Physical quantities — fundamental and derived: Question bank

Before you start, recall the two load-bearing ideas:

A physical quantity is always number × unit; the product $n u$ is what's real, so $n_{1} u_{1} = n_{2} u_{2}$ .
Fundamental quantities are the seven independent starter blocks (m, kg, s, A, K, mol, cd); derived quantities are everything built from them by multiplying, dividing, and taking powers.

Why units behave like algebra — see it, don't take it on faith. A rectangle of width $3 m$ and height $2 m$ has an area you count in unit squares: each square is $1 m \times 1 m = 1 m^{2}$ . Multiplying the numbers ( $3 \times 2 = 6$ ) and multiplying the units ( $m \times m = m^{2}$ ) are the same physical act of tiling. That's the whole reason units multiply and divide like ordinary algebra.

Figure — Physical quantities — fundamental and derived

True or false — justify

Every item below is a statement. Decide true or false, but your answer must contain the reason, not just the verdict.

Look at the radian figure below before the angle item. The angle in radians is literally arc length divided by radius — two lengths, one on top of the other — so the metres cancel and nothing is left but a pure number.

"Weight and mass are the same physical quantity."

False. Mass (kg) is a fundamental quantity — the amount of matter, unchanged anywhere. Weight is the force

W = m g

(newtons), a derived quantity that shrinks on the Moon where

g

is smaller.

"An angle in radians is a fundamental quantity because you measure it with a protractor."

False. A radian is arc length ÷ radius = length/length, so the units cancel (see the figure above): it is dimensionless. Being measurable does not make something fundamental.

"Celsius is an SI base unit of temperature."

False. The kelvin is the SI base unit; Celsius is offset and doesn't start at true zero, so ratios of Celsius readings are meaningless. Celsius is a derived, shifted scale.

"Two quantities with the same base units must be the same physical quantity."

False. Torque and energy are both

kg m^{2} s^{- 2}

, yet torque is a pseudovector (an axial quantity

τ = r \times F

whose direction flips under a mirror reflection) while energy is a plain scalar. Matching units is necessary but not sufficient for sameness.

"If you make the unit smaller, the number describing the same quantity gets larger."

True. Since

n u

is invariant,

2 m = 200 cm

: shrinking the unit forces a bigger count to describe the same physical length.

"You could add 'speed' as an eighth SI base quantity to make physics more complete."

False. Speed is already length ÷ time, so making it a base unit creates two conflicting definitions of the same thing. The base set must be minimal and independent — redundancy breaks consistency.

"Every derived unit can be written as a product of powers of the seven base units."

True. That is precisely what 'derived' means; e.g.

J = kg m^{2} s^{- 2}

Pa = kg m^{- 1} s^{- 2}

"A pure number like the ratio of two lengths is a physical quantity with a unit."

False. A ratio of like quantities is dimensionless — it carries no unit at all. It's a number, not a 'number × unit'.

Spot the error

Each line contains a flawed statement or step. Name the mistake and correct it.

The dimensional grid below is your check. Every derived unit is a row of exponents on $[kg]$ , $[m]$ , $[s]$ . Multiplying quantities adds the columns; dividing subtracts them. Read a row off the grid and the errors below become obvious.

"1 N = kg m s⁻¹ because F = ma and a is length over time."

Error: acceleration is velocity ÷ time =

(L / T) / T = L / T^{2}

, so

[a] = m s^{- 2}

, giving

1 N = kg m s^{- 2}

. The exponent on time is

- 2

, not

- 1

"To convert 72 km/h I write 72 × 1000 × 3600 m/s."

Error: hours must be divided into seconds, so multiply by

\frac{1 h}{3600 s}

(divide by 3600), giving

72 \times \frac{1000}{3600} = 20 m s^{- 1}

, not a huge number.

"Pressure has units m/m² = m, so it's basically a length."

Error: you divide the whole force unit by area:

\frac{kg m s ^{- 2}}{m ^{2}} = kg m^{- 1} s^{- 2}

. Only the length exponents combine as

1 - 2 = - 1

; the mass and time units stay.

"Since 5 m and 5 kg are both '5', they measure equal amounts."

Error: the number is meaningless without its unit. '5 m' and '5 kg' compare against completely different standards, so the physical quantities are not comparable at all.

"Energy and force have the same units because both involve force."

Error: work = force × distance, so energy is

kg m^{2} s^{- 2}

while force is

kg m s^{- 2}

— they differ by one factor of length (a metre).

"Amount of substance (mole) is just a big number, so it isn't a real base quantity."

Error: the mole is one of the seven SI base quantities. It counts elementary entities and is genuinely independent of mass, length and time.

Why questions

"Why does a physical quantity need both a number and a unit?"

The unit says which standard you compared against; the number says how many copies of that standard fit. A bare number can't tell seconds from kilograms.

"Why exactly seven base quantities — not fewer, not more?"

Seven is the minimal set that is both independent (none is buildable from the others) and sufficient (every known quantity reaches from them). Fewer leaves gaps; more is redundant.

"Why do units multiply and divide like ordinary algebra?"

Because a product of quantities is a tiling: area = length × width fills the plane with unit squares (see the opening figure), so the numbers multiply and the units multiply in the very same step. Units are just fixed factors riding along.

"Why must the temperature base scale be absolute (kelvin, not Celsius)?"

So that ratios make sense: doubling from 100 K to 200 K really doubles the thermal quantity, whereas 100 °C is not 'twice as hot' as 50 °C because Celsius's zero is arbitrary.

"Why is weight a derived quantity even though we quote it in kilograms daily?"

Weight is a force

W = m g

, measured in newtons. The everyday 'weight in kg' is really our mass; we conflate them because on Earth

g

is roughly constant.

Edge cases

The candela is weighted by your eye — see the curve. Two lamps radiating the same physical power (watts) can look very different in brightness because the eye responds most to green ( $\approx 555 nm$ ) and barely to deep red or violet. The figure shows that sensitivity curve; because it is a biological weighting, no pure energy-per-time can replace the candela.

"Is the number '2' in the ratio (10 m ÷ 5 m) a physical quantity?"

No — it's dimensionless. Dividing like-for-like cancels the metres, leaving a pure number with no unit and no fundamental character.

"What happens to a derived unit when a base quantity appears to the power zero?"

A power of zero contributes nothing (that base drops out). E.g. frequency is

s^{- 1}

— length and mass appear to the zeroth power, so they vanish from the unit.

"Can a unit carry a fractional power?"

Yes. Taking a square root halves the exponents:

area = m^{2} = m

, and the frequency of a pendulum

f \propto g / ℓ

gives

m s^{- 2} / m = s^{- 1}

. Exponents on units follow the same arithmetic as any exponent — halving is perfectly legal.

"Can a quantity be fundamental in one unit system but derived in another?"

Yes. The choice of base set is a human convention. Some systems pick different starters, so what's a base quantity here can be derived there — the physics is unchanged, only the bookkeeping differs.

"Is luminous intensity (candela) truly independent, or hidden energy per time?"

It's kept as a separate base because it is weighted by the human eye's sensitivity curve (see the figure), which no purely physical energy-rate captures. That eye-response weighting makes it independent in the SI convention.

"If two people use different units, does the underlying quantity differ?"

No. Because

n_{1} u_{1} = n_{2} u_{2}

, the product (the real quantity) is invariant. Only the label pair (number, unit) changes; nature stays the same.

Connections

Dimensional Analysis — every 'spot the error' above is caught faster by reading the exponent grid.
SI Units and Prefixes — the invariance $n_{1} u_{1} = n_{2} u_{2}$ powers all these conversions.
Newton's Second Law — the source of the newton's derivation used in the error items.
Errors and Significant Figures — real measured quantities also carry uncertainty.
Vectors — force is a vector and torque a pseudovector, distinctions beyond just their units.