Question

If the definition of SI unit 'X' is required to define SI unit 'Y', the dependency is indicated as \(X \rightarrow Y\). A valid order of dependencies, as per present convention, is __________.

• A. \(\text{metre} \rightarrow \text{second} \rightarrow \text{kilogram} \rightarrow \text{kelvin}\)
• B. \(\text{second} \rightarrow \text{metre} \rightarrow \text{kilogram} \rightarrow \text{kelvin}\)
• C. \(\text{metre} \rightarrow \text{second} \rightarrow \text{kelvin} \rightarrow \text{kilogram}\)
• D. \(\text{second} \rightarrow \text{kilogram} \rightarrow \text{kelvin} \rightarrow \text{metre}\)

Hint:

Look at the SI unit of the constant used to fix each base unit: \(\text{Hz}\) for the second, \(\text{m/s}\) for the metre (via \(c\)), \(\text{kg}\,\text{m}^2/\text{s}\) for the kilogram (via \(h\)), and \(\text{kg}\,\text{m}^2/(\text{s}^2\text{K})\) for the kelvin (via \(k_B\)). Whichever base units appear in that constant's dimensions must already be defined first.

Answer 1

The Correct Option is B

A dimensional-analysis check on each defining constant.
List the fixed constants and their units.
The 2019 SI redefinition ties each base unit to an exact numeric value of a constant: $$ \begin{array}{c|c} \text{Base unit} & \text{Fixed by (SI unit of the constant)} \\\hline \text{second (s)} & \Delta\nu_{Cs}\ (\text{Hz} = \text{s}^{-1}) \\ \text{metre (m)} & c\ (\text{m}\,\text{s}^{-1}) \\ \text{kilogram (kg)} & h\ (\text{kg}\,\text{m}^2\,\text{s}^{-1}) \\ \text{kelvin (K)} & k_B\ (\text{kg}\,\text{m}^2\,\text{s}^{-2}\,\text{K}^{-1}) \end{array} $$ Read the dependency straight off the units column.
Whatever OTHER base units appear in a constant's SI unit must already be defined before that constant can be used to fix a new unit. $\Delta\nu_{Cs}$ carries only $\text{s}^{-1}$, so the second depends on nothing else, it can be defined first. $c$ carries $\text{m}\,\text{s}^{-1}$, which needs the second, so the metre can only be defined after the second. $h$ carries $\text{kg}\,\text{m}^2\,\text{s}^{-1}$, which needs both the metre and the second, so the kilogram needs both of those defined first. $k_B$ carries $\text{kg}\,\text{m}^2\,\text{s}^{-2}\,\text{K}^{-1}$, which needs the kilogram, the metre and the second, so the kelvin must come last. Assemble the chain.
Putting units in the order that never uses an undefined unit: \[ \text{second} \rightarrow \text{metre} \rightarrow \text{kilogram} \rightarrow \text{kelvin} \] This is exactly option B. Every other ordering places metre before second, or places kilogram/kelvin before the units their defining constant's dimensions require, so they fail the same units check. \[ \boxed{\text{second} \rightarrow \text{metre} \rightarrow \text{kilogram} \rightarrow \text{kelvin}} \]