Evaluate each of the following to three significantfigures and express each answer in SI units using anappropriate prefix: (a) (212 mN)2, (b) (52 800 ms)2, and(c) [548(106)]1/2ms.

Step-by-Step Solution

Step 1

The given physical quantities are ${\left( {212\;{\rm{mN}}} \right)^2},\;{\left( {52800\;{\rm{ms}}} \right)^2}$ and ${\left[ {548\left( {{{10}^6}} \right)} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\;{\rm{ms}}$

We are required to determine the given physical quantities in SI units in three significant figures.

Step 2

The value of $1\;{\rm{mN}}$ is equal to $1 \times {10^{ – 3}}\;{\rm{N}}$.

Step 3

Now, convert the units.\begin{array}{c} {\left( {212\;{\rm{mN}}} \right)^2} = {\left( {212\;{\rm{mN}}} \right)^2} \times {\left( {\frac{{{{10}^{ – 3}}\;{\rm{N}}}}{{1\;{\rm{mN}}}}} \right)^2}\\ = 4.49 \times {10^{ – 2}}\;{{\rm{N}}^2} \end{array}

Step 4

The value of $1\;{\rm{ms}}$ is equal to $1 \times {10^{ – 3}}\;{\rm{s}}$.

Step 5

Now, convert the units.\begin{array}{c} {\left( {52800\;{\rm{ms}}} \right)^2} = {\left( {52800\;{\rm{ms}}} \right)^2} \times {\left( {\frac{{{{10}^{ – 3}}\;{\rm{s}}}}{{1\;{\rm{ms}}}}} \right)^2}\\ = 2.78 \times {10^3}\;{{\rm{s}}^2} \end{array}

Step 6

The value of $1\;{\rm{ms}}$ is equal to $1 \times {10^{ – 3}}\;{\rm{s}}$.

Step 7

Now, convert the units.\[\begin{array}{c} {\left[ {548\left( {{{10}^6}} \right)} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\;{\rm{ms}} = {\left( {548} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}} \times {\left( {{{10}^6}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}} \times \left( {\frac{{{{10}^{ – 3}}\;{\rm{s}}}}{{1\;{\rm{ms}}}}} \right)\\ = 23.4\;{\rm{s}} \end{array}\]