Question bank — SI units — seven base units and all derived units

The kilogram is the only SI base unit whose name carries a prefix. ::: True — "kilo" is baked into the base unit of mass, unlike metre or second which are prefix-free; this is a historical accident, not a logical rule. A quantity can have units yet be dimensionless. ::: True — an angle in radians is metre/metre, a pure ratio; it is dimensionless yet we still write "rad" to signal it is an angle. Two quantities with the same base-unit form must be the same physical quantity. ::: False — torque and energy are both ${\text{kg⋅m}}^2{\text{s}}^{-2}$ yet one is a rotational effect and the other is work; identical dimensions do not force identical meaning. Frequency (Hz) and radioactivity (Bq) and angular frequency all reduce to ${\text{s}}^{-1}$. ::: True in base units — but note angular frequency $\omega$ is conventionally written $\text{rad/s}$ (radian being a dimensionless name), and $\omega =2\pi f$, so a $1\text{Hz}$ spin is $2\pi \text{rad/s}$ even though both strip down to ${\text{s}}^{-1}$. Angular frequency in rad/s equals frequency in Hz. ::: False — they differ by the factor $2\pi$: $\omega =2\pi f$, because one full cycle (1 Hz) sweeps $2\pi$ radians; the radian "unit" is dimensionless so it hides this factor in the base-unit form. The mole is a base unit even though it is "just a number" ($N_A=6.022\times 10^{23}$). ::: True — amount of substance is declared an independent base quantity by convention; the mole counts entities and cannot be built from the other six. Celsius degrees and kelvin degrees are the same size. ::: True — only the zero point differs ($T_{°C}=T_K-273.15$); a rise of 1 °C equals a rise of 1 K, so temperature differences are numerically identical. Since $c$ (the speed of light) is now a fixed exact number, measuring it more carefully could change its value. ::: False — after the 2019 redefinition $c$ is defined exact; better experiments now refine the metre, not $c$. The litre is an SI unit. ::: False — the litre is an accepted non-SI unit; the coherent SI volume unit is the cubic metre ($1\text{L}=10^{-3}{\text{m}}^3$), and the litre is merely tolerated for convenience.

"Force is so fundamental in mechanics that the newton must be a base unit." ::: Error — force $F$ is derived from $F=ma$ (mass times acceleration), so its unit ${\text{kg⋅m⋅s}}^{-2}$ is built from mass, length and time; the base set of mechanics is only those three. "Weight and mass are both measured in kilograms." ::: Error — mass is in kilograms, but weight is a force measured in newtons; confusing them mixes a base quantity with a derived one. "°C is an SI base unit because temperature is a base quantity." ::: Error — the base unit of temperature is the kelvin; Celsius is a shifted convenience scale derived from it, not a base unit. "The gram is the SI base unit; the kilogram is just 1000 grams." ::: Error — officially the kilogram (prefix included) is the base unit; the gram is defined from it, not the other way round. "Pressure is force, so its unit is the newton." ::: Error — pressure is force per area, so the pascal is ${\text{N/m}}^2={\text{kg⋅m}}^{-1}{\text{s}}^{-2}$; dividing by area changes the dimensions. "A mole of electrons and a kilogram of electrons describe the same amount of stuff." ::: Error — the mole counts particles while the kilogram measures inertia/mass; they are different base quantities answering different questions. "$E=m{c}^2$ can't be right dimensionally because energy shouldn't equal mass." ::: Error — with $E$ energy, $m$ mass, $c$ speed of light, the check works: $[m{c}^2]=\text{kg}\cdot (\text{m/s}{)}^2={\text{kg⋅m}}^2{\text{s}}^{-2}=\text{J}$, exactly energy; the constant $c^2$ carries the missing dimensions. "The prefix on a kilometre and a kilogram means both are 1000 base units." ::: Error — a kilometre is $1000$ metres (base unit metre), but a kilogram is itself the base unit; you cannot stack another prefix on it, so a "kilo-kilogram" is a tonne (Mg), not a base unit. "You may write $\mu \text{k}$ (microkilo) instead of milli." ::: Error — SI forbids compound prefixes; only one prefix at a time attaches to a unit, so it must be milli (m), never micro-kilo.

Why were the base units redefined in 2019 using constants of nature instead of physical objects? ::: Objects drift, scratch, and can be lost (the old kilogram cylinder actually changed mass); fixing a constant of nature gives a standard that never varies and is reproducible in any lab. Why is luminous intensity (candela) a base quantity when it seems like a form of power? ::: The candela is weighted by the human eye's response to different wavelengths, so it cannot be reduced to raw watts; that human-centred weighting makes it independent enough to be declared base. Why can't temperature be built out of energy, even though heat and energy feel linked? ::: The link needs the Boltzmann constant $k$ to convert energy to temperature; declaring temperature independent lets $k$ do the translating rather than forcing kelvin to be a derived energy unit. Why do we bother naming derived units (newton, joule) instead of always writing base units? ::: The named units compress long strings like ${\text{kg⋅m}}^2{\text{s}}^{-2}$ into one word, making equations readable — but you can always unfold them back to base units to check dimensions. Why is a pure number like $\frac{1}{2}$ ignored in a unit check? ::: Dimensionless constants carry no units, so they cannot change the base-unit form; only the quantities with dimensions determine the unit. Why must the seven base quantities be dimensionally independent? ::: If one could be written from the others it would be redundant — you'd have more standards than necessary; independence guarantees a minimal, non-overlapping "basis" for all measurements. Why is the electronvolt (eV) allowed in physics if it is not an SI unit? ::: It is a tolerated non-SI unit sized to atomic energies ($1\text{eV}=1.602\times 10^{-19}\text{J}$); it is convenient but must be converted to joules before it fits coherently into SI equations.

What is the base-unit form of a quantity that is truly dimensionless, like refractive index or strain? ::: It has no base units at all — refractive index is a ratio of speeds and strain is length/length, so every base unit cancels and the answer is $1$ (pure number); see the amber "1" in the figure. Does adding two quantities with the same units but different meanings (e.g. torque + energy) make physical sense? ::: No — even though both are ${\text{kg⋅m}}^2{\text{s}}^{-2}$, physics forbids adding a rotational torque to an energy; matching units is necessary but not sufficient for a valid sum. If a formula gives a dimensionless argument to a function like $\sin$ or $e^x$, what does that tell you? ::: Such functions demand dimensionless inputs, so the argument's units must cancel; if they don't, the formula is wrong — a powerful consistency check. At absolute zero (0 K), what happens to the Celsius reading, and is 0 K reachable? ::: $0\text{K}=-273.15\text{°C}$; it is the theoretical floor of temperature but the third law of thermodynamics says it can be approached, never exactly reached. Can a derived unit ever "become" a base unit? ::: In principle the choice of which quantities are base is a convention — SI could have chosen force as base and mass as derived; the current seven are picked for practicality, not because they are uniquely fundamental. Is angular measure (radian) a base unit, a derived unit, or neither? ::: Neither in the usual sense — the radian is a dimensionless derived unit, defined as arc length over radius (metre/metre), so it has a name but no dimensions. Where does the steradian fit — and how is it different from the radian? ::: Same category as the radian: a dimensionless named derived unit, but for solid angle, defined as area on a sphere over radius-squared (${\text{m}}^2/{\text{m}}^2$); both carry names yet vanish in base units. What is the unit of a quantity raised to the zeroth power, e.g. $v^0$? ::: It is dimensionless ($=1$); any unit to the power zero cancels itself, leaving a pure number with no base units. Are the minute, hour, tonne and bar part of SI? ::: No — they are accepted for use with SI but are non-coherent: minute/hour ($60\text{s}$, $3600\text{s}$), tonne ($10^3\text{kg}$), bar ($10^5\text{Pa}$); a "coherent" unit needs no such conversion factor, which is why the second, kilogram and pascal are preferred. What does "coherent" mean for a derived unit? ::: A coherent unit comes out of its defining equation with a conversion factor of exactly $1$ (e.g. $1\text{N}=1{\text{kg⋅m⋅s}}^{-2}$); units like the litre or bar are non-coherent because a power-of-ten factor sneaks in.