Physics 9702 · Paper 2

Derived Units & Homogeneity

Base units are useful in two ways. They let you find the unit of any quantity, and they let you check whether an equation can be correct.
Both jobs use the same skill: you replace every symbol in an equation with its base units, then read off or compare the result.

Building a derived unit

A is any unit built from base units.
To find one, start from the quantity's defining equation and replace each symbol with its base units.
Smallest case, speed: $v = \dfrac{\text{distance}}{\text{time}}$ , so the unit is $\text{m} / \text{s} = \text{m s}^{-1}$ . No new unit name is needed.

Small coloured base-unit blocks combining with arrows into one larger block. — The small blocks stand for base units. Combining them builds one derived unit.

Force, from $F = ma$ : $\text{kg} \times \text{m s}^{-2} = \text{kg m s}^{-2}$ . This combination has the special name newton.
Energy, from $E = Fd$ : $\text{kg m s}^{-2} \times \text{m} = \text{kg m}^2\,\text{s}^{-2}$ (the joule).
Pressure, from $p = F/A$ : $\text{kg m s}^{-2} / \text{m}^2 = \text{kg m}^{-1}\,\text{s}^{-2}$ (the pascal).

You do not need to memorise derived units. Write the defining equation, swap in the base units, and simplify.

Worked example

Base units of power

Definition: $\text{power} = \dfrac{\text{work}}{\text{time}}$ .
Work has the units of energy: $\text{kg m}^2\,\text{s}^{-2}$ .
Divide by time: $\dfrac{\text{kg m}^2\,\text{s}^{-2}}{\text{s}} = \text{kg m}^2\,\text{s}^{-3}$ .

Answer

\text{power} = \text{kg m}^2\,\text{s}^{-3}

(9702/22/F/M/24 Q1(b))

Your turn— tap to reveal the worked answer (9702/13/O/N/21 Q2)

Express the ohm in SI base units, given $R = V/I$ and $V = P/I$ . This needs two swaps in a row. (9702/13/O/N/21 Q2)

First the volt: $[V] = \dfrac{\text{kg m}^2\,\text{s}^{-3}}{\text{A}} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-1}$ .
Then the ohm: $[R] = \dfrac{[V]}{\text{A}} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-2}$ .

Checking an equation: homogeneity

An equation is when every term has the same base units.
If the units do not match, the equation is wrong. This is a fast way to catch a wrong formula.
Matching units cannot prove an equation is right. A pure number like $\tfrac12$ or $2\pi$ has no units, so this check cannot see it.
Smallest case, $s = vt$ : the left side is m; the right side is $\text{m s}^{-1} \times \text{s} = \text{m}$ . Both sides match, so the equation is homogeneous.

Worked example

Testing an equation for homogeneity

A mass hangs on a spring with spring constant $k$ . Test whether $T = 2\pi\sqrt{m/k}$ is homogeneous.

Find the base units of $k$ from $F = kx$ : $[k] = \dfrac{\text{kg m s}^{-2}}{\text{m}} = \text{kg s}^{-2}$ .
Then $\left[\dfrac{m}{k}\right] = \dfrac{\text{kg}}{\text{kg s}^{-2}} = \text{s}^2$ .
The square root gives $\sqrt{\text{s}^2} = \text{s}$ , which matches $T$ . The $2\pi$ has no units, so the equation is homogeneous.

Answer

\text{homogeneous}

(9702/12/F/M/24 Q3)

Worked example

One change: find the unknown power

$T = 2\pi(L/g)^{n}$ is homogeneous. Find $n$ .

$\left[\dfrac{L}{g}\right] = \dfrac{\text{m}}{\text{m s}^{-2}} = \text{s}^2$ .
Right side: $(\text{s}^2)^{n} = \text{s}^{2n}$ . Left side: $\text{s}$ .
Match the powers: $2n = 1$ , so $n = \tfrac{1}{2}$ .

Answer

n = \tfrac{1}{2}

(9702/12/M/J/25 Q4)

Worked example

One more change: base units of a constant

The power output of a star is $P = \sigma A T^4$ , where $A$ is area and $T$ is temperature. Find the base units of the constant $\sigma$ .

Make the constant the subject: $\sigma = \dfrac{P}{A T^4}$ .
Substitute base units:
$[\sigma] = \dfrac{\text{kg m}^2\,\text{s}^{-3}}{\text{m}^2 \cdot \text{K}^4} = \text{kg s}^{-3}\,\text{K}^{-4}.$

Answer

[\sigma] = \text{kg s}^{-3}\,\text{K}^{-4}

(9702/13/O/N/24 Q3)

These three examples are the three ways the exam asks it: test an equation, find an unknown power, or find the base units of a constant. All three use the same skill: they replace each symbol with its base units, then compare.

Common mistake

Do not give units to a pure-number constant (

2\pi

and

4\pi^2

have none). And when one side has several terms, reduce every term, not just the first one.

PreviousPhysical Quantities & the SI Base Units NextPrefixes & Unit Conversions

Derived Units & Homogeneity

Base units are useful in two ways. They let you find the unit of any quantity, and they let you check whether an equation can be correct.
Both jobs use the same skill: you replace every symbol in an equation with its base units, then read off or compare the result.

Building a derived unit

A is any unit built from base units.
To find one, start from the quantity's defining equation and replace each symbol with its base units.
Smallest case, speed: $v = \dfrac{\text{distance}}{\text{time}}$ , so the unit is $\text{m} / \text{s} = \text{m s}^{-1}$ . No new unit name is needed.

Force, from $F = ma$ : $\text{kg} \times \text{m s}^{-2} = \text{kg m s}^{-2}$ . This combination has the special name newton.
Energy, from $E = Fd$ : $\text{kg m s}^{-2} \times \text{m} = \text{kg m}^2\,\text{s}^{-2}$ (the joule).
Pressure, from $p = F/A$ : $\text{kg m s}^{-2} / \text{m}^2 = \text{kg m}^{-1}\,\text{s}^{-2}$ (the pascal).

You do not need to memorise derived units. Write the defining equation, swap in the base units, and simplify.

Worked example

Base units of power

Definition: $\text{power} = \dfrac{\text{work}}{\text{time}}$ .
Work has the units of energy: $\text{kg m}^2\,\text{s}^{-2}$ .
Divide by time: $\dfrac{\text{kg m}^2\,\text{s}^{-2}}{\text{s}} = \text{kg m}^2\,\text{s}^{-3}$ .

Answer

\text{power} = \text{kg m}^2\,\text{s}^{-3}

(9702/22/F/M/24 Q1(b))

Your turn— tap to reveal the worked answer (9702/13/O/N/21 Q2)

Express the ohm in SI base units, given $R = V/I$ and $V = P/I$ . This needs two swaps in a row. (9702/13/O/N/21 Q2)

First the volt: $[V] = \dfrac{\text{kg m}^2\,\text{s}^{-3}}{\text{A}} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-1}$ .
Then the ohm: $[R] = \dfrac{[V]}{\text{A}} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-2}$ .

Checking an equation: homogeneity

An equation is when every term has the same base units.
If the units do not match, the equation is wrong. This is a fast way to catch a wrong formula.
Matching units cannot prove an equation is right. A pure number like $\tfrac12$ or $2\pi$ has no units, so this check cannot see it.
Smallest case, $s = vt$ : the left side is m; the right side is $\text{m s}^{-1} \times \text{s} = \text{m}$ . Both sides match, so the equation is homogeneous.

Worked example

Testing an equation for homogeneity

A mass hangs on a spring with spring constant $k$ . Test whether $T = 2\pi\sqrt{m/k}$ is homogeneous.

Find the base units of $k$ from $F = kx$ : $[k] = \dfrac{\text{kg m s}^{-2}}{\text{m}} = \text{kg s}^{-2}$ .
Then $\left[\dfrac{m}{k}\right] = \dfrac{\text{kg}}{\text{kg s}^{-2}} = \text{s}^2$ .
The square root gives $\sqrt{\text{s}^2} = \text{s}$ , which matches $T$ . The $2\pi$ has no units, so the equation is homogeneous.

Answer

\text{homogeneous}

(9702/12/F/M/24 Q3)

Worked example

One change: find the unknown power

$T = 2\pi(L/g)^{n}$ is homogeneous. Find $n$ .

$\left[\dfrac{L}{g}\right] = \dfrac{\text{m}}{\text{m s}^{-2}} = \text{s}^2$ .
Right side: $(\text{s}^2)^{n} = \text{s}^{2n}$ . Left side: $\text{s}$ .
Match the powers: $2n = 1$ , so $n = \tfrac{1}{2}$ .

Answer

n = \tfrac{1}{2}

(9702/12/M/J/25 Q4)

Worked example

One more change: base units of a constant

The power output of a star is $P = \sigma A T^4$ , where $A$ is area and $T$ is temperature. Find the base units of the constant $\sigma$ .

Make the constant the subject: $\sigma = \dfrac{P}{A T^4}$ .
Substitute base units:
$[\sigma] = \dfrac{\text{kg m}^2\,\text{s}^{-3}}{\text{m}^2 \cdot \text{K}^4} = \text{kg s}^{-3}\,\text{K}^{-4}.$

Answer

[\sigma] = \text{kg s}^{-3}\,\text{K}^{-4}

(9702/13/O/N/24 Q3)

These three examples are the three ways the exam asks it: test an equation, find an unknown power, or find the base units of a constant. All three use the same skill: they replace each symbol with its base units, then compare.

Common mistake

Do not give units to a pure-number constant (

2\pi

and

4\pi^2

have none). And when one side has several terms, reduce every term, not just the first one.