Time}}
Relating the derived quantity to simpler mechanical quantities.
2
Break down force into fundamental base units using Newton's second law (F=maF = ma)
Unit of Force (Newton)=kg⋅m⋅s−2\text{Unit of Force (Newton)} = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}
Mass is measured in kilograms (kg\text{kg}), acceleration in meters per second squared (m⋅s−2\text{m}\cdot\text{s}^{-2}).
3
Express work in fundamental base units
Unit of Work (Joule)=(kg⋅m⋅s−2)×m=kg⋅m2⋅s−2\text{Unit of Work (Joule)} = (\text{kg}\cdot\text{m}\cdot\text{s}^{-2}) \times \text{m} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}
Work is force multiplied by distance.
4
Divide the unit of work by the unit of time (seconds)
Unit of Power (Watt)=kg⋅m2⋅s−2s=kg⋅m2⋅s−3\text{Unit of Power (Watt)} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}
Dividing by s\text{s} decreases the exponent of seconds by 1.

Key Concept

Expressing derived SI units in terms of fundamental SI base units
Estimated Time:1m 0s
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Topics

Fundamental and Derived Units

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