Step-by-Step Solution
Step 1
We are given the units ${\rm{GN}} \cdot {\rm{\mu m}}$, ${\rm{kg/\mu m}}$, ${\rm{N/k}}{{\rm{s}}^2}$ and ${\rm{kN/\mu s}}$.
We are asked to estimate the correct SI form of the units.
Step 2
(a)
To find the SI form of unit ${\rm{GN}} \cdot {\rm{\mu m}}$ we need to reduce the combinations of units.
Since $1\;{\rm{GN}} = {\rm{1}}{{\rm{0}}^9}\,{\rm{N}}$ and ${\rm{1 \mu m}} = {\rm{1}}{{\rm{0}}^{ – 6}}\,{\rm{m}}$, then\[\begin{array}{c} {\rm{GN}} \cdot {\rm{\mu m}} = \left( {{\rm{GN}} \times \frac{{{\rm{1}}{{\rm{0}}^9}\,{\rm{N}}}}{{{\rm{1}}\;{\rm{GN}}}}} \right)\left( {{\rm{\mu m}} \times \frac{{{\rm{1}}{{\rm{0}}^{ – 6}}\,{\rm{m}}}}{{{\rm{1}}\;{\rm{\mu m}}}}} \right)\\ {\rm{GN}} \cdot {\rm{\mu m}} = \left( {{{10}^3}\;{\rm{N}} \cdot {\rm{m}} \times \frac{{1\;{\rm{kN}} \cdot {\rm{m}}}}{{{{10}^3}\;{\rm{N}} \cdot {\rm{m}}}}} \right)\\ {\rm{GN}} \cdot {\rm{\mu m}} = 1\;{\rm{kN}} \cdot {\rm{m}} \end{array}\]
Step 3
(b)
To find the SI form of unit ${\rm{kg/\mu m}}$ we need to reduce the combinations of units.
Since $1\;{\rm{\mu m}} = {\rm{1}}{{\rm{0}}^{ – 6}}\,{\rm{m}}$, then\[\begin{array}{c} {\rm{kg/\mu m}} = \left( {\frac{{{\rm{kg}}}}{{{\rm{\mu m}} \times \frac{{{{10}^{ – 6}}\,{\rm{m}}}}{{1\;{\rm{\mu m}}}}}}} \right)\\ {\rm{kg/\mu m}} = \left( {{{10}^6}\;{\rm{kg/m}} \times \frac{{1\;{\rm{Mkg/m}}}}{{{{10}^6}\;{\rm{kg/m}}}}} \right)\\ {\rm{kg/\mu m}} = 1\;{\rm{Mkg/m}} \end{array}\]
Step 4
(c)
To find the SI form of unit ${\rm{N/k}}{{\rm{s}}^2}$ we need to reduce the combinations of units.
Since $1\;{\rm{ks}} = {\rm{1}}{{\rm{0}}^3}\,{\rm{s}}$, then\[\begin{array}{c} {\rm{N/k}}{{\rm{s}}^2} = \frac{{\rm{N}}}{{{{\left( {{\rm{ks}} \times \frac{{{{10}^3}\,{\rm{s}}}}{{1\;{\rm{ks}}}}} \right)}^2}}}\\ {\rm{N/k}}{{\rm{s}}^2} = \left( {{{10}^{ – 6}}\;{\rm{N/}}{{\rm{s}}^2} \times \frac{{1\;{\rm{\mu N/}}{{\rm{s}}^2}}}{{{{10}^{ – 6}}\;{\rm{N/}}{{\rm{s}}^2}}}} \right)\\ {\rm{N/k}}{{\rm{s}}^2} = 1\;{\rm{\mu N/}}{{\rm{s}}^2} \end{array}\]
Step 5
(d)
To find the SI form of unit ${\rm{kN/\mu s}}$ we need to reduce the combinations of units.
Since $1\;{\rm{kN}} = {\rm{1}}{{\rm{0}}^3}\,{\rm{N}}$ and $1\;{\rm{\mu s}} = {\rm{1}}{{\rm{0}}^{ – 6}}\;{\rm{s}}$, then\[\begin{array}{c} {\rm{kN/\mu s}} = \frac{{\left( {{\rm{kN}} \times \frac{{{{10}^3}\,{\rm{N}}}}{{1\;{\rm{kN}}}}} \right)}}{{\left( {{\rm{\mu s}} \times \frac{{{{10}^{ – 6}}\,{\rm{s}}}}{{1\;{\rm{\mu s}}}}} \right)}}\\ {\rm{kN/\mu s}} = \left( {{{10}^9}\;{\rm{N/s}} \times \frac{{1\;{\rm{GN/s}}}}{{{{10}^9}\;{\rm{N/s}}}}} \right)\\ {\rm{kN/\mu s}} = 1\;{\rm{GN/s}} \end{array}\]