questions

Use of SI Units &amp; Their Prefixes

A high-altitude balloon carries a sensor package. The data logger records the quantities shown.
<div class="question-table-scroll"><table>
<thead>
<tr>
<th>Quantity</th>
<th align="right">Recorded value</th>
</tr>
</thead>
<tbody><tr>
<td>sampling interval</td>
<td align="right"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn><mtext> </mtext><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">s</mi></mrow></mrow><annotation encoding="application/x-tex">25\ \mathrm{ms}</annotation></semantics></math>25 ms</td>
</tr>
<tr>
<td>electric current</td>
<td align="right"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>180</mn><mtext> </mtext><mrow><mi>μ</mi><mi mathvariant="normal">A</mi></mrow></mrow><annotation encoding="application/x-tex">180\ \mathrm{\mu A}</annotation></semantics></math>180 μA</td>
</tr>
<tr>
<td>temperature</td>
<td align="right"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>268</mn><mtext> </mtext><mi mathvariant="normal">K</mi></mrow><annotation encoding="application/x-tex">268\ \mathrm{K}</annotation></semantics></math>268 K</td>
</tr>
<tr>
<td>mass of sensor package</td>
<td align="right"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>420</mn><mtext> </mtext><mi mathvariant="normal">g</mi></mrow><annotation encoding="application/x-tex">420\ \mathrm{g}</annotation></semantics></math>420 g</td>
</tr>
</tbody></table></div>
Express each recorded value in its corresponding SI base unit.

States <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn><mtext> </mtext><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">s</mi></mrow><mo>=</mo><mn>2.5</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><annotation encoding="application/x-tex">25\ \mathrm{ms}=2.5\times10^{-2}\ \mathrm{s}</annotation></semantics></math>25 ms=2.5×10−2 s.

States <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>180</mn><mtext> </mtext><mrow><mi>μ</mi><mi mathvariant="normal">A</mi></mrow><mo>=</mo><mn>1.80</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><annotation encoding="application/x-tex">180\ \mathrm{\mu A}=1.80\times10^{-4}\ \mathrm{A}</annotation></semantics></math>180 μA=1.80×10−4 A.

States <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>268</mn><mtext> </mtext><mi mathvariant="normal">K</mi></mrow><annotation encoding="application/x-tex">268\ \mathrm{K}</annotation></semantics></math>268 K is already in the SI base unit for thermodynamic temperature.

States <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>420</mn><mtext> </mtext><mi mathvariant="normal">g</mi><mo>=</mo><mn>0.420</mn><mtext> </mtext><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi></mrow></mrow><annotation encoding="application/x-tex">420\ \mathrm{g}=0.420\ \mathrm{kg}</annotation></semantics></math>420 g=0.420 kg.

The data logger records one measurement every <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn><mtext> </mtext><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">s</mi></mrow></mrow><annotation encoding="application/x-tex">25\ \mathrm{ms}</annotation></semantics></math>25 ms.
Calculate the number of measurements recorded during <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3.0</mn><mtext> </mtext><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></mrow><annotation encoding="application/x-tex">3.0\ \mathrm{min}</annotation></semantics></math>3.0 min of flight.

Converts <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3.0</mn><mtext> </mtext><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></mrow><annotation encoding="application/x-tex">3.0\ \mathrm{min}</annotation></semantics></math>3.0 min to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>180</mn><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><annotation encoding="application/x-tex">180\ \mathrm{s}</annotation></semantics></math>180 s and uses a sampling interval of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.025</mn><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><annotation encoding="application/x-tex">0.025\ \mathrm{s}</annotation></semantics></math>0.025 s.

Calculates <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>180</mn><mi mathvariant="normal">/</mi><mn>0.025</mn><mo>=</mo><mn>7200</mn></mrow><annotation encoding="application/x-tex">N=180/0.025=7200</annotation></semantics></math>N=180/0.025=7200 measurements.

During part of the flight, the balloon rises vertically through <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.8</mn><mtext> </mtext><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">m</mi></mrow></mrow><annotation encoding="application/x-tex">1.8\ \mathrm{km}</annotation></semantics></math>1.8 km in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4.5</mn><mtext> </mtext><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></mrow><annotation encoding="application/x-tex">4.5\ \mathrm{min}</annotation></semantics></math>4.5 min.
Calculate its average vertical speed in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">m</mi><mtext> </mtext><msup><mi mathvariant="normal">s</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathrm{m\,s^{-1}}</annotation></semantics></math>ms−1.

Converts the data to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1800</mn><mtext> </mtext><mi mathvariant="normal">m</mi></mrow><annotation encoding="application/x-tex">1800\ \mathrm{m}</annotation></semantics></math>1800 m and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>270</mn><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><annotation encoding="application/x-tex">270\ \mathrm{s}</annotation></semantics></math>270 s.

Calculates an average vertical speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6.7</mn><mtext> </mtext><mrow><mi mathvariant="normal">m</mi><mtext> </mtext><msup><mi mathvariant="normal">s</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mrow><annotation encoding="application/x-tex">6.7\ \mathrm{m\,s^{-1}}</annotation></semantics></math>6.7 ms−1 to two significant figures.

A student says:
<blockquote>
“The SI base unit for mass must be the gram because kilo is only a prefix.”
</blockquote>
Explain why this statement is incorrect.

States that the kilogram, symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi></mrow><annotation encoding="application/x-tex">\mathrm{kg}</annotation></semantics></math>kg, is the SI base unit for mass.

Explains that the gram is a submultiple of the kilogram, with <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mtext> </mtext><mi mathvariant="normal">g</mi><mo>=</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi></mrow></mrow><annotation encoding="application/x-tex">1\ \mathrm{g}=10^{-3}\ \mathrm{kg}</annotation></semantics></math>1 g=10−3 kg, so the prefix-like wording does not make gram the base unit.