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Difficulty: MediumFundamental and Derived Units

The force constant (stiffness) kk of a helical spring measures its resistance to elastic deformation and is defined by the relationship k=Fek = \frac{F}{e}, where FF represents the restoring force and ee represents the extension. Which of the following represents the SI unit of kk expressed strictly in terms of fundamental SI base units?

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

Answer

kg⋅s−2\text{kg} \cdot \text{s}^{-2}
The force constant kk is defined as force per unit extension (k=Fek = \frac{F}{e}). Since force has base units of kg⋅m⋅s−2\text{kg} \cdot \text{m} \cdot \text{s}^{-2} and extension has base units of m\text{m}, dividing force by extension yields kg⋅m⋅s−2m=kg⋅s−2\frac{\text{kg} \cdot \text{m} \cdot \text{s}^{-2}}{\text{m}} = \text{kg} \cdot \text{s}^{-2}.

Step-by-Step Solution

1
Identify the defining formula and constituent quantities.
The force constant formula is k=Fek = \frac{F}{e}, where force FF is measured in newtons (N\text{N}) and extension ee is measured in metres (m\text{m}).
Determining SI base units requires substituting base dimensions into the governing physical equation.
2
Express force in fundamental SI base units using Newton's second law (F=maF = ma).
1 N=1 kg⋅m⋅s−21\text{ N} = 1\text{ kg} \cdot \text{m} \cdot \text{s}^{-2}.
Mass has fundamental unit kg\text{kg}, and acceleration has fundamental unit m⋅s−2\text{m} \cdot \text{s}^{-2}.
3
Substitute the base unit expression of force into the formula for kk and simplify.
\text{SI unit of } k = \frac{\text{kg} \cdot \text{m} \cdot \text{s}^{-2}}{\text{m}} = \text{kg} \cdot \text{s}^{-2}.
The unit of length (m\text{m}) in the numerator cancels out with the unit of length in the denominator.

Key Concept

Deriving SI units of physical quantities from fundamental base units using defining equations.
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Topics

Fundamental and Derived Units

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