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The International System of Units — Question 8

Section 1-2 · Physics pages 46–47

Question 8 A notation error that changes the quantity A draft report contains the entries below. The author has separately confirmed each intended meaning. The entries and intended meanings are: mass = 5 kgs (five kilograms); temperature = 300 deg K (300 kelvins); duration = 8 ms (eight metre-seconds); area = 4 mm2 (four square millimetres); and mass = 2 mkg (two thousandths of a kilogram). Use these conventions: unit symbols are upright, unpluralized, and case-sensitive; a space separates the number from its unit; multiplication of unit symbols needs a space or centered dot. The prefix milli (m) means 10−3 and attaches directly to its unit. An exponent applies to the entire prefixed unit. Mass prefixes attach to gram (g), with 1 g = 10−3 kg; compound prefixes are not permitted. 1 mm Squared prefix ms m s milli + second one prefixed unit metre × second product of units Spacing, multiplication, and powers carry physical meaning in unit notation. (a) Rewrite all five entries correctly. For the third, also assess whether the quantity name “dura- tion” is consistent with the intended unit. (b) Explain what 8 ms actually means and why typography alone cannot repair the third entry while retaining its original quantity name. (c) Show why 4 mm2 is 4 × 10−6 m2, not 4 × 10−3 m2.
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Question 8 - Solution A notation error that changes the quantity 1 mm Squared prefix ms m s milli + second one prefixed unit metre × second product of units (a) Preserve meaning while repairing notation. m = 5 kg, T = 300 K, X = 8 m s, A = 4 mm2, m = 2 g = 0.002 kg. Here X denotes a length–time product; it cannot be labeled a duration. The symbol kg does not acquire a plural s. Kelvin is written K, without a degree sign. The proposed mkg attaches another prefix to kilogram; use gram instead. Since 1 g = 10−3 kg, 2 g preserves the intended mass. For the third entry, X is only a placeholder for a quantity the author must identify. A metre-second has length–time dimension and cannot be a duration. Fixing the unit spacing is not enough: the quantity name must also change if the confirmed intended unit is retained. (b) Prefix versus product. 8 ms = 8 × 10−3 s, [8 m s] = m s̸ = s. 8 ms means eight milliseconds, because the attached m is the prefix milli. It is a time. In 8 m s or 8 m · s, m is instead the metre symbol multiplying the second symbol. The separator therefore changes the unit’s meaning and dimension; it is not decorative spacing. (c) Square the whole prefixed unit. 1 mm2 = (1 mm)2 = (10−3 m)2 = 10−6 m2. Thus 4 mm2 = 4 × 10−6 m2 . Applying milli only once would describe a different area. These short examples test notation and prefix structure, not a general conversion procedure.

Question 8

A notation error that changes the quantity

A draft report contains the entries below. The author has separately confirmed each intended meaning. The entries and intended meanings are: mass = 5 kgs (five kilograms); temperature = 300 deg K (300 kelvins); duration = 8 ms (eight metre-seconds); area = 4 mm2 (four square millimetres); and mass = 2 mkg (two thousandths of a kilogram). Use these conventions: unit symbols are upright, unpluralized, and case-sensitive; a space separates the number from its unit; multiplication of unit symbols needs a space or centered dot. The prefix milli (m\mathrm{m}) means 10−310^{-3} and attaches directly to its unit. An exponent applies to the entire prefixed unit. Mass prefixes attach to gram (g\mathrm{g}), with 1g=10−3kg1\ \mathrm{g}=10^{-3}\ \mathrm{kg}; compound prefixes are not permitted.

Diagram shown in the original SVG worksheet.

  1. Rewrite all five entries correctly. For the third, also assess whether the quantity name “duration” is consistent with the intended unit.

  2. Explain what 8ms8\ \mathrm{ms} actually means and why typography alone cannot repair the third entry while retaining its original quantity name.

  3. Show why 4mm24\ \mathrm{mm^2} is 4×10−6m24\times10^{-6}\ \mathrm{m^2}, not 4×10−3m24\times10^{-3}\ \mathrm{m^2}.

Question 8 - Solution

A notation error that changes the quantity

Diagram shown in the original SVG worksheet.

(a) Preserve meaning while repairing notation.

m=5kg,T=300K,X=8ms,A=4mm2,m=2g=0.002kg.\begin{aligned}m&=5\ \mathrm{kg},& T&=300\ \mathrm{K},\\X&=8\ \mathrm{m\,s},& A&=4\ \mathrm{mm^2},\\m&=2\ \mathrm{g}=0.002\ \mathrm{kg}.\end{aligned}Here XX denotes a length–time product; it cannot be labeled a duration. The symbol kg\mathrm{kg} does not acquire a plural s. Kelvin is written K\mathrm{K}, without a degree sign. The proposed mkg\mathrm{mkg} attaches another prefix to kilogram; use gram instead. Since 1g=10−3kg1\ \mathrm{g}=10^{-3}\ \mathrm{kg}, 2g2\ \mathrm{g} preserves the intended mass.

For the third entry, XX is only a placeholder for a quantity the author must identify. A metre-second has length–time dimension and cannot be a duration. Fixing the unit spacing is not enough: the quantity name must also change if the confirmed intended unit is retained.

(b) Prefix versus product.

8ms=8×10−3s,[8ms]=ms≠s.8\ \mathrm{ms}=8\times10^{-3}\ \mathrm{s},\qquad[8\ \mathrm{m\,s}]=\mathrm{m\,s}\ne\mathrm{s}.

8ms8\ \mathrm{ms} means eight milliseconds, because the attached m is the prefix milli. It is a time. In 8ms8\ \mathrm{m\,s} or 8m⋅s8\ \mathrm{m\cdot s}, m is instead the metre symbol multiplying the second symbol. The separator therefore changes the unit’s meaning and dimension; it is not decorative spacing.

(c) Square the whole prefixed unit.

1mm2=(1mm)2=(10−3m)2=10−6m2.1\ \mathrm{mm^2}=(1\ \mathrm{mm})^2=(10^{-3}\ \mathrm{m})^2 =10^{-6}\ \mathrm{m^2}. Thus 4mm2=4×10−6m2\boxed{4\ \mathrm{mm^2}=4\times10^{-6}\ \mathrm{m^2}}. Applying milli only once would describe a different area. These short examples test notation and prefix structure, not a general conversion procedure.

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